1 Introduction

This paper aims to incorporate the structure of income distribution into the model presented by Shiozawa et al. (2019) by applying Miyazawa’s multiplier. Shiozawa et al. (2019) described the mechanism of quantity adjustment independently of price determination with their multisector model. They built a fundamental model that is based on the assumption of limited rationality and explained how firms make decisions during the production process from the perspective of microeconomics.Footnote 1 However, Shiozawa et al. (2019) paid little attention to the income distribution. To make up for this point, we attempt to incorporate interrelations between production, distribution, and expenditure.

Miyazawa (1963, 1976) presented the input–output model with interrelations between production, distribution, and expenditure. The general input–output model can examine how consumption expenditure affects production and the production generates income. However, it cannot analyse the production inducement from income distribution through expenditure. Miyazawa (1963, 1976) introduced the household sector as an endogenous sector into the input–output model. He regarded the household sector as an activity unit that receives income and spends it as consumption (Miyazawa 1976, 2002). Miyazawa (1963, 1976) presented the multiplier with the circular flow of macroeconomy between expenditure, production, and distribution.Footnote 2

The reason we chose Miyazawa’s multiplier is that the model in Shiozawa et al. (2019) has something in common with the input–output model. Both of them are multisector models and have stable input coefficients. The difference from the input-output model is the scope of analysis and the assumption behind the stability of input coefficients. First, Shiozawa et al. (2019) focuses on a firm’s decision-making in the production process, while input–output analysis focuses on interrelations between industries. Second, the stability of input coefficients in input–output analysis is based on the assumption that firms have a single production technique. As regards the stability of input coefficients, Shiozawa et al. (2019) adopted the minimal price theorem, according to which, firms choose a production technique that gives the minimal cost.Footnote 3 As a result, a firm will adopt the single production technique, and input coefficients will stabilize. Regarding input coefficients, the system of Shiozawa et al. (2019) is different from the Leontief system in that input coefficients can be stable even though firms have alternatives to production techniques.

This paper is organized as follows. Section 2 explains Miyazawa’s multiplier of the income distribution. We discuss the input–output model with the endogenous household sector based on Miyazawa (1963, 1976). In Sect. 3, we attempt to incorporate the impact of income distribution on the production process through expenditure into the model in Shiozawa et al. (2019) by applying Miyazawa’s multiplier. Section 4 summarises characteristics of Shiozawa et al. (2019) and Miyazawa’s multiplier. Section 5 concludes this paper.

2 Miyazawa’s model of the structure of income distribution

Miyazawa (1963) presented the multiplier to examine the structure of income distribution. The general input–output model includes repercussion effects from expenditure to production and from production to distribution, but it does not include the effects from distribution to expenditure. Miyazawa (1963) analysed the effects of distribution on expenditure and expenditure on production with his model.

Miyazawa (1963) introduced the household as an endogenous sector into the input–output model. In Miyazawa’s model, final demand is divided into two parts: household consumption and the others. Household consumption is added to a column in the transaction of intermediate goods. Likewise, value added is divided into two parts: household income and the others. Household income is added to a row in the transaction of intermediate goods. Input–output table with the endogenous household sector is shown in Fig. 1. Each sign represents the following: \(\mathbf{A}\) is the input coefficient matrix, \(\mathbf{X}\) is the vector of output, \(\mathbf{V}\) is the \(r\times n\) matrix of income ratios, where \(r\) represents the number of income groups and \(n\) represents the number of industry groups, \({\mathbf{V}}_{\mathbf{o}}\) is the row vector of other value-added ratios, \({\mathbf{F}}_{\mathbf{c}}\) is the column vector of household consumption, and \({\mathbf{F}}_{\mathbf{o}}\) is the column vector of other final demand. In the figure, household consumption and income are highlighted. From Miyazawa’s system depicted in Fig. 1, the output can be obtained as follows:

$$\begin{array}{*{20}c} {{\mathbf{X}} = {\mathbf{AX}} + {\mathbf{F}}_{{\mathbf{c}}} + {\mathbf{F}}_{{\mathbf{o}}} } \\ \end{array}$$
(1)

If we use the ratio of consumption to income, household consumption can be expressed as

$$\begin{array}{*{20}c} {{\mathbf{F}}_{{\mathbf{c}}} = {\mathbf{CVX}}} \\ \end{array}$$
(2)

where \(\mathbf{C}\) is the \(n\times r\) matrix of consumption ratios to household income that Miyazawa (1963, 1976) mentioned as consumption coefficients, \(\mathbf{V}\) is the \(r\times n\) matrix of income ratio, and \(\mathbf{X}\) is the column vector of output.Footnote 4 When we consider only one income group, \(\mathbf{C}\) will become a column vector and \(\mathbf{V}\) will become a row vector. In the case of the single income group, we obtain the Keynesian multiplier (Miyazawa 1963, 1976; Miyazawa and Masegi 1963). Substituting Eq. (2) into Eq. (1),

$$\begin{array}{*{20}c} {{\mathbf{X}} = {\mathbf{AX}} + {\mathbf{CVX}} + {\mathbf{F}}_{{\mathbf{o}}} } \\ \end{array}$$
(3)

Then, we can obtain the following equation (Miyazawa 1963, 1976):

$$\begin{array}{*{20}c} {\mathbf{X} = \left( {{\mathbf{I}} - {\mathbf{A}} - {\mathbf{CV}}} \right)^{ - 1} {\mathbf{F}}_{{\mathbf{o}}} } \\ \end{array}$$
(4)

Miyazawa (1963, 1976) transformed the inverse matrix in Eq. (4) to separate consumption activity from production activity. If we denote the Leontief inverse matrix \({\left(\mathbf{I}-\mathbf{A}\right)}^{-1}\) by \(\mathbf{B}\), the inverse matrix \({\left(\mathbf{I}-\mathbf{A}-\mathbf{C}\mathbf{V}\right)}^{-1}\) can be written as

$$\begin{aligned} \left( {{\mathbf{I}} - {\mathbf{A}} - {\mathbf{CV}}} \right)^{{ - 1}} & = \left[ {\left\{ {{\mathbf{I}} - {\mathbf{CV}}\left( {{\mathbf{I}} - {\mathbf{A}}} \right)^{{ - 1}} } \right\}\left( {{\mathbf{I}} - {\mathbf{A}}} \right)} \right]^{{ - 1}} \\ & = \left( {{\mathbf{I}} - {\mathbf{A}}} \right)^{{ - 1}} \left[ {{\mathbf{I}} - {\mathbf{CV}}\left( {{\mathbf{I}} - {\mathbf{A}}} \right)^{{ - 1}} } \right]^{{ - 1}} \\ & = \begin{array}{*{20}c} {{\mathbf{B}}\left[ {{\mathbf{I}} - {\mathbf{CVB}}} \right]^{{ - 1}} } \\ \end{array} \\ \end{aligned}$$
(5)

The input–output model can be expressed as

$$\begin{array}{*{20}c} {\mathbf{X} = {\mathbf{B}}\left[ {{\mathbf{I}} - {\mathbf{CVB}}} \right]^{ - 1} {\mathbf{F}}_{{\mathbf{o}}} } \\ \end{array}$$
(6)
Fig. 1
figure 1

Input–output table with the endogenous household sector. This figure is based on Miyazawa (1963, 1976), Hewings et al. (1999), Ishiro (2006), Hayashi and Takahashi (2007), and Tahara (2022)

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Miyazawa (1976: p. 5) called the inverse matrix \({\left[\mathbf{I}-\mathbf{C}\mathbf{V}\mathbf{B}\right]}^{-1}\) as the “subjoined inverse matrix.”

The subjoined inverse matrix is a multiplier including the repercussion effects from distribution to consumption. When final demand \({\mathbf{F}}_{\mathbf{o}}\) increases, the output will increase according to the repercussion process. Figure 2 shows the repercussion process via \({\left[\mathbf{I}-\mathbf{C}\mathbf{V}\mathbf{B}\right]}^{-1}\). An increase in \({\mathbf{F}}_{\mathbf{o}}\) will induce output of each sector by \(\mathbf{B}{\mathbf{F}}_{\mathbf{o}}\). The additional output will increase household income by \(\mathbf{V}\mathbf{B}{\mathbf{F}}_{\mathbf{o}}\). This increment in income will increase consumption by \(\mathbf{C}\mathbf{V}\mathbf{B}{\mathbf{F}}_{\mathbf{o}}\). The consumption will stimulate production and output will increase by \(\mathbf{B}\mathbf{C}\mathbf{V}\mathbf{B}{\mathbf{F}}_{\mathbf{o}}\). This process continues for some time. The process in Fig. 2 shows that the inverse matrix proposed by Miyazawa (1963, 1976) incorporates the repercussion effects from production to distribution, distribution to consumption, and consumption to production. The row of production in Fig. 2 shows that the repercussion effects of final demand on production through \({\left[\mathbf{I}-\mathbf{C}\mathbf{V}\mathbf{B}\right]}^{-1}\) can be expressed as follows (Hayashi and Takahashi 2007):

$$\Delta \mathbf{X}=\mathbf{B}\Delta {\mathbf{F}}_{\mathbf{o}}+\mathbf{B}\mathbf{C}\mathbf{V}\mathbf{B}\Delta {\mathbf{F}}_{\mathbf{o}}+\mathbf{B}\mathbf{C}\left(\mathbf{V}\mathbf{B}\mathbf{C}\right)\mathbf{V}\mathbf{B}\Delta {\mathbf{F}}_{\mathbf{o}}+\mathbf{B}\mathbf{C}{\left(\mathbf{V}\mathbf{B}\mathbf{C}\right)}^{2}\mathbf{V}\mathbf{B}\Delta {\mathbf{F}}_{\mathbf{o}}+\mathbf{B}\mathbf{C}{\left(\mathbf{V}\mathbf{B}\mathbf{C}\right)}^{3}\mathbf{V}\mathbf{B}\Delta {\mathbf{F}}_{\mathbf{o}}$$
$$\begin{array}{c}+{\mathbf{BC}}{\left(\mathbf{V}\mathbf{B}\mathbf{C}\right)}^{4}{\mathbf{VB}}\Delta {\mathbf{F}}_{\mathbf{o}}+\cdots \end{array}$$

where \(\Delta\) represents an increment. Then,

$$\begin{array}{*{20}c} {\Delta {\mathbf{X}} = {\mathbf{B}}\Delta {\mathbf{F}}_{{\mathbf{o}}} + {\mathbf{BC}}\left[ {{\mathbf{I}} + \left( {{\mathbf{VBC}}} \right) + \left( {{\mathbf{VBC}}} \right)^{2} + \left( {{\mathbf{VBC}}} \right)^{3} + \cdots } \right]{\mathbf{VB}}\Delta {\mathbf{F}}_{{\mathbf{o}}} } \\ \end{array}$$
(7)

Equation (7) can be rewritten as

$$\begin{array}{*{20}c} {\Delta {\mathbf{X}} = {\mathbf{B}}\Delta {\mathbf{F}}_{{\mathbf{o}}} + {\mathbf{BC}}\left( {{\mathbf{I}} - {\mathbf{VBC}}} \right)^{ - 1} {\mathbf{VB}}\Delta {\mathbf{F}}_{{\mathbf{o}}} } \\ \end{array}$$
(8)

Miyazawa (1976: p. 6) called the inverse matrix \({\left(\mathbf{I}-\mathbf{V}\mathbf{B}\mathbf{C}\right)}^{-1}\) as the “interrelational income multiplier.” If we denote the interrelational income multiplier \({\left(\mathbf{I}-\mathbf{V}\mathbf{B}\mathbf{C}\right)}^{-1}\) as \(\mathbf{K}\), the induced output can be expressed as follows (Miyazawa 1963, 1976):

$$\begin{array}{*{20}c} {{\mathbf{X}} = {\mathbf{B}}\left( {{\mathbf{I}} + {\mathbf{CKVB}}} \right){\mathbf{F}}_{{\mathbf{o}}} } \\ \end{array}$$
(9)

Equation (9) is essentially the same as Eq. (6), but Eq. (9) has an advantage over Eq. (6) in that the total induced income is obvious. The row of distribution in Fig. 2 shows the repercussion effects on income. An increase in output will induce \(\mathbf{V}\mathbf{B}{\mathbf{F}}_{\mathbf{o}}\) of income in the first round. In the next round, the income will increase by \((\mathbf{V}\mathbf{B}\mathbf{C})\mathbf{V}\mathbf{B}{\mathbf{F}}_{\mathbf{o}}\). In the repercussion process, income will increase by \(\mathbf{V}\mathbf{B}\mathbf{C}\) times. Hence, the total induced income will become \(\mathbf{K}\) times \(\mathbf{V}\mathbf{B}{\mathbf{F}}_{\mathbf{o}}\).

Fig. 2
figure 2

Source: Miyazawa (1963: p. 22, 1976: p. 8), Miyazawa and Masegi (1963: p. 94), and Hayashi and Takahashi (2007: p. 168)

Repercussion process including the repercussion effects from distribution to consumption.

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3 Application of Miyazawa’s model to the multisector model of evolutionary economics

This section focuses on the simplest case of stationary point in Shiozawa et al. (2019) to show a possibility of applying Miyazawa’s multiplier of income distribution to the model in Shiozawa et al. (2019).Footnote 5 The multiplier developed by Miyazawa takes into account how income distribution affects the production process through expenditure. The output will generate new income, which in turn induces more consumption. The increased consumption stimulates the output and this increment in output will generate more income.

Shiozawa et al. (2019) explained that output is determined by inventories and demand forecasts. If firms’ demand forecasts are constant, the sales volume in period \(t\) is expressed as follows (Shiozawa et al. 2019):Footnote 6

$$\begin{array}{*{20}c} {{\mathbf{S}}\left( t \right) = {\mathbf{S}}\left( {t - 1} \right){\mathbf{A^{\prime}}} + {\mathbf{F^{\prime}}}} \\ \end{array}$$
(10)

where \(\mathbf{S}\left(t\right)\) is the row vector of sales volume in period \(t\), \(\mathbf{S}\left(t-1\right)\) is the row vector of sales volume in period \(t-1\), \(\mathbf{A}\boldsymbol{^{\prime}}\) is the transposed matrix of input coefficients, and \(\mathbf{F}\mathbf{^{\prime}}\) is the row vector of final demand. The superscript comma in the equation denotes that the matrix is transposed. The input coefficient matrix in Shiozawa et al. (2019) is technically the transposed matrix of input coefficients. They explained the input coefficient matrix as follows: “Let \(\mathbf{A}\) be an ordinary \(n\times n\) input coefficient matrix. Its \((i, j)\) element \({a}_{i,j}\) represents the input of the product \(j\) required per unit output of product \(i\)” (Shiozawa et al. 2019: p. 196). Intermediate goods are shipped from sector \(j\) to sector \(i\) according to the explanation. By contrast, the element \({a}_{i,j}\) of input coefficient matrix in the input–output analysis means that intermediate goods are shipped from sector \(i\) to sector \(j\). This indicates that while in Shiozawa et al. (2019), intermediate goods are shipped from column to row, in the input–output analysis, intermediate goods are shipped from sectors in the row to sectors in the column. Accordingly, the vector of final demand becomes a row vector in Shiozawa et al. (2019). Equation (10) represents sales volume is determined by the past sales volume and final demand in the case of constant demand forecasts. Expectations of sales volume are significant for firms to make decisions during the production process (Shiozawa et al. 2019).

At the stationary point, \(\mathbf{S}\left(t\right)\) is equal to \(\mathbf{S}\left(t-1\right)\) (Shiozawa et al. 2019). We can obtain the following equation at the stationary point.

$$\begin{array}{*{20}c} {{\mathbf{S}} = {\mathbf{SA^{\prime}}} + {\mathbf{F^{\prime}}}} \\ \end{array}$$
(11)

According to Miyazawa (1963, 1976), final demand can be divided into two parts: consumption expenditure and the others.

$${\mathbf{S}}={\mathbf{SA}}^{\prime}+{\mathbf{F}}_{{\mathbf{c}}}^{\prime}+{\mathbf{F}}_{{\mathbf{o}}}^{{{\prime}}}$$
(12)

where \({\mathbf{F}}_{\mathbf{c}}^{\mathbf{^{\prime}}}\) is the \(1\times n\) vector of household consumption and \({\mathbf{F}}_{\mathbf{o}}^{\mathbf{^{\prime}}}\) is the \(1\times n\) vector of other demand. If we use the \(r\times n\) transposed matrix of consumption coefficients \(\mathbf{C}\boldsymbol{^{\prime}}\) and the \(n\times r\) transposed matrix of income ratio \(\mathbf{V}\boldsymbol{^{\prime}}\), household consumption \({\mathbf{F}}_{\mathbf{c}}^{\mathbf{^{\prime}}}\) can be expressed as

$$\begin{array}{*{20}c} {{\mathbf{F}}_{{\mathbf{c}}} ^{\prime} = {\mathbf{SV}}^{\prime}{\mathbf{C}}^{\prime}} \\ \end{array}$$
(13)

Substituting Eq. (13) into Eq. (12), we obtain

$${\mathbf{S}} = {\mathbf{SA^{\prime}}} + {\mathbf{SV}}^{\prime}{\mathbf{C}}^{\prime} + {\mathbf{F}}_{{\mathbf{o}}}^{\varvec{^{\prime}}}$$

Then,

$$\begin{array}{*{20}c} {{\mathbf{S}} = {\mathbf{F}}_{{\mathbf{o}}}^{\varvec{^{\prime}}} \left( {{\mathbf{I}} - {\mathbf{A^{\prime}}} - {\mathbf{V^{\prime}C^{\prime}}}} \right)^{ - 1} } \\ \end{array}$$
(14)

We can transform the inverse matrix in Eq. (14) by the same procedure in Eq. (5). If we denote the Leontief inverse matrix \({\left(\mathbf{I}-\mathbf{A}\mathbf{^{\prime}}\right)}^{-1}\) by \(\mathbf{B}\mathbf{^{\prime}}\), the inverse matrix \({\left(\mathbf{I}-\mathbf{A}\mathbf{^{\prime}}-\mathbf{V}\mathbf{^{\prime}}\mathbf{C}\mathbf{^{\prime}}\right)}^{-1}\) can be written as follows:

$$\begin{array}{*{20}c} { \left( {{\mathbf{I}} - {\mathbf{A^{\prime}}} - {\mathbf{V^{\prime}C^{\prime}}}} \right)^{ - 1} = {\mathbf{B^{\prime}}}\left[ {{\mathbf{I}} - {\mathbf{V^{\prime}C^{\prime}B^{\prime}}}} \right]^{ - 1} } \\ \end{array}$$
(15)

Equation (14) can be expressed as

$$\begin{array}{*{20}c} {\mathbf{S} = {\mathbf{F}}_{{\mathbf{o}}}^{\varvec{^{\prime}}} {\mathbf{B^{\prime}}}\left[ {{\mathbf{I}} - {\mathbf{V^{\prime}C^{\prime}B^{\prime}}}} \right]^{ - 1} } \\ \end{array}$$
(16)

Equation (16) is essentially the same as Eq. (6). The inverse matrix \({\left[\mathbf{I}-{\mathbf{V}}^{\mathbf{^{\prime}}}{\mathbf{C}}^{\mathbf{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}\right]}^{-1}\) is the subjoined inverse matrix in Miyazawa (1976). The impact of income distribution on expenditure is included in Eq. (16). The repercussion process is essentially the same as in Fig. 2. An increase in expenditure will stimulate sales volume by \({\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}\). The additional sales volume will increase income by \({\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}\mathbf{V}^{\prime}\). An increase in income creates additional consumption by \({\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}\mathbf{V}^{\prime}\mathbf{C}^{\prime}\). This consumption will induce sales volume by \({\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}\mathbf{V}^{\prime}\mathbf{C}^{\prime}\mathbf{B}^{\prime}\). This process will continue for some time, and the sales volume will increase as a result of interactions between production, distribution, and consumption.

We can transform Eq. (16) in the same way as we used to obtain Eq. (9). The repercussion effects of a change in final demand on sales volume through \({\left[\mathbf{I}-{\mathbf{V}}^{\mathbf{^{\prime}}}{\mathbf{C}}^{\mathbf{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}\right]}^{-1}\) can be expressed as:

$$\Delta \mathbf{S}=\Delta {\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}\mathbf{B}\mathbf{^{\prime}}+\Delta {\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}\mathbf{B}\mathbf{^{\prime}}\mathbf{V}\mathbf{^{\prime}}\mathbf{C}\mathbf{^{\prime}}\mathbf{B}\mathbf{^{\prime}}+\Delta {\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}\mathbf{B}\mathbf{^{\prime}}\mathbf{V}\mathbf{^{\prime}}(\mathbf{C}\mathbf{^{\prime}}\mathbf{B}\mathbf{^{\prime}}\mathbf{V}\mathbf{^{\prime}})\mathbf{C}\mathbf{^{\prime}}\mathbf{B}\mathbf{^{\prime}}+\Delta {\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}\mathbf{B}\mathbf{^{\prime}}\mathbf{V}\mathbf{^{\prime}}{\left({\mathbf{C}}^{\mathbf{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}{\mathbf{V}}^{\mathbf{^{\prime}}}\right)}^{2}\mathbf{C}\mathbf{^{\prime}}\mathbf{B}\mathbf{^{\prime}}+\Delta {\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}\mathbf{B}\mathbf{^{\prime}}\mathbf{V}\mathbf{^{\prime}}{\left({\mathbf{C}}^{\mathbf{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}{\mathbf{V}}^{\mathbf{^{\prime}}}\right)}^{3}\mathbf{C}\mathbf{^{\prime}}\mathbf{B}\mathbf{^{\prime}}+\Delta {\mathbf{F}}_{\mathbf{o}}^{\boldsymbol{^{\prime}}}\mathbf{B}\mathbf{^{\prime}}\mathbf{V}\mathbf{^{\prime}}{\left({\mathbf{C}}^{\mathbf{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}{\mathbf{V}}^{\mathbf{^{\prime}}}\right)}^{4}\mathbf{C}\mathbf{^{\prime}}\mathbf{B}\mathbf{^{\prime}}+\cdots$$

where \(\Delta\) represents an increment. Then,

$$\begin{array}{*{20}c} {\Delta {\mathbf{S}} = \Delta {\mathbf{F}}_{{\mathbf{o}}}^{\varvec{^{\prime}}} {\mathbf{B}}^{\prime} + \Delta {\mathbf{F}}_{{\mathbf{o}}}^{\varvec{^{\prime}}} {\mathbf{B^{\prime}V^{\prime}}}\left[ {{\mathbf{I}} + \left( {{\mathbf{C^{\prime}B^{\prime}V^{\prime}}}} \right) + \left( {{\mathbf{C^{\prime}B^{\prime}V^{\prime}}}} \right)^{2} + \left( {{\mathbf{C^{\prime}B^{\prime}V^{\prime}}}} \right)^{3} + \cdots } \right]{\mathbf{C^{\prime}B^{\prime}}}} \\ \end{array}$$
(17)

Equation (17) can be rewritten as

$$\begin{array}{*{20}c} {\Delta {\mathbf{S}} = \Delta {\mathbf{F}}_{{\mathbf{o}}}^{\varvec{^{\prime}}} {\mathbf{B^{\prime}}} + \Delta {\mathbf{F}}_{{\mathbf{o}}}^{\varvec{^{\prime}}} {\mathbf{B^{\prime}V^{\prime}}}\left( {{\mathbf{I}} - {\mathbf{C^{\prime}B^{\prime}V^{\prime}}}} \right)^{ - 1} {\mathbf{C^{\prime}B^{\prime}}}} \\ \end{array}$$
(18)

The inverse matrix \({\left(\mathbf{I}-{\mathbf{C}}^{\mathbf{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}{\mathbf{V}}^{\mathbf{^{\prime}}}\right)}^{-1}\) is essentially the same as the “interrelational income multiplier” put forward by Miyazawa (1976: p. 6). If we denote the inverse matrix \({\left(\mathbf{I}-{\mathbf{C}}^{\mathbf{^{\prime}}}{\mathbf{B}}^{\mathbf{^{\prime}}}{\mathbf{V}}^{\mathbf{^{\prime}}}\right)}^{-1}\) as \(\mathbf{K}\mathbf{^{\prime}}\), the sales volume can be expressed as follows:

$$\begin{array}{*{20}c} {{\mathbf{S}} = {\mathbf{F}}_{{\mathbf{o}}}^{\varvec{^{\prime}}} \left( {{\mathbf{I}} + {\mathbf{B^{\prime}V^{\prime}K^{\prime}C^{\prime}}}} \right)} \\ \end{array} {\mathbf{B^{\prime}}}$$
(19)

Equation (19) is essentially the same as Eq. (9). The point of this result is that we derived Eq. (19). If we could use the firm-level data, we could calculate the induced sales volume including beneficial repercussions of distribution for consumption. This result shows that the impact of income distribution on sales volume can be introduced into the simplest case in Shiozawa et al. (2019) by applying Miyazawa’s model.

4 Characteristics of Shiozawa et al. (2019) and Miyazawa’s model

This section first summarises main points of Shiozawa et al. (2019) and Miyazawa’s model on the basis of the author’s understanding, then describes their similarities and differences.

Shiozawa et al. (2019) provided multisector models focusing on the production activity. The purpose of Shiozawa et al. (2019) is to show that the quantity adjustment can work properly in the economy from the point of view of microeconomics. Their model is based on the continuous decisions of firms with the limits of capabilities. It includes input–output relations between firms. The main point of the models is the separation of quantity adjustment from fluctuations in prices. The separation from price changes is based on the minimal price theorem. The minimal price theorem is concerned with choosing production techniques. The minimal price theorem assures that a firm selects a production technique which costs the most reasonable price. Prices play an important role not in the adjustment of supply and demand but in selecting production techniques. As a result, the minimal price theorem assures that input coefficients can be stable even when the final demand changes. Shiozawa et al. (2019) have explained that changes in final demand are followed by changes in production in an economy with a variety of goods and services. They have shown that the quantity adjustment can control the gap between production and demand under fixed prices. Another point is that Shiozawa et al. (2019) is centered on the process analysis. The process analysis focuses on the continuous decision-making of economic agents with the limited capabilities.

Miyazawa (1963, 1976) and Miyazawa and Masegi (1963) extended the input–output model and introduced the structure of income distribution into the multiplier. The most noted is that Miyazawa’s multiplier includes the income generation process. The input–output model can analyse how much output will be induced by changes in final demand and how much household income will be generated by the newly produced output. However, it cannot explain how much expenditure will be induced by changes in distribution. Miyazawa (1963, 1976) and Miyazawa and Masegi (1963) introduced the household sector as an endogenous one and filled the gap between distribution and expenditure in the input–output model. The endogenous consumption activity is included in Miyazawa’s model. Accordingly, his model can show the circular flow of income which is the process where expenditure increases production, the induced production generates more income, this additional income stimulates consumption, which in turn induces production again. The main point of Miyazawa’s multiplier is that it can analyse the inducement effects of final demand on production including beneficial repercussions of distribution for expenditure. Miyazawa’s model has empirically and theoretically shown the interrelations between expenditure, production and distribution.

Shiozawa et al. (2019) has several things in common with Miyazawa’s model. They both consider the demand-constrained economy, and both models are multisector models based on the input–output model. In both models, input coefficients are assumed to be stable. The stability of input coefficients represents a fixed production technique. The stability of input coefficients is required to apply the input–output model. As regards the stability of input coefficients, Shiozawa et al. (2019: p. 112) says, “Our system is basically an extension of Leontief system, as we assume linear production techniques without joint products.” The assumption of fixed production technique is adopted not only by the input–output model but also by post-Keynesian models. Another similarity is the quantity adjustment which is a characteristic of Keynesian theories. When final demand changes, production is adjusted to demand in both models.

Although Shiozawa et al. (2019) and Miyazawa’s model have several similarities, their foundations are different. First, in both models, the input coefficients are considered stable and the production technique is fixed. However, the assumptions about the stability of the coefficients are different. Miyazawa’s model assumed that each industry has only one production technique and the technique does not change during the period. By contrast, Shiozawa et al. (2019) adopted the minimal price theorem. In the minimal price theorem, firms have several options of production technique. Firms chose the production technique which allows the lowest cost. Accordingly, one production technique is chosen by each firm. The second difference is about the quantity adjustment. The product inventory plays an important role in the adjustment process in Shiozawa et al. (2019). They assume that firms try to keep a proper level of inventory to avoid stockout. Miyazawa’s model does not take the product inventory into account. Third, Shiozawa et al. (2019) provides a firm-level model which is based on decision-making of economic agents, but Miyazawa’s model is at industry-level. This difference is not so critical. If we could obtain firm-level data of final demand, transaction of inputs and so on, Miyazawa’s model could calculate the firm-level inducement effects of final demand on production. Fourth, Shiozawa et al. (2019) adopts the process analysis, but Miyazawa’s model assumes equilibrium. There is a possibility of adapting Miyazawa’s model to the process analysis. Miyazawa (2002) mentioned that his model can be extended to a dynamic model and the idle equipment and inventories can be considered by turning equalities into inequalities. Lastly, Shiozawa et al. (2019) focuses on the production process, but Miyazawa’s model focused on the distribution structure.

Evolutionary economics aims to establish a theoretical foundation of economics with an emphasis on the change process. Shiozawa et al. (2019) provides the microfoundations of evolutionary economics. To be widely accepted as a fundamental framework, an empirical analysis is necessary as much as the theoretical foundation. Applying Miyazawa’s model to the model of Shiozawa et al. (2019) can strengthen the foundation of evolutionary economics by providing an empirical analysis.

The application of Miyazawa’s model can also help us with understanding the circular flow of income that is an evolutionary process of the economy. In the economy, changes in final demand induce changes in production. The production generates income, and the income stimulates expenditures, which in turn induce production. The economy develops through this process. Miyazawa’s model can fill in the missing piece of the model by Shiozawa et al. (2019) by adding an explanation of this channel from production to distribution and from distribution to expenditure. The microfoundations of evolutionary economics has potential to become more comprehensive by such integration.

5 Concluding remark

This note has shown that we can incorporate beneficial repercussions of income distribution for production through expenditure into the model of Shiozawa et al. (2019) by using Miyazawa’s model. Shiozawa et al. (2019) developed the multisector model which is based on the continuous decisions of firms with the limited capabilities. They explained that the quantity adjustment works properly under fixed prices. They succeeded in building the model with the separation of quantity adjustment from fluctuations in prices, but they paid little attention to the income distribution. When production increases by an increase in final demand, additional income accrues. This additional income stimulates consumption and changes in consumption influence production. To include the influence of income distribution on production through expenditure, we applied Miyazawa’s multiplier to the simplest case in Shiozawa et al. (2019). The attempt of this study has shown a possibility of extending the model of Shiozawa et al. (2019) to empirical analysis.

Shiozawa et al. (2019) extended the input–output model to explain how production can be adjusted to changing demand flows as a result of the continuous decisions of firms. Miyazawa (1963, 1976) extended the input–output model to analyse the effects of the structure of income distribution on production activity. Their purposes and assumptions are different, but Miyazawa’s multiplier seems applicable to Shiozawa et al. (2019). First, the stability of input coefficients is necessary to apply Miyazawa’s model. Shiozawa et al. (2019) adopted the minimal price theorem and the input coefficients are stable. Second, the quantity adjustment works in both models, although the adjustment mechanism is not the exact same thing. Production is adjusted to changes in final demand via multiplier processes. Lastly, Miyazawa’s model is industry-level analysis, and Shiozawa et al. (2019) provided a model for firm-level analysis. When some fundamental points such as the stability of input coefficients are satisfied, the applicability of Miyazawa’s model to microeconomic analysis depends on the availability of the data of individual firms.

This note is based on the simplest case of stationary point in Shiozawa et al. (2019) where the firm’s demand forecast was constant. Given that Shiozawa et al. (2019) focused on the process analysis, the result in this note is limited. To fill this gap, we may need to extend Miyazawa’s model to a dynamic model and take into account the idle equipment and inventories by turning equalities into inequalities for further work. We believe that Miyazawa’s model can help incorporate the structure of income distribution and extend the microfoundations of evolutionary economics to empirical analysis.