Generalization of Technological Propagation/Diffusion Externalities Based on the General Equilibrium Model of Formation Process of Marshallian External Economies: Number 4

Abstract

As mentioned in the Sect. 8.1.1 of Chap. 8, we have treated selectively the independently existing indirect economic effects, different from the transferred effects from the direct effects. In Chap. 9, we have coped with the task that the following “two” will coincide with each other (transferred effects ≒ indirect effects) or come to be incidence base effects > transferred effects, and then the independently existing theory can be proved, in the form of contrasting the effects in generation base (this is the direct effects) with the effects in incidence base (which include the pure indirect effects existing independently except the transferred effects). This is, so to speak, that we have tackled on the “existence demonstration” concerning the pure indirect effects which exist independently. This chapter is based on [17], [10, pp. 57–63], [11], [13], [15], etc.

Appendix

The contents of this appendix (policy model) are discussed in detail in Chap. 9 (Sects. 9.2.1 and 9.2.2).

10.1.1 Policy Model for the Proof of Existence of Pure Indirect Economic Effects —Generation Base vs. Incidence Base Criteria

10.1.1.1 A1. Definition of Benefits

What are the benefits (= economic effects)? The benefits are theoretically measured in terms of variations of the household utility. The benefits are measured by investigating how much the household utility should change, before and after the project implement ([5], [7], and [8–16]).

In this model, the utility is the function of leisure time (hour) and consumption vector. So, the benefits will be obtained as the difference between the utility gotten from the leisure time and consumption goods vector before the project implement and that after the project implement.

In this chapter, the benefits are defined by using the consumer surplus mentioned below. In case of the utility function being the logarithmic linear type, this consumer surplus coincides with the variation of utility (though the benefits in the dimension of utility have been averted in Sect. 1.1).

• 1. Logarithmic linear type utility function

The following utility function is used here, which is what the Cobb-Douglas type of utility function U(T − Z, X) is transformed logarithmically:

$$ {U}_L\left(T-Z,\ X\right)= \log U\left(T-Z,\ X\right)={\theta}_0 \log \left(T-Z\right)+{\displaystyle \sum_{k=1}^n{\theta}_k \log {X}_k}, $$

(10.14)

where, from the maximization of household utility, the following equilibrium values are obtained:

$$ Z=T-\left(\frac{\theta_0}{w}\right)I,\kern0.75em {X}_k=\left(\frac{\theta_k}{p_k}\right)I\kern1.5em \left(k=1,\cdots, n\right)\kern0.5em \mathrm{and}\ \lambda =1/I $$

(10.15)

here, $I\equiv {}_w{}T+{D}_v+{D}_u$.

• 2. Consumer surplus

The consumer surplus of leisure s ₀(T − Z) and the consumer surplus of the kth consumption goods s _k (X _k ) are defined as follows:

$$ {s}_0\left(T-Z\right)={\displaystyle \int }-\lambda wd\left(T-Z\right);\kern0.5em {s}_k\left({X}_k\right)={\displaystyle \int}\lambda {p}_kd{X}_k\kern1.25em \left(k=1,\cdots, n\right). $$

(10.16)

This consumer surplus coincides with variations of household utility.

10.1.1.2 A2. Benefits of Incidence Base Criterion and Those of Generation Base Criterion

• 1. Benefits of incidence base criterion

The benefits based on the incidence base B ^inb is defined as follows: The value obtained by subtracting the consumer surplus before the implement of project from the consumer surplus after the implement of project is defined to be benefits:

$$ {B}^{inb}\equiv \left[{s}_0\left(T-\widehat{Z}\right)+{\displaystyle {\sum}_{k=1}^n{s}_k\left(\widehat{X}\right)}\right]-\left[{s}_0\left(T-Z\right)+{\displaystyle {\sum}_{k=1}^n{s}_k(X)}\right]={U}_L\left(T-\widehat{Z},\widehat{X}\right)-{U}_L\left(T-Z,X\right). $$

(10.17)

The labor supply in the equilibrium state before the implement of project is Z; the vector of consumer goods X is denoted. The sign “^” is affixed to the variable when the new equilibrium has come into existence in all the markets, where the benefits based on the incidence base ought to be equivalent to the variations of logarithmic linear type utility function.

• 2. Benefits of the generation base criteria

The variables before and after the implement of project are defined as follows:

• (i) Prior to the implement of project

The economy prior to the implement of project is in the equilibrium state, where any sign to the variables is not annexed.

• (ii) Implement of the project

The direct technological change brought about by the impact of implement of project is expressed with the variables with the sign of tilt “^~.” In this chapter, this technological changes are specialized to be only ${\tilde{\alpha}}_{21},{\tilde{\alpha}}_1$ and ${\tilde{\beta}}_1$.

• (iii) Influence posterior to the implement of project in terms of the benefits in generation base

The benefits in generation base are measured in the state where the first market related directly to the project only has been in equilibrium after the short-run adjustment. Here, this price changes in the first market and the sphere of influence brought about by which are shown by the variables with the sign of asterisk “*.”

The equilibrium price of the first market and the prices of the other unchanging goods markets are expressed to be “p ^*₁ ,” and to be “${p}_i\left(i=2,\cdots,\ n\right)$,” respectively.

Owing to the change of price of the 1st goods to the p ^*₁ , it is supposed that the consumption quantity of the 1st goods by household should be changed to be X ^*₁ , and the intermediate input quantity of the 1st goods of the enterprise belonging to the ith industry is changed to be x ^*_1i ; moreover, receiving the influence of which, the output of the enterprise pertaining to the ith industry is changed to be y ^* _i and the profits to be π ^* _i .

At this point, the benefits B ^ob of the generation base criteria are expressed as follows:

$$ {B}^{ob}\equiv \Delta {s}_1+\Delta {s}_{\pi },\kern0.5em \Delta {s}_1\equiv {s}_1\left({X}_1^{*}\right)-{s}_1\left({X}_1\right),\kern0.5em \Delta {s}_{\pi}\equiv \frac{1}{I^{*}}\left[{\displaystyle \sum_{i=1}^n\left({\pi}_i^{*}-{\pi}_i\right){n}_i}\right]. $$

(10.18)

Δs ₁ is the increase of consumer surplus formed by the variations of consumer goods of the 1st goods due to the implement of project, which is defined to be the differences between the consumer surplus of the 1st goods before the implement of project s ₁(X ₁) and the consumer surplus of the 1st goods after the implement of project s ₁(X ^*₁ ). The consumption quantity of the 1st goods after the implement of project X ^*₁ is defined as follows, reflecting the price p ^*₁ under the partial equilibrium of the 1st goods and the total income I before the implement of project:

$$ {X}_1^{*}\equiv \left(\frac{\theta_1}{p_1^{*}}\right)/I,\kern1em here,\kern0.5em I\equiv \left(wT+{D}_v+{D}_u\right). $$

(10.19)

The Δs _π is the amounts equivalent to the consumer surplus corresponding to the total of the increase of profits in the whole industries, when the partial equilibrium of the 1st goods (transport service market) has attained and the initial benefits received by the 1st industry have done forward shifting, supposing the equilibrium being not broken as to all the markets except the 1st goods, which is defined to be differences between the benefits obtained from the enterprise profits before the implement of project π _i and those obtained from the enterprise profits after the implement of project, π ^* _i . The 1/I ^* in the equation of Δs _π do mean the marginal utility of the income; the reason that is divided by which is so as to convert the monetary unit to the utility unit (the increase of consumer surplus ) [6].

This total income I ^* is defined as follows, considering the influences of profit variations due to the price change after the implement of project:

$$ {I}^{*}\equiv \left(wT+{\displaystyle \sum_{i=1}^n{\pi}_i^{*}{n}_i}+{D}_u\right). $$

(10.20)

Here, it should be noticed that the number of enterprises of each industry after the implement of project has not been changed.

The profit of enterprise π ^* _i , products y ^* _i , and intermediate input x ^*₁₁ pertaining to the 1st industry after the implement of project are defined as follows:

$$ {\pi}_i^{*}\equiv {p}_1^{*}{y}_1^{*}-\left(w-{\delta}_1\right){z}_1-{p}_1^{*}{x}_{11}^{*}-{\displaystyle \sum_{m=2}^n{p}_m{x}_{m1}} $$

(10.21)

$$ {y}_i^{*}\equiv {\tilde{A}}_1{\Lambda}_1{z}_1^{a_{01}}{\left({x}_{11}^{*}\right)}^{a_{11}}{x}_{21}^{{\tilde{a}}_{21}}{\displaystyle \prod_{m=3}^n{x}_{m=1}^{am1}} $$

(10.22)

$$ {x}_{11}^{*}=\left(\frac{a_{11}}{p_1^{*}}\right){\left({p}_1^{*}{\tilde{A}}_1{\Lambda}_1\right)}^{\frac{1}{{\overset{\sim }{\mu}}_1}}{\left(\frac{\left(w-{\delta}_1\right)}{a_{01}}\right)}^{\frac{-{a}_{01}}{{\overset{\sim }{\mu}}_1}}{\left(\frac{p_1^{*}}{a{}_{11}}\right)}^{\frac{-{\overset{\sim }{a}}_{11}}{{\overset{\sim }{\mu}}_1}}{\left(\frac{p_2}{{\tilde{a}}_{\mathbf{21}}}\right)}^{\frac{-{\overset{\sim }{a}}_{21}}{{\overset{\sim }{\mu}}_1}}{{\displaystyle \prod_{m=3}^n\left(\frac{p_m}{a_{m1}}\right)}}^{\frac{-{a}_{m1}}{{\overset{\sim }{\mu}}_1}}. $$

(10.23)

Here, the function of externality of the industrial scale enlargement à ₁ and the parameter indicating the scale economies ${\tilde{\mu}}_1$ are defined as follows:

$$ {\tilde{A}}_1\equiv {\tilde{\alpha}}_1{\left({z}_1{n}_1\right)}^{{\tilde{\beta}}_1},\ \mathrm{and}\ {\tilde{\mu}}_1\equiv 1-{\tilde{\alpha}}_{21}-{\displaystyle \sum_{m=0,m\ne 2}^n{\alpha}_{m1}}. $$

(10.24)

Here, the variable with the sign of “^~” at the upper right-hand side indicates the technological change due to the impact after the implement of project, and the variable with the sign of “*” at the upper right-hand side indicates that it has received the influence of price change of the 1st goods .

The enterprise profits from the 2nd industry to the nth industry π ^* _i , product y ^* _i , and intermediate input x ^*_1i are defined as follows:

$$ {\pi}_i^{*}\equiv {p}_i^{*}{y}_i^{*}-\left(w-{\delta}_i\right){z}_i-{p}_1^{*}{x}_{1i}^{*}-{\displaystyle \sum_{m=2}^n{p}_m{x}_{mi}\left(i=2,\cdots, n\right)}, $$

(10.25)

$$ {y}_i^{*}\equiv {A}_i{\Lambda}_i{z}_i^{a_{0i}}{\left({x}_{1i}^{*}\right)}^{a_{1i}}{\displaystyle \prod_{m=2}^n{x}_{mi}^{a_{mi}}\left(i=2,\cdots, n\right)}, $$

(10.26)

$$ {x}_{1i}^{*}=\left(\frac{a_{1i}}{p_1^{*}}\right){\left({p}_i{A}_i{\Lambda}_i\right)}^{{}^{\frac{1}{\mu_1}}}{\left(\frac{\left(w-{\delta}_i\right)}{a_{0i}}\right)}^{\frac{-{a}_{0i}}{\mu_i}}{\left(\frac{p_1^{*}}{a_{1i}}\right)}^{\frac{-{a}_{1i}}{\mu_i}}{\displaystyle \prod_{m=2}^n{\left(\frac{p_m}{a_m}\right)}^{\frac{-{a}_{mi}}{\mu_i}}\left(i=2,\cdots, n\right)}. $$

(10.27)

The variables with the mark “*” indicate that they receive the influences from the price change of the 1st goods .

The partial equilibrium price of the 1st goods p ^*₁ is defined to satisfy the following demand‐supply equilibrium equation of the 1st goods:

$$ {y}_1^{*}{n}_1-{x}_{11}^{*}{n}_1-{\displaystyle \sum_{i=2}^n{x}_{1i}^{*}{n}_i-{X}_1^{*}}=0. $$

(10.28)

Here, the variables with the sign “*” should be payed attention to be the function of price p ^*₁ , respectively. The equilibrium price p ^*₁ is obtained so as to satisfy this equation.

As mentioned above, it is conspicuously characteristic that the influence of price change of the 1st goods p ^*₁ has exerted on the π ^* _i , y ^* _i , x ^*_1i in the other markets excepting the 1st industry. But, it is nevertheless that the number of enterprises of each industry after the implement of project has not changed, as the design model of the benefits in generation base. This point is almost the same as the previous model, which states that the influence of the initial impact is only to the 1st industry (transport services) and the 4th industry (resource); but this time the model exerts influence over the more extensive industries including those of m = 3, …, n, which is essentially different from the model of previous chapter (see, Chap. 9 or [17]).