Verification of Independent Existence Theory Depended on the Market Equilibrium Model: Based on the Great Discrepancy of the Benefits in Generation Base vs. the Benefits in Incidence Base

Abstract

Jan Tinbergen’s, “The Appraisal of Road Construction: Two Calculation Schemes [18],” that appeared in RE & Stat. on August 1957, which gave us a great profound impression, coincided with the mentality of the times when we are about to study on the real economic research of expressways, right on the heels of Report on Kobe–Nagoya Expressway Survey (for the Ministry of Construction, Government of Japan, August 8, 1956, 188 pp.) by a group of experts headed by Ralph J. Watkins. (10 months later, the introductory paper of [18] by Yukihide Okano came out in the periodical: Expressways, Express Highway Research Foundation of Japan.)

This chapter is written based on [3, 6–8].

Appendix

9.1.1 Result of Simulation

Simulation results of the models I, II, and III are as follows:

[Model I (M = 4)]

$$ \widehat{p} \equiv \left({\widehat{p}}_1,{\widehat{p}}_2,\dots, {\widehat{p}}_M\right)=\left(4.0223,\ 3.7626,\ 3.5795,\ 3.5202\right); $$

(9.1a)

$$ \tilde{p} \equiv \left({\tilde{p}}_1,{\tilde{p}}_2,\dots, {\tilde{p}}_M\right)=\left(3.6674,\ 3.7489,\ 3.5398,\ 3.4184\right); $$

(9.2a)

$$ {\widehat{Y}}_I \equiv \left({\widehat{Y}}_1,{\widehat{Y}}_2,\dots, {\widehat{Y}}_M\right)=\left(6.5326,\ 0.40133,\ 0.71896,\ 1.3065\right); $$

(9.3a)

$$ {\tilde{Y}}_I \equiv \left({\tilde{Y}}_1,{\tilde{Y}}_2\dots, {\tilde{Y}}_M\right)=\left(7.1077,\ 0.40806,\ 0.72624,\ 1.3372\right); $$

(9.4a)

$$ {\widehat{x}}_0 \equiv \left({\widehat{x}}_{01},{\widehat{x}}_{02},\dots, {\widehat{x}}_{0M}\right)=\left(15.77,\ 0.68959,\ 1.3468,\ 2.7134\right); $$

(9.5a)

$$ {\tilde{x}}_0 \equiv \left({\tilde{x}}_{01},{\tilde{x}}_{02},\dots, {\tilde{x}}_{0M}\right)=\left(15.64,\ 0.69860,\ 1.3453,\ 2.6970\right); $$

(9.6a)

$$ {\widehat{x}}_I \equiv \left({\widehat{x}}_{21},{\widehat{x}}_{32},\dots, {\widehat{x}}_{M,M-1},{\widehat{x}}_{1M}\right)=\left(2.0956,\ 0.18703,\ 0.27537,\ 0.35436\right); $$

(9.7a)

$$ {\tilde{x}}_I \equiv \left({\tilde{x}}_{21},{\tilde{x}}_{32},\dots, {\tilde{x}}_{M,M-1},{\tilde{x}}_{1M}\right)=\left(2.4336,\ 0.19159,\ 0.28326,\ 0.38639\right); $$

(9.8a)

$$ \widehat{y} \equiv \left({\widehat{y}}_1,{\widehat{y}}_2,\dots, {\widehat{y}}_M\right)=\left(29.537,\ 126.33,\ 99.599,\ 67.518\right); $$

(9.9a)

$$ \tilde{y} \equiv \left({\tilde{y}}_1,{\tilde{y}}_2,\dots, {\tilde{y}}_M\right)=\left(32.053,\ 125.42,\ 99.625,\ 68.774\right); $$

(9.10a)

$$ \widehat{\pi} \equiv \left({\widehat{\pi}}_1,{\widehat{\pi}}_2,\dots, {\widehat{\pi}}_M\right)=\left(2.628,\ 0.151,\ 0.2574,\ 0.4599\right); $$

(9.11a)

$$ \tilde{\pi} \equiv \left({\tilde{\pi}}_1,{\tilde{\pi}}_2,\dots, {\tilde{\pi}}_M\right)=\left(1.303,\ 0.153,\ 0.2571,\ 0.4571\right); $$

(9.12a)

$$ \widehat{s} \equiv \left({s}_1\left({\widehat{y}}_1\right),{s}_2\left({\widehat{y}}_2\right),\dots, {s}_M\left({\widehat{y}}_M\right)\right)=\left(0.33856,\ 1.9356,\ 1.3803,\ 0.84248\right); $$

(9.13a)

$$ \tilde{s} \equiv \left({s}_1\left({\tilde{y}}_1\right),{s}_2\left({\tilde{y}}_2\right),\dots, {s}_M\left({\tilde{y}}_M\right)\right)=\left(0.34674,\ 1.9327,\ 1.3804,\ 0.84617\right); $$

(9.14a)

$$ {\widehat{I}}_y=1188.372; $$

(9.15a)

$$ {\tilde{I}}_y=1175.499; $$

(9.16a)

$$ {\widehat{\mu}}_y \equiv 1/{\widehat{I}}_y=0.84149\times {10}^{-3}; $$

(9.17a)

$$ {\tilde{\mu}}_y \equiv 1/{\tilde{I}}_y=0.85700\times {10}^{-3}; $$

(9.18a)

$$ {p}_1^{\ast }=3.96893; $$

(9.19a)

$$ {Y}_1^{\ast }=6.7788; $$

(9.20a)

$$ {Y}_4^{\ast }=1.3293; $$

(9.21a)

$$ {x}_{14}^{\ast }=0.37471; $$

(9.22a)

$$ {y}_1^{\ast }=29.942; $$

(9.23a)

$$ {\pi}_1^{\ast }=3.2498; $$

(9.24a)

$$ {\pi}_4^{\ast }=0.47872; $$

(9.25a)

$$ {s}_1^{\ast } \equiv {s}_1\left({y}_1^{\ast}\right)=0.33993; $$

(9.26a)

$$ {I}_y^{\ast }=1196.489; $$

(9.27a)

$$ {\mu}_y^{\ast }=0.83578\times {10}^{-3}; $$

(9.28a)

$$ \widehat{\Omega}=\left[\begin{array}{cccc}\hfill -6.4674\times {10}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1.1453\times {10}^2\hfill \\ {}\hfill -7.1251\times 10\hfill & \hfill -1.4140\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -3.2489\times {10}^2\hfill & \hfill -1.6319\times {10}^3\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -2.5560\times {10}^2\hfill & \hfill -1.2932\times {10}^3\hfill \end{array}\right]; $$

(9.29a)

$$ \tilde{\Omega}=\left[\begin{array}{cccc}\hfill -1.4145\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1.2488\times {10}^2\hfill \\ {}\hfill -1.8009\times {10}^2\hfill & \hfill -1.4377\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -3.3283\times {10}^2\hfill & \hfill -1.6484\times {10}^3\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -2.6292\times {10}^2\hfill & \hfill -1.3236\times {10}^3\hfill \end{array}\right]; $$

(9.30a)

Below, models II and III:

[Model II (M = 5)]

$$ \widehat{p} = \left(3.8614,\ 3.7253,\ 3.5636,\ 3.3815,\ 3.3546\right); $$

(9.31a)

$$ \tilde{p} = \left(3.5901,\ 3.7239,\ 3.5511,\ 3.3522,\ 3.2779\right); $$

(9.32a)

$$ {\widehat{Y}}_{{}_1} = \left(4.6510,\ 0.30312,\ 0.49204,\ 0.67183,\ 0.96192\right); $$

(9.33a)

$$ {\tilde{Y}}_{{}_1} = \left(4.9708,\ 0.30703,\ 0.49401,\ 0.67523,\ 0.97904\right); $$

(9.34a)

$$ {\widehat{x}}_{{}_0} = \left(10.776,\ 0.49685,\ 0.85917,\ 1.2268,\ 1.9039\right); $$

(9.35a)

$$ {\tilde{x}}_{{}_0} = \left(10.707,\ 0.50307,\ 0.85958,\ 1.2223,\ 1.8934\right); $$

(9.36a)

$$ {\widehat{x}}_{{}_1} = \left(1.4463,\ 0.14576,\ 0.21216,\ 0.24379,\ 0.25906\right); $$

(9.37a)

$$ {\tilde{x}}_{{}_1} = \left(1.6773,\ 0.14811,\ 0.21456,\ 0.24859,\ 0.27711\right); $$

(9.38a)

$$ \widehat{y}=\left(20.604,\ 106.79,\ 89.306,\ 70.586,\ 47.434\right); $$

(9.39a)

$$ \tilde{y}=\left(21.998,\ 106.04,\ 88.961,\ 70.678,\ 48.187\right); $$

(9.40a)

$$ \widehat{\pi}=\left(1.7960,\ 0.1129,.\mathrm{0.1753},\ 0.2272,\ 0.3227\right); $$

(9.41a)

$$ \tilde{\pi} = \left(0.8923,\ 0.1143,\ 0.1754,\ 0.2264,\ 0.3209\right); $$

(9.42a)

$$ \widehat{s}=\left(0.20170,\ 1.5569,\ 1.1979,\ 0.85137,\ 0.51458\right); $$

(9.43a)

$$ \tilde{s} = \left(0.20606,\ 1.5546,\ 1.1969,\ 0.85163,\ 0.51668\right); $$

(9.44a)

$$ {\widehat{I}}_y = 1193.434; $$

(9.45a)

$$ {\tilde{I}}_y=1184.6469; $$

(9.46a)

$$ {\widehat{\mu}}_y = 0.83792\times {10}^{-3}; $$

(9.47a)

$$ {\tilde{\mu}}_y=0.84413\times {10}^{-3}; $$

(9.48a)

$$ {p}_1^{*}=3.8355; $$

(9.49a)

$$ {Y}_1^{*}=4.7376; $$

(9.50a)

$$ {Y}_5^{*} = 0.9702; $$

(9.51a)

$$ {x}_{15}^{*} = 0.26632; $$

(9.52a)

$$ {y}_1^{*} = 20.744; $$

(9.53a)

$$ {\pi}_1^{*} = 2.0074; $$

(9.54a)

$$ {\pi}_5^{*} = 0.32933; $$

(9.55a)

$$ {s}_1^{*} = 0.20215; $$

(9.56a)

$$ {I}_y^{*} = 1196.214; $$

(9.57a)

$$ {\mu}_y^{*} = 0.83597\times {10}^{-3}; $$

(9.58a)

$$ \widehat{\varOmega}=\left[0\begin{array}{ccccc}\hfill -4.6200\times {10}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -8.2036\times 10\hfill \\ {}\hfill -4.8211\times {10}^1\hfill & \hfill -1.1721\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -2.8376\times {10}^2\hfill & \hfill -1.4368\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -2.7425\times {10}_2\hfill & \hfill -1.3168\times {10}_3\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1.8203\times {10}^2\hfill & \hfill -9.4910\times {10}^2\hfill \end{array}\right]; $$

(9.59a)

$$ \tilde{\varOmega}=\left[\begin{array}{ccccc}\hfill -9.9088\times {10}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -8.7753\times 10\hfill \\ {}\hfill -1.2301\times {10}^2\hfill & \hfill -1.1872\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -2.8832\times {10}^2\hfill & \hfill -1.4425\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -2.7678\times {10}^2\hfill & \hfill -1.3235\times {10}^3\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1.8562\times {10}^2\hfill & \hfill -9.6600\times {10}^2\hfill \end{array}\right]; $$

(9.60a)

[Model III (M = 6)]

$$ \widehat{p}=\left(3.7251,\ 3.6899,\ 3.5665,\ 3.3725,\ 3.2136,\ 3.2101\right); $$

(9.61a)

$$ \tilde{p}=\left(3.5210,\ 3.6923,\ 3.5626,\ 3.3630,\ 3.1923,\ 3.1531\right); $$

(9.62a)

$$ {\widehat{Y}}_I=\left(3.4628,\ 0.24567,\ 0.38262,\ 0.48758,\ 0.58389,\ 0.72342\right); $$

(9.63a)

$$ {\tilde{Y}}_I=\left(3.6456,\ 0.24837,\ 0.38351,\ 0.48786,\ 0.58558,\ 0.73318\right); $$

(9.64a)

$$ {\widehat{x}}_0=\left(7.7396,\ 0.38979,\ 0.64136,\ 0.8361,\ 1.0320,\ 1.3701\right); $$

(9.65a)

$$ {\tilde{x}}_0=\left(7.7017,\ 0.39433,\ 0.64216,\ 0.83674,\ 1.0281,\ 1.3639\right); $$

(9.66a)

$$ {\widehat{x}}_{{}_{{}_I}}=\left(1.0488,\ 0.11946,\ 0.17399,\ 0.19956,\ 0.20458,\ 0.19326\right); $$

(9.67a)

$$ {\tilde{x}}_I=\left(1.2168,\ 0.12098,\ 0.17470,\ 0.20044,\ 0.20750,\ 0.20354\right); $$

(9.68a)

$$ \widehat{y}=\left(15.303,\ 92.695,\ 79.918,\ 67.613,\ 53.217,\ 35.517\right); $$

(9.69a)

$$ \tilde{y}=\left(16.105,\ 92.147,\ 79.584,\ 67.446,\ 53.290,\ 35.968\right); $$

(9.70a)

$$ \widehat{\pi}=\left(1.2900,\ 0.0907,\ 0.1365,\ 0.1644,\ 0.1876,\ 0.2322\right); $$

(9.71a)

$$ \tilde{\pi}=\left(0.6418,\ 0.0917,\ 0.1366,\ 0.1641,\ 0.1869,\ 0.2312\right); $$

(9.72a)

$$ \widehat{s}=\left(0.12991,\ 1.2941,\ 1.0431,\ 0.80263,\ 0.56777,\ 0.3400\right); $$

(9.73a)

$$ \tilde{s}=\left(0.13234,\ 1.2924,\ 1.0421,\ 0.80216,\ 0.56796,\ 0.3412\right); $$

(9.74a)

$$ {\widehat{I}}_y=1197.118; $$

(9.75a)

$$ {\tilde{I}}_y=1190.8111; $$

(9.76a)

$$ {\widehat{\mu}}_y=0.83534\times {10}^{-3}; $$

(9.77a)

$$ {\tilde{\mu}}_y=0.83976\times {10}^{-3}; $$

(9.78a)

$$ {p}_1^{*}=3.7218; $$

(9.79a)

$$ {Y}_1^{*}=3.4711; $$

(9.80a)

$$ {Y}_6^{*}=0.72422; $$

(9.81a)

$$ {x}_{16}^{*}=0.19394; $$

(9.82a)

$$ {y}_1^{*}=15.317; $$

(9.83a)

$$ {\pi}_1^{*}=1.3095; $$

(9.84a)

$$ {\pi}_6^{*}=0.23284; $$

(9.85a)

$$ {s}_1^{*}=0.12995; $$

(986a)

$$ {I}_y^{*}=1197.377; $$

(9.87a)

$$ {\mu}_{{}_y}^{*}=0.83516\times {10}^{-3}; $$

(9.88a)

$$ \widehat{\Omega}=\left[\begin{array}{cccccc}\hfill -3.4464\times {10}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -6.083\times 10\hfill \\ {}\hfill -3.4460\times 10\hfill & \hfill -1.0023\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -2.4776\times {10}^2\hfill & \hfill -1.2699\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -2.6565\times {10}^2\hfill & \hfill -1.2436\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -2.0976\times {10}^2\hfill & \hfill -1.036\times {10}^3\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1.324\times {10}^2\hfill & \hfill -7.1653\times {10}^2\hfill \end{array}\right]; $$

(9.89a)

$$ \tilde{\Omega}=\left[\begin{array}{cccccc}\hfill -7.2745\times {10}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -6.4067\times 10\hfill \\ {}\hfill -8.8657\times 10\hfill & \hfill -1.0134\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -2.5092\times {10}^2\hfill & \hfill -1.2729\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill -2.6673\times {10}^2\hfill & \hfill -1.2443\times {10}^3\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -2.1069\times {10}^2\hfill & \hfill -1.0390\times {10}^3\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1.3428\times {10}^2\hfill & \hfill -7.2620\times {10}^2\hfill \end{array}\right]; $$

(9.90a)

Note that, in all the models, $\widehat{\Omega}$ and $\tilde{\Omega}$ are dominant diagonal matrixes which have negative diagonal elements, which imply that the equilibrium of the market is locally stable about both $\widehat{P}$ and $\tilde{P}$.