{ "cells": [ { "cell_type": "markdown", "id": "f2bcccd4-fde4-45f7-9c8e-00927bb69641", "metadata": {}, "source": [ "# 予算配分について\n", "\n", "Ryo Ikota (Jan. 14th, 2022)\n", "\n", "10兆円の大学ファンドなるものがあるらしいので、予算配分について簡単な考察をしてみた。\n", "目的は **予算の極端な集中配分は効果的ではない可能性がある** ことを示すことである。\n", "初等的なミクロ経済学の教科書(大学1、2年生レベル)に載っていそうな内容と思われるが、自分はその辺りよく知らない。\n", "経済学教授の目に留まると不可の成績を頂戴するのかも知れない。" ] }, { "cell_type": "markdown", "id": "e0fa2e6f-6fb1-463e-9131-ac96606656b1", "metadata": {}, "source": [ "## 準備" ] }, { "cell_type": "code", "execution_count": 1, "id": "fffcdeea-b2f6-40af-adab-996c5ffe9528", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import sympy as sy" ] }, { "cell_type": "markdown", "id": "43ffae99-84e4-41a7-83cd-cdf21b081cca", "metadata": {}, "source": [ "## 予算と研究成果の量についての仮定" ] }, { "cell_type": "markdown", "id": "2c1036cf-7378-40ef-9df8-11c7aa49066b", "metadata": {}, "source": [ "ある大学の予算を1000倍にすれば研究成果も1000倍になるかというと、そんなことはないだろう(**収穫逓減**)。どこかで頭打ちになるはずだ。\n", "そこで、予算 $x$ を与えられた時の研究成果の量 $f(x)$ を\n", "\n", "$$\n", "f(x) = \\frac{c x}{1+x}\n", "$$\n", "\n", "とおく。ここで $c$ は正のパラメータ。上に凸な関数ならどれでも同じような結果になるだろう。パラメータが $c=1$ の時の関数 $f(x)$ のグラフは以下の通り:" ] }, { "cell_type": "code", "execution_count": 2, "id": "31fd8b77-fa9f-45ef-9799-88c2b9df5f4e", "metadata": {}, "outputs": [], "source": [ "def f(x, c=1):\n", " return c * x / (1 + x)" ] }, { "cell_type": "code", "execution_count": 3, "id": "0a237a3f-607b-428c-8430-315e592ae67b", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "xn = np.linspace(0., 2., 41)\n", "yn = f(xn)\n", "\n", "plt.plot(xn, yn)\n", "plt.xlabel(r\"$x$\")\n", "plt.ylabel(r\"$f(x)$\")\n", "plt.title(r\"$f(x)$ for $c=1$\")\n", "plt.grid(True);" ] }, { "cell_type": "markdown", "id": "f037d99c-cac9-4de7-8a2a-503c56574ec9", "metadata": {}, "source": [ "ここではA大学とB大学の間で総予算 $Y$ を分け合うものとする(大学の数が $N$ 個でも同様の結果が成り立つだろう)。\n", "総額 $Y$ のうち $m$ がA大学に配分されるものとする。A大学は $c=2$, B大学は $c=1$ とする。\n", "つまり、 **同じ予算があるならA大学はB大学よりも2倍の成果を出す** と仮定する。\n", "\n", "この時、 **A大学にほとんどの予算を割り振るのが効果的に思える人もいるかも知れないが、そうとも限らない** ことを示す。" ] }, { "cell_type": "markdown", "id": "f6c3cd6e-580f-49bb-b857-d644f6c5a671", "metadata": {}, "source": [ "## 最適な配分の計算" ] }, { "cell_type": "markdown", "id": "0cdc0a8b-99a4-4d84-bf6b-c8a2657ac2d4", "metadata": {}, "source": [ "以上の仮定から、二つの大学の研究成果の合計は\n", "\n", "$$\n", "Q(m, Y) = f(m, 2) + f(Y-m,1), \\qquad (0 \\leq m \\leq Y)\n", "$$\n", "\n", "となる。" ] }, { "cell_type": "markdown", "id": "8c67f244-3e82-4732-82b7-7c621c2f50b1", "metadata": {}, "source": [ "簡単な計算から、 $Y$ が与えられた時に $Q(m,Y)$ を最大化する $m=m_{\\rm opt}$ は次式で与えられる:\n", "\n", "$$\n", "m_{\\rm opt} = \n", "\\begin{cases}\n", "Y & {\\text for} \\; 0 \\leq Y \\leq \\sqrt{2} - 1\\\\\n", "- \\sqrt{2} Y + 2 Y - 2 \\sqrt{2} + 3 \\qquad & {\\text for}\\; Y \\geq \\sqrt{2} - 1 \n", "\\end{cases}\n", "$$" ] }, { "cell_type": "markdown", "id": "4d718ded-fa8e-4069-acfa-86596cc4933a", "metadata": {}, "source": [ "総額を $Y=6$ とした時の $Q(m,Y)$ のグラフは以下の通りで、 $m_{\\rm opt} \\neq Y$, つまりA大学だけに全ての予算を割り当てることは最適ではない。" ] }, { "cell_type": "code", "execution_count": 4, "id": "4970a28a-69f7-49d9-ac7b-5b69b375ecf8", "metadata": {}, "outputs": [], "source": [ "def Q(m,Y):\n", " return f(m,2) + f(Y - m, 1)" ] }, { "cell_type": "code", "execution_count": 5, "id": "7339a835-a3d1-444b-ab4f-135dbbd34062", "metadata": {}, "outputs": [ { "data": { "image/png": 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i7n1lOW+s3M4vrziRnp1ygi4tphQKEjPuztRFm/nx1KWs33mAc4b04I4LBtO/h64Zl/hQkJPJL68YzfgBhXz3uUWcf98Mfnb5KM4dWhR0aTGjUJCYWLRhD9//5xLeXrOTQUUF/OnGUzhjQPegyxL5EDPj8rJiTurTmVueepfPPzGH68eV8u2PDSEzPfmf4aBQkHa1c18N90xdyt/KK+iSm8UPPzmcK8aUkJEC/7gksR1fmM+zXx7HT6a+x6NvrGbltioeuPokCnJiO7FirCkUpF24O/+Yt4G7X1jK3gO13HRGP245ewAdk/wflCSX7Ix0vvvxoQzuWcCdf1/Ipx58k0evH5vUV8YpFKTNrd+5n2//YxEzlm9jdEln7rlshO41kIT26bEl9OrcgS/9qZxLfvMGj14/luG9OwVdVrvQOby0mbr6EA/NWMm5v/gP5Wt28r+fGMYzXxqnQJCkcMaA7jz9pXFkpBmf/t2bvLp0S9AltQuFgrSJ97dUcskDb/CjKcs4o38hr9x2FteNK03J67wleQ3qWcA/bj6d4wvz+PwTc3jizTVBl9Tm1H0kx8TdeeLNtfxoylLyszN44OqTuGB4T81RJEmrR8cc/vqF0/jqU+/y3ecWc7A2xOfPPD7ostqMQkGitrWymm8+vYDp721j4qBCfnr5KAoLNBWxJL/crAwe/GwZt06exw+nLCU7M41rTysNuqw2oVCQqPx7yRZuf2YBVQfr+P7Fw7jm1L46O5CUkpGexi+vHE1NfYjvPreYrPQ0rjy5T9BlHTOFghyV6tp67n5hCX9+ax1DjuvI5CtH6ylWkrIy09O4/zMnMumJcr7194VkpqdxWVlx0GUdE4WCtFrFrv188U/lLNqwl0lnHs9/nTdQD7uRlJedkc7vrinjxsff4b+fnk9WRhofH9Ur6LKiplCQVpn5/nZueWoudfXOI9eN4ewhqTMXjMiR5GSm8/trx3D9o+/wtb/MIzM9jfOH9wy6rKjE5JJUMysxs2lmttTMFpvZrc2sY2b2KzNbYWYLzOykWNQmh+fuPPiflVz76FsUFmTz/C1nKBBEmpGblcGjN4xlZHEnbnlqLm+v3hl0SVGJ1X0KdcB/ufsQ4FTgZjMb2mSdC4ABkdck4Lcxqk1asO9gHV958l3umbqMC4Yfx9+/fDr9uucFXZZI3MrPzuCxG06mpGsuX/5zORt3Hwi6pKMWk1Bw903uPjfycyWwFOjdZLWLgSc8bDbQ2cyOi0V98mHrduznkt+8wdRFm/jWBYO5/zMnpvxjCkVao1OHTB66ZgzVtSG+8Mdyqmvrgy7pqJi7x/aAZqXADGC4u+9t9PkLwD3uPjPy/lXgdnef02T7SYTPJCgqKiqbPHlyVHVUVVWRn58f1bbxpq3bsnpPPb8or6be4ebROQztFrvBZP29xJ9kaQfEti3vbq3jvrkHOa1XOpNGZLf5JdvH0paJEyeWu/uYZhe6e8xeQD5QDlzazLIXgTMavX8VKDvc/srKyjxa06ZNi3rbeNOWbXlt2RYf8p2pfvo9r/r7WyrbbL+tpb+X+JMs7XCPfVvu+/dy73v7C/7w66vafN/H0hZgjrfwvRqzuY/MLBN4Bvizuz/bzCoVQEmj98XAxljUJmF/eWcdNz0+h37d83j2y+Po3yM5fjsUCcpXJvbno8OK+NGUpbyxYnvQ5bRKrK4+MuARYKm7/7yF1Z4Hro1chXQqsMfdN8WivlTn7vzileXc/sxCTu/fnb984TR6FKTWc2lF2kNamnHvp0dzfPc8bn5yLut37g+6pCOK1ZnC6cA1wEfMbF7kdaGZfdHMvhhZZwqwClgB/B74coxqS2l19SHueGYh9736PpeXFfPIdWPI14CySJvJz87g99eOIRRyPv/EHPbX1AVd0mHF5F+/hwePDzvKEunnujkW9UhYbX2IWye/y5SFm/nqR/rz9XMHav4ikXZQ2j2PX111Itf/4R1++tJ7fO8Tw4IuqUV6nkKKqq0P8dWnwoHwPx8bwm3nDVIgiLSjCYN6cN1pfXn8zTVxfWObQiEFHQqEqYvCgXDT+OSZC14knn3z/MEUd+nAN5+ez4Ga+Lx/QaGQYmrrQ9zyZDgQvnPRUAWCSAzlZWfwk0tHsmbHfu59+b2gy2mWQiGF1NSF+MqTc3lp8Wa+e9FQbjyjX9AliaSccf27c/UpfXjkjdWUr42/biSFQoqoqQtxy1Nz+dfiLdz18aF8ToEgEphvXTiEXp068N9PL4i7aTAUCikgFHK+8bf5/GvxFr738aHccLoCQSRI+dkZ3HPZCFZt28cv/r086HI+QKGQAu55aRnPz9/IN88fxPUKBJG4MH5AIVeOLeH3M1bx7rpdQZfTQKGQ5B6duZqHZqzi2tP68qWzTgi6HBFp5M6PDaGoYw7fjKNuJIVCEntxwSbufnEJHx1WxF0fH6b7EETiTMecTH506Qje31rFIzNXB10OoFBIWrNX7eDrf5lHWZ8u3HfliaSnKRBE4tHEQT04e3APHvzPSnbvrwm6HIVCMnpvcyWff2IOJV078PB1Y8jJjN3zEETk6P33+YOoOljHA9NXBl2KQiHZbNpzgOv/8DYdMtN5/HMn0zk3K+iSROQIBvfsyCdP7M1js9YE/ghPhUISqa6t5wt/LKeyuo7HbjiZ4i65QZckIq1027kDweEXrwR7iapCIYl87/nFLKjYw88/PYqhvToGXY6IHIXiLrlcc1pfnplbwftbKgOrQ6GQJCa/vY7J76zn5okncN6wnkGXIyJRuHlif/KyMvjpv4KbF0mhkARW7annu88tZvyA7tx27qCgyxGRKHXNy+ILZx3PK0u2BDYvkkIhwe2oOsj97x6ksCCbX+nSU5GE97kz+lFYkM09U5cRfvZYbCkUElhdfYivTn6XvTXO764po0uerjQSSXS5WRl89ewBvLNmF68t2xrz48ckFMzsUTPbamaLWljeycz+aWbzzWyxmd0Qi7oS3f+9vJw3VuzguqFZDO/dKehyRKSNXDm2hNJuufz0pfeoD8X2bCFWZwqPAecfZvnNwBJ3HwVMAO41M/3aexgvLdrMg/9ZyVUn92F8cWbQ5YhIG8pMT+MbHx3Ee1sq+ef8jTE9dkxCwd1nAIcbNXGgwMKT8+RH1q2LRW2JaMveau54dgEjizvxvU8MDbocEWkHHxtxHAN65PPoG6tjOrYQL2MK9wNDgI3AQuBWdw8FW1J8cndufyY8o+IvrhhNdoamsBBJRmbGteNKWVCxh3fX747dcWOVQGZWCrzg7sObWXY5cDpwG3AC8Aowyt33NrPuJGASQFFRUdnkyZOjqqeqqor8/Pyotg3S9PW1PLa4hs8OyeKcvuFuo0RtS3PUlviTLO2AxGtLdZ3z9en7GVWYzhdH5Xxg2bG0ZeLEieXuPqbZhe4ekxdQCixqYdmLwPhG718DTj7SPsvKyjxa06ZNi3rboKzZXuVDvjPVr/79bK+vDzV8nohtaYnaEn+SpR3uidmW/31+sfe/80XfsvfABz4/lrYAc7yF79V46T5aB5wNYGZFwCBgVaAVxZn6kPNff51Peprx08tHkqb7EURSwrWn9aUu5Dz51rqYHC9Wl6Q+BbwJDDKzCjO70cy+aGZfjKxyNzDOzBYCrwK3u/v2WNSWKH7/+irmrN3F9y8eRq/OHYIuR0RipLR7HhMGFvLnt9ZRU9f+Q60Z7X4EwN2vOsLyjcB5saglES3dtJefv7ycC4b35JLRvYMuR0Ri7LpxpVz/h3eYumgTF7fzd0C8dB9JCw7W1fP1v8yjY4dMfnDJcD1SUyQFnTmgkH7d83hs1pp2P5ZCIc79+tUVLNtcyU8uG0G3/OygyxGRAKSlGdee1pd31+1mQcXu9j1Wu+5djsmKrZU8+J+VXHZSMWcPKQq6HBEJ0OVlxeRlpbf72YJCIU65O3c9v5jcrHTuvHBw0OWISMAKcjK5rKyYF+ZvYnvVwXY7jkIhTk1ZuJk3VuzgGx8dpG4jEQHg2tNKqakP8VQ7Xp6qUIhD+w7W8YMXlzD0uI5cfUrfoMsRkTjRv0c+4wd0509vraWunWZPVSjEoV+/toJNe6q5+5JhemiOiHzA9eNK2bL3IHO31LfL/hUKcWbF1ioembmKy8uKKevbNehyRCTOTBjUgwE98tl+oH1uZFMoxBF353vPLyYnM507LtDgsoh8WHqaMfXW8Vx4fPs8ckahEEemLtrMzBXb+cZ5g+iuwWURaUFGevt9dSsU4sT+mjrufmEJQ47ryNWn9Am6HBFJUUcdCmaWZ2Z6sksbu//Q4PLFw9r1twARkcM54rePmaWZ2WfM7EUz2wosAzaZ2WIz+5mZDWj/MpPb5j3VPDxzNZ88sTdjSjW4LCLBac2vpNMIPw3tW0BPdy9x9x7AeGA2cI+ZfbYda0x69097n1DIue3cgUGXIiIprjVTZz/s7n9s+qG77wSeAZ4xs8w2ryxFrN+5n7+8s54rxpZQ0jU36HJEJMW15kzhs2b2y8ONI7h7bRvWlFJ+9er7mBm3fES9cCISvNaEwvlANfCamfVo53pSysptVTwzt4JrTu1Lz045R95ARKSdHTEUIs95vgO4D5hhZpPM7GQzU1/HMfrlv98nJzOdL004IehSRESAVl6SamYXATcBNcBJwP8B681sRTvWltSWbtrLP+dv5IbTS3WjmojEjSMONJvZKmAp8At3f6XJsuLWHMTMHgUuAra6+/AW1pkA/BLIBLa7+1mt2Xei+sUryynIyWDSeJ0liEj8aM3VRxe6+7LmFrh7RSuP8xhwP/BEcwvNrDPwAHC+u69L9rGL+et38/KSLdx27kA65erCLRGJH60ZU2g2EI6Gu88Adh5mlc8Az7r7usj6W4/1mPHs3leW0yU3k8+d0S/oUkREPiBe5lMYCHQxs+lmVm5m1wZdUHt5e/VOZizfxpcmnEB+dmtO1EREYsfco3t6j5kdB+x091Y9LNTMSoEXmhtTMLP7gTHA2UAH4E3gY+6+vJl1JwGTAIqKisomT54cVf1VVVXk5+dHte2xuOftA2za5/z0zA5kp7fNA3SCakt7UFviT7K0A9SWQyZOnFju7mOaXejuUb2AfwOrgf9r5fqlwKIWlt0BfK/R+0eATx1pn2VlZR6tadOmRb1ttOat2+V9b3/Bfz9jZZvuN4i2tBe1Jf4kSzvc1ZZDgDnewvdq1N1H7n4OcDzwh2j30chzwHgzy4jc/3AK4SueksojM1dTkJ3BFWNLgi5FRKRZR9WpbWaZ3mhKi0jiLG7Fdk8BE4DuZlYB3EX40lPc/UF3X2pmLwELgBDh+ZYWHU1t8W7TngNMWbiJ68eVUpCjK45EJD61OhTM7GHgUjPbB2wk/AW+wN1/faRt3f2qVqzzM+Bnra0n0Tw+ay0hd64bVxp0KSIiLTqaM4XxQJG715pZb2AUMLJ9ykou+w7W8eRbazl/eE/NhCoice1oQmE20IXwXckbgA3AlHapKsk8M7eCvdV13HjG8UGXIiJyWEcz0PwQ8B8z+4aZjTezTu1VVDIJhZw/vLGG0SWdKevbJehyREQO62hC4U/AXwmfXXwZmGVmK9ulqiTy2rKtrN6+jxt197KIJICj6T6qcPe7Gn9gZpre8wgembmaXp1yuGB4z6BLERE5oqM5U5hnZrc2/sBbeTdzqlq8cQ9vrtrB9aeXkpEeLzOKiIi07GjOFIqAc8zsdmAuMB+Y5+5/a5fKksAjM1eTm5XOFWP7BF2KiEirtDoU3P3T0NBlNAwYQfjOY4VCM7bureaf8zdy9Sl96dRBN6uJSGJozUN2LHLnMtDQZTQ38mp2HYE/zl5LXci54fTSoEsREWm11nR0TzOzW8zsA30gZpZlZh8xs8eB69qnvMR0sK6eP7+1jnOHFNG3W17Q5YiItFpruo/OBz4HPGVm/YDdhKe3TgNeJvyYznntVWAiem3pVnbuq+HqU/sGXYqIyFE5Yii4e7WZ/QOoAjoD24CX3H1X+5aWuJ4ur6CoYzZn9O8edCkiIkfliN1HZnYeUE74jOEk4JvAbDMb1861JaRtlQeZvnwbl55UTHpa2zxER0QkVlrTffQDYLy7rzj0gZmdBvzezG4E9iXbNNfH4rl5G6gPOZedVBx0KSIiR601oZDVOBAA3P1NM7sUeAE4SPjy1JTn7vxtTgWjSzrTv0dyPPJPRFJLa64+qjazwqYfevj5yfWEn6sswOKNe3lvSyWXl+ksQUQSU2tC4WfAP8ysV+MPzaw7cNDdt7ZLZQno6fIKsjLS+PjIXkdeWUQkDrXm6qNnIncxv2lm5YSnt8gCPk14vEGAmroQz83bwLlDi+iUqzuYRSQxtWqWNnd/EhhCeAyhE1ALfMbdH2/H2hLKa8u2smt/rbqORCShHc3cR/uBR6M5iJk9ClxE+Kltww+z3ljCT3i7wt2fjuZYQXm6vIIeBdmM170JIpLAYjWf82OE73NokZmlAz8B/hWLgtrStsqDTHtvK588sbemyBaRhBaTbzB3nwHsPMJqtwDPAAk3cN1wb4K6jkQkwcXFr7Vm1hv4JPBg0LVE4+nyCkYVd2JgUUHQpYiIHBOL1YzXZlYKvNDcmIKZ/Q24191nm9ljkfWaHVMws0nAJICioqKyyZMnR1VPVVUV+fnHfoPZ2r313DWrmmuGZnF2n2CuOmqrtsQDtSX+JEs7QG05ZOLEieXuPqbZhe4ekxdQCixqYdlqYE3kVUW4C+mSI+2zrKzMozVt2rSot23se88v8gF3TvFd+w62yf6i0VZtiQdqS/xJlna4qy2HAHO8he/Vo3kcZ7tx936Hfm50pvCPwApqpdr6EM/N28g5Q3vQOTcr6HJERI5ZTELBzJ4CJgDdzawCuAvIBHD3hBxHAHhr1U527qvh4tG9gy5FRKRNxCQU3P2qo1j3+nYspU29vGQzOZlpnDngQ1NDiYgkpLi4+igRuTuvLNnC+AGFdMhKD7ocEZE2oVCI0qINe9m0p5rzhhYFXYqISJtRKETp5SWbSTM4e4hCQUSSh0IhSi8v3sLY0q50zdNVRyKSPBQKUVizfR/vbankvGE9gy5FRKRNKRSi8MqSLQAaTxCRpKNQiMIrS7YwuGcBJV1zgy5FRKRNKRSO0vaqg8xZu1NdRyKSlBQKR+m1pVsJubqORCQ5KRSO0stLNtO7cweG9eoYdCkiIm1OoXAU9tfU8fr72zl3aBFmFnQ5IiJtTqFwFGYs387BupC6jkQkaSkUjsLLSzbTqUMmY/t1DboUEZF2oVBopbr6EK8u3crZg3uQma4/NhFJTvp2a6W31+xkz4FazhumriMRSV4KhVZ6efEWsjPSOHOgnp0gIslLodAK///ZCd3JzYqLJ5iKiLQLhUIrvLelkg27D3COpskWkSSnUGiFWSt2AHDGgO4BVyIi0r5iEgpm9qiZbTWzRS0sv9rMFkRes8xsVCzqaq1ZK3fQp2suxV00AZ6IJLdYnSk8Bpx/mOWrgbPcfSRwN/BQLIpqjfqQ89bqHYw7oVvQpYiItLuYjJq6+wwzKz3M8lmN3s4Gitu9qFZavHEPldV1nKZQEJEUEI9jCjcCU4Mu4pA3V4bHE047XqEgIsnP3D02BwqfKbzg7sMPs85E4AHgDHff0cI6k4BJAEVFRWWTJ0+Oqp6qqiry8/OPuN69c6rZfiDEj8fH73hCa9uSCNSW+JMs7QC15ZCJEyeWu/uYZhe6e0xeQCmw6DDLRwIrgYGt3WdZWZlHa9q0aUdcp6au3od8Z6r/z98XRn2cWGhNWxKF2hJ/kqUd7mrLIcAcb+F7NS66j8ysD/AscI27Lw+6nkMWVOxmf029xhNEJGXEZKDZzJ4CJgDdzawCuAvIBHD3B4HvAt2AByLPKajzlk5tYujQeMKpGk8QkRQRq6uPrjrC8puAm2JRy9GYtXIHg3sW0DUvK+hSRERiIi66j+JRdW095Wt3Me4E3cUsIqlDodCCd9ft5mBdSOMJIpJSFAoteHPVDtIMTtZT1kQkhSgUWvDmyu0M792JTh0ygy5FRCRmFArN2F9Tx7z1u9V1JCIpR6HQjDlrdlFb75raQkRSjkKhGW+u2kFGmjG2VOMJIpJaFArNmLVyB6NLOpOXrUdvikhqUSg0sbe6loUVGk8QkdSkUGjindU7CTkKBRFJSQqFJt5cuYOsjDRO6tMl6FJERGJOodDErJU7KOvThZzM9KBLERGJOYVCI7v21bBk0151HYlIylIoNDJn7S4ATtHUFiKSohQKjSys2E2awYjiTkGXIiISCIVCI/Mr9jCwqIDcLN2fICKpSaEQ4e4s3LCHEb11liAiqUuhEFGx6wA799UwsqRz0KWIiARGoRCxoGIPAKM0niAiKSwmoWBmj5rZVjNb1MJyM7NfmdkKM1tgZifFoq7GFmzYTWa6MahnQawPLSISN2J1pvAYcP5hll8ADIi8JgG/jUFNH7Bg/R6GHNeR7AzdtCYiqSsmoeDuM4Cdh1nlYuAJD5sNdDaz42JRG0Ao5CzasIeR6joSkRQXL2MKvYH1jd5XRD6LidU79lF5sI6RvTvH6pAiInEpXi7It2Y+82ZXNJtEuIuJoqIipk+fHtUBq6qqGradtbEOgIOb32f69JVR7S9IjduS6NSW+JMs7QC1pTXiJRQqgJJG74uBjc2t6O4PAQ8BjBkzxidMmBDVAadPn86hbac/v5gOmeu56sIJZKTHy8lT6zVuS6JTW+JPsrQD1JbWiJdvwOeBayNXIZ0K7HH3TbE6+MINexjWq2NCBoKISFuKyZmCmT0FTAC6m1kFcBeQCeDuDwJTgAuBFcB+4IZY1AVQVx9i8cY9fObkvrE6pIhI3IpJKLj7VUdY7sDNsailqeVbqqiuDTGqRFceiYikfH/Jwg27ATTnkYgICgXmV+yhICeD0m55QZciIhK4lA+FhRXhm9bS0pq7KlZEJLWkdCgcrKtn2ea9jNBNayIiQIqHwtJNldTWu2ZGFRGJSOlQWFixG0DPUBARiUjpUJhfsYdueVn06pQTdCkiInEhpUNhQcVuRhZ3wkyDzCIikMKhUF3nrNhaxcjizkGXIiISN1I2FNbuDRFy9AwFEZFGUjYUVu8JAehMQUSkkZQNhTV76+nVKYfCguygSxERiRspGwqr94QYoa4jEZEPSMlQ2LO/li37XV1HIiJNpGQoLNywB4BRCgURkQ9IyVDIzkxjVGG6pssWEWkiJUNhbGlXvl6WQ6fczKBLERGJKykZCiIi0jyFgoiINIhZKJjZ+Wb2npmtMLM7mlneycz+aWbzzWyxmd0Qq9pERCQsJqFgZunAb4ALgKHAVWY2tMlqNwNL3H0UMAG418yyYlGfiIiExepM4WRghbuvcvcaYDJwcZN1HCiw8JSl+cBOoC5G9YmICLELhd7A+kbvKyKfNXY/MATYCCwEbnX3UGzKExERAHP39j+I2aeAj7r7TZH31wAnu/stjda5HDgduA04AXgFGOXue5vsaxIwCaCoqKhs8uTJUdVUVVVFfn5+VNvGG7UlPiVLW5KlHaC2HDJx4sRydx/T7EJ3b/cXcBrwr0bvvwV8q8k6LwLjG71/jXBwtLjfsrIyj9a0adOi3jbeqC3xKVnakiztcFdbDgHmeAvfqxlRxczRewcYYGb9gA3AlcBnmqyzDjgbeN3MioBBwKrD7bS8vHy7ma2NsqbuwPYot403akt8Spa2JEs7QG05pG9LC2ISCu5eZ2ZfAf4FpAOPuvtiM/tiZPmDwN3AY2a2EDDgdnc/bIPdvTDamsxsjrd0+pRg1Jb4lCxtSZZ2gNrSGrE6U8DdpwBTmnz2YKOfNwLnxaoeERH5MN3RLCIiDVI5FB4KuoA2pLbEp2RpS7K0A9SWI4rJJakiIpIYUvlMQUREmkjJUDjS5HyJwsweNbOtZrYo6FqOhZmVmNk0M1samQzx1qBripaZ5ZjZ240mdvzfoGs6VmaWbmbvmtkLQddyLMxsjZktNLN5ZjYn6HqiZWadzexpM1sW+TdzWpvuP9W6jyKT8y0HziU83cY7wFXuviTQwqJgZmcCVcAT7j486HqiZWbHAce5+1wzKwDKgUsS9O/EgDx3rzKzTGAm4SlbZgdcWtTM7DZgDNDR3S8Kup5omdkaYMyRLnWPd2b2OPC6uz8cmTQ01913t9X+U/FMoTWT8yUEd59BeOLAhObum9x9buTnSmApH54bKyFEbhitirzNjLwS9jcvMysGPgY8HHQtAmbWETgTeATA3WvaMhAgNUOhNZPzSUDMrBQ4EXgr4FKiFulumQdsBV5x94RtC/BL4JtAMkxO6cDLZlYemUMtER0PbAP+EOnSe9jM8tryAKkYCtbMZwn7m1wyMbN84Bnga95kIsRE4u717j4aKAZONrOE7Nozs4uAre5eHnQtbeR0dz+J8HNdbo50vyaaDOAk4LfufiKwD2jTcdFUDIUKoKTR+2LC03VLgCL9788Af3b3Z4Oupy1ETuunA+cHW0nUTgc+EemLnwx8xMz+FGxJ0YvMmoC7bwX+TrgrOdFUABWNzj6fJhwSbSYVQ6Fhcr7IIM2VwPMB15TSIoOzjwBL3f3nQddzLMys0Mw6R37uAJwDLAu0qCi5+7fcvdjdSwn/O3nN3T8bcFlRMbO8yEUMRLpbzgMS7qo9d98MrDezQZGPzgba9IKMmM19FC9ampwv4LKiYmZPEX50aXczqwDucvdHgq0qKqcD1wALI33xAHdG5stKNMcBj0eucksD/uruCX0pZ5IoAv4e/v2DDOBJd38p2JKidgvw58gvtauANn2efcpdkioiIi1Lxe4jERFpgUJBREQaKBRERKSBQkFERBooFEREpIFCQUREGigURESkgUJBpI2Y2d/M7H4zm2lma83sDDN7wsyWm1ki3lQoKUihINJ2RgCr3P0M4HHCU3fcDgwHLjWz7CCLE2mNlJvmQqQ9mFkO0JnwVNMAB4BH3H1TZPl+oCaQ4kSOgs4URNrGMGCuux967sAoIs+EiDyoZqNrThlJAAoFkbYxApjf6P1IYEHk51GNfhaJawoFkbYxApgHDV1JHdx9V2RZ44AQiWuaJVVERBroTEFERBooFEREpIFCQUREGigURESkgUJBREQaKBRERKSBQkFERBooFEREpMH/AzsirKAj3C4rAAAAAElFTkSuQmCC", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "Y_sample = 6.0\n", "m_sample = np.linspace(0., Y_sample, 51)\n", "Q_sample = Q(m_sample, Y_sample)\n", "plt.plot(m_sample, Q_sample)\n", "plt.xlabel(r\"$m$\")\n", "plt.ylabel(r\"$Q(m, Y)$\")\n", "plt.title(r\"$Q(m,Y)$ for $Y=6$\")\n", "plt.grid(True);" ] }, { "cell_type": "markdown", "id": "5bedba89-484d-439c-96b8-a3aa0cca9789", "metadata": {}, "source": [ "総額 $Y$ を横軸にした時の最適なA大学への予算配分率 $m_{\\rm opt}/Y$ のグラフは以下の通りで、$m_{\\rm opt}/Y \\rightarrow 2 - \\sqrt{2} \\simeq 0.586$ に漸近する。" ] }, { "cell_type": "code", "execution_count": 6, "id": "5129b901-c5c5-4137-bc9f-dd5be03eac40", "metadata": {}, "outputs": [], "source": [ "def mopt(Y):\n", " Y = np.asarray(Y)\n", " sqrt2 = np.sqrt(2)\n", " return np.where(Y < sqrt2 - 1.0, Y, -sqrt2*Y + 2.0*Y - 2*sqrt2 + 3.0)" ] }, { "cell_type": "code", "execution_count": 7, "id": "c254d0e4-3d5e-427b-8c4f-0f65d3b5e57d", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "Yn = np.linspace(0.1, 10.0, 101) # starting at 0.1 to avoid dividing by zero.\n", "plt.plot(Yn, mopt(Yn)/Yn)\n", "plt.xlabel(r\"$Y$\")\n", "plt.ylabel(r\"$m_{\\rm opt}/Y$\")\n", "plt.title(\"The allocation ratio\")\n", "plt.grid(True);" ] }, { "cell_type": "markdown", "id": "46d519cb-6e96-4e13-b4f3-8b562d24a6fd", "metadata": {}, "source": [ "配分が最適である時の二大学の合計の研究成果のグラフは以下の通り:" ] }, { "cell_type": "code", "execution_count": 8, "id": "16790f43-d630-4da5-ab06-ebf8924554d3", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "Qopt_np = Q(mopt(Yn), Yn)\n", "plt.plot(Yn, Qopt_np)\n", "plt.xlabel(r\"$Y$\")\n", "plt.ylabel(r\"$Q(m_{\\rm opt}(Y), Y)$\")\n", "plt.title(\"The total output when the allocation is optimal\")\n", "plt.grid(True);" ] }, { "cell_type": "markdown", "id": "2b30d240-ab90-45fe-ad0d-b907d511f6a5", "metadata": {}, "source": [ "## 結語" ] }, { "cell_type": "markdown", "id": "0a627d56-54ba-45f5-aeb9-174cae11abb0", "metadata": {}, "source": [ "以上の考察は極めてシンプルなもので、現実はこれほど単純ではないだろうが、一つの示唆を与える。\n", "それは、**収穫逓減があれば投資の極端な選択と集中は非効率である可能性がある**ということだ。\n", "\n", "選択と集中が望ましくないと考えられる理由は他にもある。選択と集中はギャンブルであり、ミクロのレベルで行うべきで、マクロのレベルで行うべきではない。\n", "例えば、国全体の経済からすれば企業はミクロな存在であり、それぞれの企業が経営判断によって選択と集中を行った場合、成功して成長する企業もあれば失敗して大きな損失を被る企業もあるだろう。しかしながらそういったランダムネスは国全体のレベルではランダムさが互いに打ち消しあって(**大数の法則**)、マクロなレベルではギャンブル性は薄れるだろう。他方、国全体の大学研究の予算を選択集中した場合は失敗した場合の損失は極めて大きなものになる。企業というミクロの視点で国というマクロな存在の政策を考えることは慎むべきではなかろうか。" ] }, { "cell_type": "markdown", "id": "2d3c04ef-60c4-4b93-927d-21e548ae7640", "metadata": {}, "source": [ "## 付録(SymPyによる計算)" ] }, { "cell_type": "code", "execution_count": 9, "id": "c0ec8c3b-57fb-4a66-ae7e-d9de294ec433", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(Y, m)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sy.var('Y m', real=True)" ] }, { "cell_type": "code", "execution_count": 10, "id": "648a7aa3-24ba-4643-bbc9-6c1b4c112abc", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle - \\frac{2 m}{\\left(m + 1\\right)^{2}} + \\frac{Y - m}{\\left(Y - m + 1\\right)^{2}} - \\frac{1}{Y - m + 1} + \\frac{2}{m + 1}$" ], "text/plain": [ "-2*m/(m + 1)**2 + (Y - m)/(Y - m + 1)**2 - 1/(Y - m + 1) + 2/(m + 1)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dQdm = sy.diff(Q(m,Y),m)\n", "dQdm" ] }, { "cell_type": "code", "execution_count": 11, "id": "9666d712-92d1-45aa-a744-2873c15718e6", "metadata": {}, "outputs": [], "source": [ "m1, m2 = sy.solve(dQdm, m)" ] }, { "cell_type": "code", "execution_count": 12, "id": "dacde34d-3cb5-4197-a5d6-b8a0010797aa", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle - \\sqrt{2} Y + 2 Y - 2 \\sqrt{2} + 3$" ], "text/plain": [ "-sqrt(2)*Y + 2*Y - 2*sqrt(2) + 3" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m1" ] }, { "cell_type": "code", "execution_count": 13, "id": "906d65e5-8353-4528-a8d0-d2cde400b809", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\sqrt{2} Y + 2 Y + 2 \\sqrt{2} + 3$" ], "text/plain": [ "sqrt(2)*Y + 2*Y + 2*sqrt(2) + 3" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m2" ] }, { "cell_type": "markdown", "id": "b5e5e987-1984-487f-ad85-1e50cf79db30", "metadata": {}, "source": [ "$0 \\leq m \\leq Y$ であるから m1 が解。" ] }, { "cell_type": "code", "execution_count": 14, "id": "3a77d37a-40f4-48b0-8247-574cf487aaeb", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle -1 + \\sqrt{2}$" ], "text/plain": [ "-1 + sqrt(2)" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Y0, = sy.solve(m1 - Y, Y)\n", "Y0" ] }, { "cell_type": "code", "execution_count": 15, "id": "946ca0b9-3f66-4488-bab6-7db46bfc260b", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\frac{- 12 Y + 9 \\sqrt{2} Y - 24 + 17 \\sqrt{2}}{- 4 Y + 3 \\sqrt{2} Y - 8 + 6 \\sqrt{2}}$" ], "text/plain": [ "(-12*Y + 9*sqrt(2)*Y - 24 + 17*sqrt(2))/(-4*Y + 3*sqrt(2)*Y - 8 + 6*sqrt(2))" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Qopt = Q(m1,Y).simplify()\n", "Qopt" ] }, { "cell_type": "code", "execution_count": 16, "id": "2afa9cc0-0ff0-4293-a321-83afe2a2cd98", "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 3$" ], "text/plain": [ "3" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sy.limit(Qopt, Y, sy.oo)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.4" } }, "nbformat": 4, "nbformat_minor": 5 }