高级微观经济学中的效用极大化与支
AB1 . 1 5 最 优 解1 . 8 5 效 用 函 数 极 大 化 模 型 图 解4321001234x 1x 2B1 . 1 5 最 优 解1 . 8 5 A支 出 函 数 极 小 化 模 型 图 解4321001234x 1x 2p 1 = 2 . 6p 2 = 3wxptsu+=21)1( .(marrrrpwx21+=-* rrpx21+=-* vxtspe+=)1(212( .minrrvpxr1201)(-=12+1 . 2 2 最 优 解2 . 1 2 支 出 函 数 极 小 化 模 型 ( C-D函 数 )4321001234x 1x 2vxtsxpe+=-a122 .minvpxpaa)1()(212 121-=* -*B1 . 1 5 最 优 解1 . 8 5 A支 出 函 数 极 小 化 模 型 图 解4321001234x 1x 2vxtspe+=)1(212( .minrrvpxr1201)(-=12+r 0.7产品1 产品2 e, u 最大支出w价格 2.6 3 0数量 0 0 2支出 0 0 0 7 TRUE效用函数 0 100产品1 产品2 e, u 最小效用v价格 2.6 3 0数量 0 0 2支出 0 0 0 FALSE效用函数 0 4 100wxptsu+=21)( .marvxtspe+=)1(21( .minrr等效用线 0.2381 4 1 2 3 4 5r / u0 0.7 3 0 4 1 2 3 4 5x1 / x2 2.3 0.2381 0.1 3.5751 0.7277 1.6585 2.6117 3.5751 4.5445预算约束线 0.2 3.3171 0.5714 1.4553 2.3775 3.3171 4.2668p1 / p2 2.6 3 # 0.3 3.1015 0.4474 1.2883 2.183 3.1015 4.0339w 8.2511 # 0.4 2.9106 0.3436 1.1428 2.0116 2.9106 3.8271与横轴交点坐标 3.1735 0 0.5 2.737 0.2552 1.0128 1.8567 2.737 3.6383与纵轴交点坐标 0 2.7504 0.6 2.5765 0.1796 0.8948 1.7142 2.5765 3.4632最优解 0.7 2.4266 0.1157 0.7868 1.582 2.4266 3.2991r / p1r+p2r -2.333 0.1846 0.8 2.2856 0.0631 0.6873 1.4584 2.2856 3.1443p1r-1 / p2r-1 0.0414 0.0257 0.9 2.1523 0.0229 0.5954 1.3422 2.1523 2.9972x*1 / x*2 1.8492 1.1477 1 2.0256 2E-23 0.5104 1.2326 2.0256 2.857umax 4 1.1 1.9049 #NUM! 0.4318 1.129 1.9049 2.7231.2 1.7896 #NUM! 0.3592 1.0309 1.7896 2.59441.3 1.6793 #NUM! 0.2925 0.9378 1.6793 2.47080 1.1477 1.4 1.5735 #NUM! 0.2314 0.8495 1.5735 2.35171.8492 1.1477 1.5 1.472 #NUM! 0.1759 0.7656 1.472 2.2371.8492 0 1.6 1.3746 #NUM! 0.1263 0.686 1.3746 2.12621.7 1.2809 #NUM! 0.0827 0.6104 1.2809 2.0191等效用线有效数据高度 40 1.8 1.1908 #NUM! 0.0458 0.5388 1.1908 1.91551.9 1.1041 #NUM! 0.0168 0.4711 1.1041 1.81532 1.0208 #NUM! #NUM! 0.4072 1.0208 1.71822.1 0.9406 #NUM! #NUM! 0.347 0.9406 1.62412.2 0.8636 #NUM! #NUM! 0.2907 0.8636 1.5332.3 0.7895 #NUM! #NUM! 0.2381 0.7895 1.44462.4 0.7184 #NUM! #NUM! 0.1894 0.7184 1.3592.5 0.6502 #NUM! #NUM! 0.1448 0.6502 1.2762.6 0.5849 #NUM! #NUM! 0.1044 0.5849 1.19552.7 0.5224 #NUM! #NUM! 0.0687 0.5224 1.11762.8 0.4627 #NUM! #NUM! 0.0382 0.4627 1.0422.9 0.4059 #NUM! #NUM! 0.0141 0.4059 0.96893 0.3519 #NUM! #NUM! #NUM! 0.3519 0.89813.1 0.3007 #NUM! #NUM! #NUM! 0.3007 0.82963.2 0.2526 #NUM! #NUM! #NUM! 0.2526 0.76343.3 0.2074 #NUM! #NUM! #NUM! 0.2074 0.69953.4 0.1654 #NUM! #NUM! #NUM! 0.1654 0.63783.5 0.1267 #NUM! #NUM! #NUM! 0.1267 0.57843.6 0.0916 #NUM! #NUM! #NUM! 0.0916 0.52133.7 0.0604 #NUM! #NUM! #NUM! 0.0604 0.46643.8 0.0336 #NUM! #NUM! #NUM! 0.0336 0.41383.9 0.0124 #NUM! #NUM! #NUM! 0.0124 0.36364 #NUM! #NUM! #NUM! #NUM! #NUM! 0.3157A B C D E F G H I J K L M N O12345678910111213141516171819202122232425262728293031323334353637383940414243446 7 8 9 in E4: =(E3D3-D4D3)(1/D3)6 7 8 95.5181 6.4946 7.4735 8.4541 in D8: =E7/D65.2234 6.1849 7.1501 8.1184 in E9: =E7/E64.9756 5.924 6.8775 7.83514.7551 5.6915 6.6342 7.5819 in D11: =D3/(D3-1)4.5534 5.4784 6.411 7.3494 in E11: =D6D11+E6D114.3659 5.2801 6.203 7.1326 in D12: =D6($D$11-1)4.1899 5.0936 6.0071 6.9283 in D13: =$E$7*D12/$E$114.0233 4.9168 5.8212 6.7342 in E14: =(D13D3+E13D3)(1/D3)3.8648 4.7483 5.6439 6.54893.7133 4.5871 5.474 6.3711 in E21: =COUNT(J3:J48)3.5681 4.4322 5.3106 6.2 YY =OFFSET(效用最大模型!$J$3,0,0,效用最大模型!$E$21,1)3.4284 4.2831 5.153 6.03493.2939 4.1391 5.0007 5.87523.164 3.9999 4.8533 5.72033.0384 3.865 4.7102 5.572.9167 3.7341 4.5713 5.42382.7988 3.607 4.4361 5.28142.6844 3.4834 4.3045 5.14272.5733 3.3631 4.1762 5.00732.4653 3.246 4.0511 4.87512.3603 3.1318 3.929 4.7462.2581 3.0204 3.8097 4.61972.1587 2.9118 3.6932 4.49612.0618 2.8057 3.5792 4.37511.9675 2.7022 3.4677 4.25661.8757 2.601 3.3585 4.14041.7862 2.5021 3.2517 4.02661.699 2.4055 3.147 3.91491.614 2.311 3.0445 3.80541.5312 2.2186 2.9441 3.69791.4505 2.1283 2.8456 3.59241.3719 2.0399 2.7491 3.48881.2954 1.9535 2.6545 3.38711.2208 1.869 2.5617 3.28731.1483 1.7864 2.4708 3.18911.0777 1.7056 2.3816 3.09271.009 1.6265 2.2941 2.9980.9422 1.5493 2.2083 2.9050.8774 1.4737 2.1241 2.81350.8144 1.3999 2.0416 2.7236)1(0rxu-=rpw21+-*r1AB2)(maxP Q R S T U V W X Y Z AA1234567891011121314151617181920212223242526272829303132333435363738394041424344AB1 . 1 5 1 . 8 5 效 用 函 数 极 大 化 模 型 图 解4321001234x 1x 2p 1 = 2 . 6p 1 = 2 . 6p 2 = 3 0)1(21(ux=+rrwxptsu+21)1( .(maxrrwxptsu+=21)1( .(maxrr1.8492 1.1477 1.8492 1.1477r / u0 0.7 0.0001 82511 1E-10 0.0001 2E-10 82511p1 / p2 2.6 3 0.1001 82.399 0.001 0.1001 0.0016 82.387w 8.2511 0.2001 41.161 0.005 0.2001 0.008 41.131最优解 # 0.3001 27.367 0.0127 0.3001 0.0205 27.317r / p1r+p2r -2.333 0.1846 0.4001 20.437 0.0248 0.4001 0.0398 20.364p1r-1 / p2r-1 0.0414 0.0257 0.5001 16.25 0.0414 0.5001 0.0664 16.154x*1 / x*2 1.8492 1.1477 0.6001 13.435 0.0629 0.6001 0.1004 13.314umax 4 0.7001 11.403 0.0892 0.7001 0.1419 11.2580.8001 9.8611 0.1204 0.8001 0.1907 9.69290.9001 8.6458 0.1563 0.9001 0.2463 8.45531.0001 7.66 0.1968 1.0001 0.3083 7.44881.1001 6.8418 0.2415 1.1001 0.376 6.61171.2001 6.1502 0.2901 1.2001 0.4487 5.90331.3001 5.5568 0.3422 1.3001 0.5255 5.29561.4001 5.0415 0.3975 1.4001 0.6058 4.76831.5001 4.5896 0.4554 1.5001 0.6886 4.30681.6001 4.19 0.5156 1.6001 0.7733 3.90011.7001 3.8343 0.5775 1.7001 0.859 3.53971.8001 3.516 0.6407 1.8001 0.945 3.21881.9001 3.2298 0.7047 1.9001 1.0308 2.9322.0001 2.9715 0.7693 2.0001 1.1158 2.67492.1001 2.7377 0.8339 2.1001 1.1994 2.4442.2001 2.5255 0.8983 2.2001 1.2814 2.2362.3001 2.3324 0.9621 2.3001 1.3614 2.04842.4001 2.1565 1.0251 2.4001 1.4391 1.87892.5001 1.9959 1.0871 2.5001 1.5143 1.72552.6001 1.8491 1.1478 2.6001 1.5868 1.58662.7001 1.7147 1.2071 2.7001 1.6566 1.46062.8001 1.5916 1.2648 2.8001 1.7237 1.34622.9001 1.4787 1.3209 2.9001 1.7879 1.24223.0001 1.3751 1.3752 3.0001 1.8493 1.14763.1001 1.2798 1.4278 3.1001 1.9079 1.06143.2001 1.1923 1.4786 3.2001 1.9638 0.98283.3001 1.1116 1.5275 3.3001 2.0171 0.91113.4001 1.0374 1.5746 3.4001 2.0678 0.84553.5001 0.9689 1.6199 3.5001 2.116 0.78553.6001 0.9058 1.6634 3.6001 2.1619 0.73063.7001 0.8475 1.7051 3.7001 2.2054 0.68033.8001 0.7936 1.7452 3.8001 2.2467 0.63413.9001 0.7437 1.7835 3.9001 2.286 0.59174.0001 0.6976 1.8202 4.0001 2.3233 0.55264.1001 0.6549 1.8553 4.1001 2.3586 0.51674.2001 0.6153 1.8889 4.2001 2.3922 0.48364.3001 0.5786 1.9211 4.3001 2.4241 0.45314.4001 0.5445 1.9518 4.4001 2.4544 0.42494.5001 0.5128 1.9812 4.5001 2.4831 0.39894.6001 0.4833 2.0093 4.6001 2.5104 0.37484.7001 0.4559 2.0361 4.7001 2.5364 0.35254.8001 0.4303 2.0618 4.8001 2.561 0.33184.9001 0.4065 2.0863 4.9001 2.5844 0.31265.0001 0.3843 2.1098 5.0001 2.6067 0.2947rpwx21+=-*r21*等支出线 1.281 -0.33 0 4 in D3:p1 / p2 2.6 3 1E-04 3.997 8.251 2.75 -0.72 in D7:e 4.2 0.1 3.575 1 0.333 -3.13x1 / x2 2 -0.33 0.2 3.317 3 1 -2.47 in E7:固定效用函数约束曲线 0.3 3.101 5 1.667 -1.8r / v 0.7 4 0.4 2.91 7 2.333 -1.13 in E5:x1 / x2 1.7 1.281 0.5 2.737 9 3 -0.47 in E8:最优解 0.6 2.576 11 3.667 0.2r / (p2/p1)r -2.33 0.716 0.7 2.426 13 4.333 0.867 in D10:x10 / x20 1.849 1.148 0.8 2.285 15 5 1.533 in E10:emin 8.251 0.9 2.152 17 5.667 2.2 in D11:1 2.025 19 6.333 2.8671.1 1.905 21 7 3.533 in E12:0 1.148 1.2 1.789 23 7.667 4.2 in E13:1.849 1.148 1.3 1.679 25 8.333 4.867 in D14:1.849 0 1.4 1.573 27 9 5.533 in E14:1.5 1.472 29 9.667 6.240 1.6 1.374 31 10.33 6.8673.9 0.012 1.7 1.281 33 11 7.5331.8 1.191 35 11.67 8.2， 1.9 1.1042 1.0212.1 0.9412.2 0.8642.3 0.7892.4 0.7182.5 0.652.6 0.5852.7 0.5222.8 0.4632.9 0.4063 0.3523.1 0.3013.2 0.2533.3 0.2073.4 0.1653.5 0.1273.6 0.0923.7 0.063.8 0.0343.9 0.0124 #NUM!A B C D E F G H I J K L M N O P Q R S1234567891011121314151617181920212223242526272829303132333435363738394041424344=效用最大问题计算模型!D6 in E3: =效用最大问题计算模型!E6=效用最大问题计算模型!D3=效用最大问题计算模型!E14=(E4-D3*D5)/E3=(E7D7-D8D7)(1/D7)=D7/(D7-1)=(E3/D3)D10=(1+E10)(-1/D7)*E7 in E11: =(1+1/E10)(-1/D7)*E7=D3*D11+E3*E11=(E3/D3)(1/(D7-1)*D11=1/D7*(1-D7)/D7)(D7-1)=D14*D3D7*E3(1-D7)*E7212)(pxe-=uxtspe+=)1(2 .minr)(rr120)(-r120)(-T U V W X Y Z1234567891011121314151617181920212223242526272829303132333435363738394041424344212)(pxe-=uxtspe+=)1(2 .minr)(rr120)(-r120)(-AA AB AC AD1234567891011121314151617181920212223242526272829303132333435363738394041424344B1 . 1 5 最 优 解1 . 8 5 A支 出 函 数 极 小 化 模 型 图 解4321001234x 1x 2p 1 = 2 . 6p 2 = 32 . 6p 1 =两 种 商 品 需 求 量 随 商 品 1 价 格 的 变 化5432100123456商 品 1 价 格商 品 1 需 求 量商 品 2 需 求 量3p 2 =两 种 商 品 需 求 量 随 商 品 2价 格 的 变 化5432100123456商 品 2价 格商 品 1 需 求 量商 品 2 需 求 量p 1 = 2 . 6p 2 = 3p 1 = 2 . 6p 2 = 33p 2 =两 种 商 品 需 求 量 随 商 品 2价 格 的 变 化5432100123456商 品 2价 格商 品 1 需 求 量商 品 2 需 求 量等支出线 0.267 0 4 1.332p1 / p2 2.6 3 9.174 3.058 -0.41 1E-044E+06e 6 1 0.333 -3.13 0.1 119x1 / x2 2 0.267 3 1 -2.47 0.2 42.1效用函数约束线 5 1.667 -1.8 0.3 22.92a / u 0.6 1.7 7 2.333 -1.13 0.4 14.89x1 / x2 2 1.332 9 3 -0.47 0.5 10.65最优解 11 3.667 0.2 0.6 8.106(1-a)p1/ap2 0.578 13 4.333 0.867 0.7 6.433x1 / x2 2.117 1.223 15 5 1.533 0.8 5.265e 9.174 17 5.667 2.2 0.9 4.41319 6.333 2.867 1 3.768A / e 1.96 9.174 21 7 3.533 1.1 3.26623 7.667 4.2 1.2 2.86625 8.333 4.867 1.3 2.5420 1.223 27 9 5.533 1.4 2.2742.117 1.223 29 9.667 6.2 1.5 2.0512.117 0 31 10.33 6.867 1.6 1.86233 11 7.533 1.7 1.735 11.67 8.2 1.8 1.561.9 1.4392 1.3322.1 1.2382.2 1.1552.3 1.082.4 1.0132.5 0.9532.6 0.8992.7 0.8492.8 0.8042.9 0.7633 0.7253.1 0.693.2 0.6583.3 0.6293.4 0.6013.5 0.5753.6 0.5523.7 0.5293.8 0.5093.9 0.4894 0.4714.1 0.4544.2 0.4384.3 0.4234.4 0.4084.5 0.395)1(2a-=uxupxupxaa)1( ,)1(22-=-=*212)(pxe-=)(-=aAupxupxaa)1( ,)1(22-=-=*1 . 2 2 2 . 1 2 等 支 出 线 与 效 用 函 数 约 束 线4321001234x 1x 2p 1 = 2 . 6p 2 = 3uxtse+-a12 .min
