/Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 39 0 obj << Policonomics » Article > Microeconomics - A > Lagrangian Oct 28.
�N��*���n�L;�5
��3�Aw�ևS���c��%����_4A���F��.7Ł۶�jM��3^)���W? /FirstChar 33 571 285.5 314 542.4 285.5 856.5 571 513.9 571 542.4 402 405.4 399.7 571 542.4 742.3
562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6
/Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8
<<
Show all steps. How does this compare to the utility you found in part C of this question? 0000058568 00000 n
545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091
/Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 44 0 obj /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 endobj
0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5
The Lagrangian method uses a somewhat different method of modifying the objective function being maximized to account for (equality) constraints that restrict the feasible range of the objective function. 0000014986 00000 n
0000010216 00000 n
0000003901 00000 n
0000007233 00000 n
0000014256 00000 n
0000005943 00000 n
/Widths[314.8 527.8 839.5 786.1 839.5 787 314.8 419.8 419.8 524.7 787 314.8 367.3 0000002735 00000 n
0000003364 00000 n
In the optimum, this slope should be -5. 0000014054 00000 n
0000015007 00000 n
0000069876 00000 n
>> 0000060508 00000 n
How long can capitalism survive given the rising poverty and inequality in younger generations? 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 F) What is the Marginal Rate of Substitution for the utility curve at the point the curve intersects the budget line when X=10.
0000013214 00000 n
/BaseFont/EKBGWV+CMR6 /FontDescriptor 29 0 R 0000009461 00000 n
0000003011 00000 n
D) Show the Marginal Rate of Substitution and the slope of the budget line at the optimal choice. 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 9 0 obj xڵY[�ۺ~�p���W�E��4�i�����)�nQhm�VcK{$9��_߹��,y��H_,�fH��|C/�8I�}~\�p��C�Il��f�X�*V�$����w����/�;�,W���,�y�P��P Y�T�ծ,*���}
�4ڠ����X,�8?�׀.���_�,zj��H�4Ib�$�������+���p���)���~%B�-�Wqm��Q��������-��v9ކZ�^8�5�'))�REݾl��ƥ�Ma�B��a)L�^4�hW���K��=���;�fQ}�|�?���X�߆3�L���zu5���OTt�2�]/W*M�������V�r�}j;w�2M�SZT�E�����^*�\��� ��Á�h 6��2�X��]������lю&���S�:��O-n��[�G&Mg�?p�p�)��������$��Tt%�4�r6�f���m=4e��"�Ը��j�ۈ8Ƀ���Ɇ��
� A) Use the Lagrange Multiplier method to solve for the quantity of X and Y when I=$1,000, Px=$25, and Py=$5, and Utility = X*Y. Indicate x and y intercepts, as well as optimal bundle of x and y. /LastChar 196 /FirstChar 33 /LastChar 196 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 >> 600.2 600.2 507.9 569.4 1138.9 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 The method operates as follows: i. Transform all of the constraints into a form that equals zero.
/Subtype/Type1
/BaseFont/JOREEP+CMR9 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 endobj /Type/Font 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 << /BaseFont/YQHBRF+CMR7 /Name/F3 ��l���+��2X��4�J!$����w�|��-(6}�@º��:�B��txzwD'pS�e��5�u��������i�������8�,���:��7�X�8���8�� :�r6��m;��|8�����X֜������x�e��� <<
18 0 obj /LastChar 196 511.1 511.1 511.1 831.3 460 536.7 715.6 715.6 511.1 882.8 985 766.7 255.6 511.1] 41 0 obj
<<
/Linearized 1
/O 43
/H [ 1821 707 ]
/L 115136
/E 71450
/N 9
/T 114198
>>
endobj
xref
41 69
0000000016 00000 n
36 0 obj >> 0000008042 00000 n
endobj /FontDescriptor 8 0 R known as the Lagrange Multiplier method. %PDF-1.3
%����
351.8 935.2 578.7 578.7 935.2 896.3 850.9 870.4 915.7 818.5 786.1 941.7 896.3 442.6 323.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 569.4 323.4 323.4 >> /LastChar 196 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000007821 00000 n
0000061979 00000 n
/Subtype/Type1 >> 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 277.8 500]
0000014460 00000 n
0000056055 00000 n
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 << endobj All of the following are related to medium of exchange, right? 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 0000001728 00000 n
472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 826.4 295.1 531.3] C) How many utils are obtained at the optimal choice? /FontDescriptor 17 0 R /Name/F12 /FirstChar 33 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 /FirstChar 33
/BaseFont/UTOXGI+CMTI10
/FontDescriptor 26 0 R 770.7 628.1 285.5 513.9 285.5 513.9 285.5 285.5 513.9 571 456.8 571 457.2 314 513.9 351.8 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 351.8 351.8 << 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 753.7 1000 935.2 831.5
endobj /Subtype/Type1 /LastChar 196 stream A) Use the Lagrange Multiplier method to solve for the quantity of X and Y when I=$1,000, Px=$25, and Py=$5, and Utility = X*Y.
Ze�}j�U���cĭe[�. endobj endobj 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 /LastChar 196 0000069130 00000 n
388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6
0000010529 00000 n
791.7 777.8] X, Y and lambda to get: So now we know that in the optimum, Y will be 5 times larger than X, Use the expression for Y to find X by using (3), C) Simply put the optimal values for X and Y into the utility function, D) MRS = (marginal utility of X) / (marginal utility of Y), Slope of the budgetline => rewrite budget line (solve for Y), Y = 200 - 5X => slope is -5, which is the same as -MRS, Utility of 1500 is lower then the utility found at the optimum (=2000), F) From question D we know that MRS = Y/X, So when X=10 and Y=150, the MRS = 150/10 = 15, While the slope of the budget line = -5 (always), In this case, the indifference curve won't touch the budget line, it will cut right through it from above.
0000008063 00000 n
0000014439 00000 n
/LastChar 196 /Filter[/FlateDecode] 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 742.3 799.4 0 0 742.3 599.5 571 571 856.5 856.5 285.5 314 513.9 513.9 513.9 513.9 /Widths[323.4 569.4 938.5 569.4 938.5 877 323.4 446.4 446.4 569.4 877 323.4 384.9 Compare this Marginal Rate of Substitution to the slope of the budget line. 0000005293 00000 n
384.3 611.1 675.9 351.8 384.3 643.5 351.8 1000 675.9 611.1 675.9 643.5 481.5 488 0000005073 00000 n
This method involves adding an extra variable to the problem called the lagrange multiplier, or λ. Show how you obtain your answer. 0000015723 00000 n
0000044090 00000 n
endobj
500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 (6.1).4 Given any function x(t), we can produce the quantity S.We’ll just deal with one coordinate, x, for now. E) What is the utility along the budget line when X=10? /Type/Font /LastChar 196 525 768.9 627.2 896.7 743.3 766.7 678.3 766.7 729.4 562.2 715.6 743.3 743.3 998.9