Question

Cursue a consumer with preferences described by (x1, x2) = x1 + x2 Suppose she faces prices pi 1 and P2 = 1/2 and that she has an income of I = 2. For your reference, the marginal utilities at a bundle (x1, x2) in this setting are given by MU (x1, x2) = 1 MU?(x), x2) = 2V x2 3(a) Write down the two equations which characterize the consumer's utility-maximizing bundle (X1.3) in this situation. In other words, write both the optimality condition and the constraint that must be satisfied at this consumer's optimal choices of X, and X2.) Show your work and/or explain 3(b) Using the two equations from part (a), find the bundle (,x) that maximizes this consumer's utility subject to her budget constraint. Show your work.

Answer 1

=) u (91%) = 2, + JA P= 1 and P2 = 1/2 Income knez (D= 2. o budget constraint - P,:d, + Pe. Az I2.. i 1.8, + 1.22 E 2 . til. 2, + de s2 . 2 | Hence, Marcinige U (4,%) = 4, +502 . Subject to a +22 22 . set up a Lagrangean (4) L: 411,20) + (21+ 22 - 2). 4: 2,+V2, + ?(91+ 2y. -2).

First order Equations 22. 27, - да, 1+ - 2L , 2 x м, + 2 + + ЭХя, ая, 222, дя, 2 x 9а, 9. - 1+O+x+0 - 4+ . for of timgation 2 = 0 . |+5 го — XL - 92, + 27, 237, + 2X3, , 21, 21.. дя, - D+ , +2+ -о ааваче е 1, +5) a for optimigation to y = 0 Nom 02, , 7,

36). The tuvo optimality equations are itd: 0 - © . o Uking these 2 Equations . | From 6. HA=0 in te-1 Putting t= -1 in Equation G . & Clothint (-1)) = 0 in Lol OR Vay - 1

lothing , * = 1 in Budget covilaint Hit me knome from Budget lenitusaint :- %+ N = 2 . .. + 1 - 2 ** = 3/2 OR 1979=105 3(6): How the utility mariniging bundle is (1-5, 1)

Summary,

3(a). The two First order condition equations are :

( i )   
1+X=0
  

( ii )
1/2(1/VT, + 1) = 0

3(b). Using these two equations we get the optimal bundle for utility maximization as

(.1, r3) = (1.5, 1)

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