Question

Derive the Marshallian demand functions for Goods X, and X, by maximizing following utility-maximizing problem. What restrictions does a Cobb-Douglas lity function (preferences) impose on demand functions? Explain your answer. marks) 1/4 Maximize u = x;"/4x2 4x, + 2x, = 100 Subject to - Use the information in above to derive the consumer's indirect utility anction (value function) and then prove Roy's identity (10 marks)

Answer 1

2, 4 14 L = U (2,,H2) = 2, 3/4 x 44 subject to : 401+2x₂=100 Oh, 2x1+2=50 Setting up the langsangian function 2 3/4 x 14+ also-2x, H2). af - 22,4xY4+ (-2) = 0 )Y-22 on, a = 2/3 (2) 4 - 0 ak 42,742,3/4 + ( -1) = (3) - © 222 22 24 = 50 - 281-22 =0 241 +22=50 - ③ Equaling ① and ② osito (%) & ) = 03; = § 1 34 =20 -

putting this value (4) in equation 3 2x1 + x = 50; 3x2 = 24 - 422=50 241 +2 x1=50 T22 = 12.5 8x1=50 /x = 18.75/ u = (16-75)34(12.5344 (9.01854) (1.8603) [u= 16.94251 andirect- citility. . , 3)4 (12-5)/4 -

314 x 14 Assume, ulx, 22) = x, 314x 'Pai + P₂ 2 = M. MRS= ( 2 ) 2 1/4 x 14 (+1 2 2 3/4 2,314 MRS = 32 at Cequilibrium : MRS 372 322 42 = Pa are 7 Budget constraint: 3 P2 Z2 + P222=M 22 = M 4.12 Pixit M = m Pi*} = 3M/4 (2) 4P1) (4. M uta,Ma) = MC4 (7)

ū=U* (3) 14 4 и ( 4 ) 2 м (17ssi) & (1)" ( 4 м/ а 279 ӘfИ t, ән /ər () (E" (3) 3) әй әм гом (А) / 11, а* - (?)" (2) | t = 3 1 1 (Roy's identity)