a representative consumer with preferences over goods x1 and x2 given by U(x1,x2

Question

a representative consumer with preferences over goods x1 and x2 given by U(x1,x2)=1/2 log X1+1/2 Log X2. assume the consumer has an exogenous income of M=100. suppose initial prices are p1=3 and p2=5 calculate the optimal consumption of both goods

Explanation / Answer

as Pi =3 and P2 =5 => 3x1+5x2 = 100 so max utility with constraint 3x1+5x2 =100 => using lagrangers eqn 1/2 log X1+1/2 Log X2 -k( 3x1+5x2 -100) differating wrt x1 and x2 we get 1/2x1 -3k =0 and 1/2x2 -5k =0 => 1/2X1 = 3 and 1/2X2 =5 2X1/2X2 = 5/3 => X1/X2 =5/3 as 3X1+5X2 =100 and X1 =5/3 X2 so solving these 2 eqns 3(5/3 X2) +5X2 =100 => X2 =10 so X1 = 5/3 *10 = 50/3 = 16.666 =17 goods (apppx) X2 =10 goods

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