Consider the following utility function over goods 1 and 2, plnx1 +3lnx2: (a) [15 points] Derive...

Answer:

Consumer's objective is to maximize utility subject to expenditure function, from which we can get the marshallian demand function.

U (x_1, x_2) = (log x_1 + 3 log x_2)^1/2 Let, the budget line is mu = p_1 x_1 + p_2 x_2 a) objective is to maximize U (x_1, x_2) s. t mu = p_1 x_1 + p_2 x_2 So, Logarithm equation is alpha = (log x_1 + 3 log x_2)^1/2 + lambda (mu - p_1 x_1 - p_2 x_2) First order condition, partial differential alpha/partial differential x_1 = 1/2 (log x_1 + 3 log x_2)^- 1/2 1/x_1 - lambda p_1 = 0 (I) partial differential alpha/partial differential x_2 = 1/2 (log x_1 + 3 log x_2)^- 1/2 3/x_2 - lambda p_2 = 0 (II) partial differential alpha/partial differential lambda = mu - p_1 x_1 - p_2 x_2 = 0 (III) from (I) and (II) 1/2 (log x_1 + 3 log x_2)^- 1/2 1/x_1/1/2 (log x_1 + 3 log x_2)^- 1/2 3/x^2 = lambda p_1/lambda p_2 rightarrow x_2/3 x_1 = p_1/p_2 rightarrow x_2 = p_1/p_2 3 x_1, put this value in (III) p_1 x_1 + p_2 p_1/p_2 3 x_1 = mu