Question

Consider the following utility function over goods 1 and 2, plnx1 +3lnx2: (a) [15 points] Derive the Marshallian demand functions and the indirect utility function. (b) [15 points] Using the indirect utility function that you obtained in part (a), derive the expenditure function from it and then derive the Hicksian demand function for good 1. (c) [10 points] Using the functions you have derived in the above, show that i. the indirect utility function is homogeneous of degree zero in prices and income; ii. the Hicksian demand function for goods 1 is homogeneous of degree zero in prices.

1. Consider the following utility function over goods 1 and 2, (a) [15 points] Derive the Marshallian demand functions and the indirect utility (b) 15 points] Using the indirect utility function that you obtained in part (a), (c) [10 points Using the functions you have derived in the above, show that function derive the expenditure function from it and then derive the Hicksian demand function for good i. the indirect utility function is homogeneous of degree zero in prices and income ii. the Hicksian demand function for goods 1 is homogeneous of degree zero in prices.

Answer 1

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