5. (15 points) (Cobb-Douglas Utility) A consumer has the utility function of u($1, X2) = X1 X2 /4 3/4 The price of each good is denoted by p1 and p2, and the consumer's income is denoted by m. (a) Mathematically express the consumer's constrained utility maximization problem. and write down the corresponding Lagrangian function. (b) Derive the demand function for each good by using the Lagrangian optimization method. (c) Based on (b), calculate the price and income elasticities of good 2 (i.e., find Or2 P1 0x2 p2 ap1 x2 ' Op2 12' and ox2 m (d) Suppose now there is an ad-valorem tax t2 on good 2. Answer (a) and (b) again. (e) Based on (d), what is the optimal consumption bundle when p1 = 2, p2 = 2, t2 = 0.1 and m = 100