1. Hailey lives in city A and consumes two goods, 1 and 2, with the following utility function, 1 3 1 a($1,1'2,h-)= Mfr; 5.512 where A > U is a constant and h is the number of hours to work. The goods prices in city A are pf and p3, and the wage rate mg. (a) [20 points] Suppose h is xed. Use the Lagrangean method to derive the Marshallian demand functions, taking h as given. (b) [10 points] Now suppose Hailey can choose h. Solve for the optimal h. Then derive the Marshallian demand functions and the indirect utility function. (c) [10 points] Show that the Marshallian functions are homogeneous of degree zero in prices and wage rate. That is, $5 [tpil'tpg7tw) = 'Ti (Pillawai: 1": 1: 2 for anyt) 0. (d) [10 points] Now a new opportunity shows up in city 13 and Hailey considers moving from A to B. Suppose Hailey's utility of living in city 13 is given by 1 2 _h'\" Hip-I Huh-I ab ($1,332,:l1) = Ar 5.\" where ha is taken as given and is the same as the optimal choice of h for city A. All goods prices and the wage rate in city 13 are the same as those in city A. Derive the Marshallian demand functions given ha, without using the Lagrangean method. (e) [10 points] Compare the maidiuized utilities in the two cities and answer: Should Hailey move to B? [Hintt This part need not involve a lot of math. Try to study your answers in the previous parts carefully and nd a simple way to answer this.]