please help me with solve the hw3

1 A New Keynesian Model Instead of considering sticky prices in a New Keynesian model as we did in class, we consider sticky wages in this problem. A representatitve household has a continuum of members indexed by 16 [0,1]. Each member con- sumes C; and supplies a dierentiated type of labor L,,. The representative household maximizes its life-time utility 00 10 1 l+tp C L- . t t _ it (11 may [10 V/o1+90 ] where 3 is the discount factor, 0' governs the risk aversion, 3! measures the utility weight for leisure, and cp is the inverse of the elasticity of hours worked. The budget constraint is l 1 0t++f oitdi = [witdnna D 0 BtIRtl Pt I Pt where H t is the prots received from rms the household owns. Each household member sets his / her real wage We, but subject to an adjustment cost , 2 $2,, = 9( W't 1)Y;. E Wit1 There is a competitive labor union aggregating the differentiated types of labor Lit, according to 1 51 Eil Lt: (f LitE di) , 0 where E is the elasticity of substitution among labor types. The labor union rent the labor L, to rms at the real wage Wt. The labor union takes the wage W\" and W, as given and determines the optimal demand for Lg, by maximizing its prots 1 ma / witch 0 1) Derive the demand function for type-i labor. 2) Derive the rstorder conditions for C, and 3,. 3) Derive the optimal wage setting W\" for typei labor. ( ( ( ( 4) Loglinearize the equation obtained in (3)