Intertemporal choice problem with internal and external habit

Intertemporal choice problem with internal and external habit. It is standard practice to introduce habit information into the household's problem of macroeconomic models. Consider the following household's problem 1-0 1 Max Eo 1 Ct - not-1 t=0 S.t. Citi=rkitwk and KHI=if (1-6)k. where all the variables and parameters have the usual interpretation aside from the variable S, , which denotes habit information that can be either internal habit or external habit(i.e., catching- up-with-the-Joneses utility function). a) Derive the Euler equation and the optimal labor supply when the household has internal habit (i.e., St-1=Ct-1). b) Repeat (a) above when the household has external habit (i.e., St-1=Ct-1). c) Briefly explain, using the optimality condition in (a) or (b), whether the wealth effect exists. If you think there is the wealth effect, explain how you can eliminate it. d) Linearize the optimality conditions in (a) and (b)