construct the Lagrangian and derive the first order kuhm-tucker conditions for a problem

Question

(Consumption under Certainty) Consider aunit mass of identical consumers with the instantaneous utility function u(C_t ). At time t = 0, the representative consumer maximizes her expected lifetime utility,

Subject to At 0 given initial wealth A0 0, and

 ,    0

(a) Using the law of motion (*) derive a single lifetime budget constraint. Be specific about what you did for At

(b) Construct the Lagrangian for this problem, and derive the first order kuhm-tucker conditions for the interior solution. Combining them obtain the Euler equation and interpret it

(c)  Show that MRS is given by

_{t+1, t}) = 1/      0             (*)

(d) Assume that ρ= (1-β)/ β (the rate of time preference) equals r. show that

              i) (*) implies the lifetime consumption smoothing:

           ii) Show that C above is given by   C=

Summary

This question belongs to operations management, given utility and expected life time functions under certain conditions. You are required to understand the concept and answer the given questions.

Total Word Count 189