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


(Consumption under Certainty) Consider aunit mass of identical consumers with the instantaneous utility function u(Ct ). 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=


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.

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