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1. Consider the following (real valued) sequence (x, ).2,, defined by: xo = 1, x1 = 1+a, x=ltata , X, = 1+ at a'+... + a" and so on. Prove that if (a| q'( cits - Wits) 8=0 Use this definition to show that the problem above has a recursive representa- tion. No formal proof is necessary, but explain the logic underlying Theorem 1 in this context. What interest rate do the "bonds" a, pay? (c) In a deterministic economy, such as the one above, it appears as if a one period bond is sufficient to allocate resources efficiently. Is this also true in a stochastic economy? Why or why not?2. (10 points) Consider the following Ramsey growth model augmented with a labor-leisure choice, and habits. max SAW(C, C-1,4) s.t a +kit = S ( ke, l. ) ko, C-1, 60 given In class, however, we often considered problems of the type max s.t To given (a) Define It, F(., .) and I(.) such that these two problems coincide exactly. (b) What are the first order conditions to the problem in (1)-(3)? (c) What is the Bellman equation corresponding to the problem in (1)-(3)? (no proof needed)