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Problem 3 Consider the following two-period model with production: max U = Inct + Blnct+1 Ct , Ct+1, bt+1, kt+1 subject to ct + bt+1 + kt+1 =yt Ct+1 = yt+1+ (1+r) be+1 + (1 -6) kt+1 yt+1 = 2 . kt+1 where bt+1 denotes bonds, kt+1 denotes physical capital invested in the household's firm and d E [0, 1] stands for capital depreciation rate. (a). Write down the problem in the form of a Lagrangian. (b). Find the optimal value of kt+1. (c). Find the optimal values of ct, Ct+1 and bt+1. (d). Calculate the derivative of optimal kt+1 with respect to r. Provide intuition for this result. (e). In equilibrium the rental rate for capital rk is equal to the marginal product of capital Oyt+1/Okt+1. Show that rk - 8 = r