Please Solve

4. Consider a household in three-period version of the dynamic consumption-sarong model with the preferences: [61,(32., c3) 2 loge; + logq + ,82 log (:3 Where ,8 is the household's subjective discount factor. In each of the three periods t = 1,2, 3, they enter with a predetermined amount of real wealth n_1, earn interest income rink; and exogenous real labor income yt. They also choose the ow of consumption q and the stock of wealth a: to carryr to the next period. You may.r assume that the interest rate is constant so that r = r1 = r2 = r3. {:1} Write down the real period-1. period-2 and period-3 budget constraints. {b} Using the sequential Lagrangian formulation, derive the relevant rst-order condi- tions in terms of the logarithmic utility function. (c) Using your rstorder conditions from part (b). derive two consmnptionsavings op timalit}r conditions. One optimality condition should be for periods 1 and 2., and the other for periods 2 and 3. {d} Solve for the optimal choices (c3, c5. c5] in terms of the exogenous variables. ,3. r. m, ya, and 3,3. Assume initial and terminal conditions of on = as = 0. [e] Briey explain in words how the permanent income hypothesis is reected in the consumption functions from part (d)