4. Consider a household in three-period version of the dynamic consumption-savings model with the preferences: u(C1, (2, (3) = loge + Blog c2 + Blog ca where B 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 at-1, earn interest income red-1 and exogenous real labor income yr. They also choose the flow of consumption & and the stock of wealth a, to carry to the next period. You may assume that the interest rate is constant so that r = r = 12 = ry. (a) Write down the real period-1, period-2 and period-3 budget constraints. (b) Using the sequential Lagrangian formulation, derive the relevant first-order condi- tions in terms of the logarithmic utility function. (c) Using your first-order conditions from part (b), derive two consumption-savings op- timality 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 (cj, c;, c;) in terms of the exogenous variables, B, r, y1, 12, and y3. Assume initial and terminal conditions of ag = a3 = 0. (e) Briefly explain in words how the permanent income hypothesis is reflected in the consumption functions from part (d)