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4. Consider a household in three-period version of the dynamic consumption-savings model with the preferences: u(c1, C2, C3) = log ci + Blog C2 + B2 log c3 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 rtat-1 and exogenous real labor income yt. They also choose the flow of consumption Ct and the stock of wealth at to carry to the next period. You may assume that the interest rate is constant so that r=r1 = r2 = 13. (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 (C1,c, c) in terms of the exogenous variables, B, r, y1, 42, and y3. Assume initial and terminal conditions of ao = a3 = 0. (e) Briefly explain in words how the permanent income hypothesis is reflected in the consumption functions from part (d). 4. Consider a household in three-period version of the dynamic consumption-savings model with the preferences: u(c1, C2, C3) = log ci + Blog C2 + B2 log c3 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 rtat-1 and exogenous real labor income yt. They also choose the flow of consumption Ct and the stock of wealth at to carry to the next period. You may assume that the interest rate is constant so that r=r1 = r2 = 13. (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 (C1,c, c) in terms of the exogenous variables, B, r, y1, 42, and y3. Assume initial and terminal conditions of ao = a3 = 0. (e) Briefly explain in words how the permanent income hypothesis is reflected in the consumption functions from part (d)

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