1. (a) (5 marks) Write down the budget constraint for an individual when old. (b) (8 marks) Write down the constrained maximisation problem for an individual born in period t, when choosing the optimal consumption path over their lifetime (Cyt, Cmt+1, Cot+2), their optimal housing level when middle-aged { Hmt+1}, their optimal borrowing when young { Byt } and their optimal lending when middle-aged { Bmt+1} . (c) (8 marks) Write down the FOC of your constrained optimisation problem in (b) and solve for an Euler equation that describes the optimal demand for housing. (Hint: Use the Lagrangian method and use one lagrange multiplier for each con- straint in your problem. ) (d) (5 marks) Provide the intuition underlying the Euler equation that describes the optimal demand for housing. (e) (8 marks) Use your FOC from (c) to derive an Euler equation relating consumption when young to consumption when middle-aged and a separate Euler equation relating consumption when middle-aged to consumption when old.Consider a overlapping generations model Where individuals live for three periods of life: 'young', 'middle-aged' and 'old'. Assume there are N individuals in each generation over time (is. no population growth) and that individuals when young have the following lifetime utility function: Ur : log Cyt Jr 13 [log Cmt+1 + V (Hmt+1 * Hmin)] + .52 log Cot+2 where Cy: is consumption when young, Cm\" is consumption when middle-aged and Cut\" is consumption when old, 3 is the rate of time preference, HmHJ is the quantity of housing consumed when middle aged, Hm,\" is the minimal level of housing services that must be consumed, and V is a concave function governing the utility from housing with the properties that V' (H) > 0,)?\" (H)