I need a solution for the following macroeconomics problem

Financial Macroeconomics Problem Set 2 Due Date: March 30th 2023 1 The Permanent Income Hypothesis Revisited Consider a consumer who maximizes a quadratic utility function. She or he can freely borrow or save at the constant risk-free rate, r. The consumer has a finite lifetime. She/he starts working as soon as she/he is born, works until age 40, retires at age 41 and die at the end of her/his 60th year. Both working life and retirement are deterministic: the consumer knows at t = 0 when she/he starts working, when she/he retires and when she/he dies (a life full of surprises as you can see). So this individual at age t maximizes [ "- xu( C,). (1) with u(c) = - x (c-c)", where c is large enough and is never reached. Assume that the wage that the consumer receives follows a random walk 1074 = We-1 + Et (2) with &, being a white noise, i.e. for all t, EI = 0 and Vare-1(@ ) = Eur (e - Bite])] = Bul [=]] = 1 The retired worker receives no salary (we also assume that she or he receives no pension when retired). In other words, we = wt-ite if ts 40, if t > 40 (3) For simplicity, assume that r = 0 and 8 = 1.(a) What is the (dynamic) budget constraint of the consumer and that must hold at each point in time ? Distinguish the two cases, i.e. when the consumer works and when she/he is retired. (b) Write down the consumer's full maximization problem. What is the ap- propriate terminal solvency constraint? (c) Derive the unique (inter-temporal) [Hint: expected] budget constraint for a consumer of age t. (d) Derive the Bellman equation associated with this maximization problem. [Hint: if expressed in terms of the state variables only, there are two different Bellman equations depending on whether the consumer works or is retired for the current period]. (e) Derive the Euler equation. (f) What is the optimal consumption function for the consumer with an arbitrary age ? Express it as the current wage it is facing wy. Distinguish be- tween working and retired worker. Extra points: do we have that consumption is smoother than wage, i.e. compare Var,(ct+1) and Var,(we+1)? (g) What is the optimal savings/financial wealth function for the consumer with an arbitrary age ? Distinguish between working and retired worker. (h) Derive the relationship between the change in consumption Ac = q - "-1 and the innovation in the labor income/wage s, (Hint: express c as a func- tion of my and c-1 as a function of wt-1). This takes the form Act = ft X et- How is ft related to age ? What is its value at t = 0 and t = 40? (i) What is the relationship between Ac, and et for the permanent income model seen in class (cf. slides 62-65) with infinite lifetime and the wage process following a random walk like in equation (2)