need to solve the exercise 175 from the recursive macroeconomic theory by lars ljungqvistexercis

a. Given the hint, we can start by assuming that the optimal policy has bt+1 = 0 for all t ≥ 0.

Question

Exercise 17.5 Trade A consumer orders consumption streams according to $sum_{t=0}^{infty} eta^t E_0 frac{c_t^{1-gamma}}{1-gamma}$ where $gamma > 1$ and $E_0$ is the mathematical expectation conditional on time 0 information. The consumer can borrow or lend a one-period risk-free security that bears a fixed rate of return of $R = eta^{-1}$. The consumer's budget constraint at time $t$ is $c_t + R^{-1}b_{t+1} = y_t + b_t$ where $b_t$ is the level of the asset that the consumer brings into period $t$. The household is subject to a "natural" borrowing limit. The household's initial asset level is $b_0 = 0$ and his endowment sequence $y_t$ follows the process $y_{t+1} = y_t exp(sigma_epsilon epsilon_{t+1} + mu)$ where $epsilon_{t+1}$ is an i.i.d. Gaussian process with mean zero and variance 1, $mu = .5gammasigma_epsilon^2$, and $sigma_epsilon > 0$. The consumer chooses a process ${c_t, b_{t+1}}_{t=0}^infty$ to maximize the utility function subject to the budget constraint and the natural borrowing limit. a. Give a closed-form expression for the consumer's optimal consumption and asset accumulation plan. Hint 1: If $log x$ is $N(mu, sigma^2)$, then $Ex = exp(mu + sigma^2/2)$. Hint 2: You could start by trying to verify the following guess: the optimal policy has $b_{t+1} = 0$ for all $t geq 0$. b. Discuss the solution that you obtained in part a in terms of Friedman's permanent income hypothesis. c. Does the household engage in precautionary savings?

          Exercise 17.5 Trade

A consumer orders consumption streams according to $sum_{t=0}^{infty} eta^t E_0 frac{c_t^{1-gamma}}{1-gamma}$ where $gamma > 1$ and $E_0$ is the mathematical expectation conditional on time 0 information. The consumer can borrow or lend a one-period risk-free security that bears a fixed rate of return of $R = eta^{-1}$. The consumer's budget constraint at time $t$ is $c_t + R^{-1}b_{t+1} = y_t + b_t$ where $b_t$ is the level of the asset that the consumer brings into period $t$. The household is subject to a "natural" borrowing limit. The household's initial asset level is $b_0 = 0$ and his endowment sequence $y_t$ follows the process $y_{t+1} = y_t exp(sigma_epsilon epsilon_{t+1} + mu)$ where $epsilon_{t+1}$ is an i.i.d. Gaussian process with mean zero and variance 1, $mu = .5gammasigma_epsilon^2$, and $sigma_epsilon > 0$. The consumer chooses a process ${c_t, b_{t+1}}_{t=0}^infty$ to maximize the utility function subject to the budget constraint and the natural borrowing limit.

a. Give a closed-form expression for the consumer's optimal consumption and asset accumulation plan.
Hint 1: If $log x$ is $N(mu, sigma^2)$, then $Ex = exp(mu + sigma^2/2)$.
Hint 2: You could start by trying to verify the following guess: the optimal policy has $b_{t+1} = 0$ for all $t geq 0$.

b. Discuss the solution that you obtained in part a in terms of Friedman's permanent income hypothesis.

c. Does the household engage in precautionary savings?

Added by Michelle H.

Question

Please give Ace some feedback