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2. Consider the following consumption-savings problem: ${c_{t},x_{t+1}}_{t=0}max?t=0????_{t}1??c_{t}?$ subject to $c_{t}+x_{t+1}=Rx_{t}$, with $x_{0}>0$ given. Assume that $0<?<1,?>1$ and $R=1/?$. (i) Set up the problem recursively, and find the policy functions explicitly. (ii) Parameter values are given by $?=2$ and $?=0.9$. Numerically approximate the value function and policy functions using the following procedure: a. Discretize the state space using the grid $x_{i}=0.8+(i?1)?$, where $?=0.01$ and $i$ takes on values from 1 to 41 . This defines the state variable on a grid of $n=41$ equally spaced points. b. Define the value function as an $n×1$ vector, and set your initial guess, $v_{0}$, as an $n×1$ vector of zeros. Find the new value function $v_{j+1}$ by solving $v_{j+1}(x_{i})=Tv_{j}(x_{i})$ for $i=1,…,41$. c. Iterate on $v_{j+1}$ until $x_{i}max??v_{j+1}(x_{i})?v_{j}(x_{i})?<0.001$. d. Compute the policy functions $c=h(x)$ and $x_{?}=g(x)$ using the approximate fixed point $v_{?}$. Do not hand in your code or any numerical answers. Please just submit two figures: one that plots the optimal policy for consumption, $c=h(x)$, and another that plots the optimal policy for savings, $x_{?}=g(x)$.

(i) To set up the problem recursively, we can star

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