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## Assignment Description

Assignment 6

Solving an RBC model using Discrete State-Space

Consider a neoclassical growth model with aggregate shocks. Use the following functional assumptions: Per-period utility is represented by u(c) = (c1−σ −1)/(1−σ) and production function is zF(k,1) = zkα. The representative agent discounts future at rate 0 < β < 1 and capital depreciates at rate 0 < δ < 1. The aggregate shock z is estimated to follow the following AR(1) process

where

For the entire assignment, you will use a discretized version of this process.

Use the following parameter values:,

(Stopping rule).

(a)     Use the Euler equation from a version of this model without uncertainty (z = 1 for all periods) to find the steady-state capital stock kss numerically. Use n = 1000 equidistant points between 0.85kss and 1.15kss for the discretized state-space K = {k1,...,kn}.

(b)     Use Tauchen’s method to discretize the continuous AR(1) process into m = 17 discrete states Z = {z1,...,zm}. Use 3 standard deviations on each side of the unconditional mean of logz for the interval.

(Do not forget to convert logz grids into z grids at the end.)

(c)     Use

for each i,j

as the initial guess for the value function, where f(z,k) = zF(k,1) + (1 − δ)k.

(d)     Solve the model using any of the discrete state method you learned so far. It is recommended that you use Howard’s Improvement algorithm. You can use very high Nh (e.g. Nh=100) to speed up the computation. (Warning: This will take a long time if you do not use any acceleration methods.)

(e)     On the same graph, plot the policy function for each value of z separately. (g(k,z1),g(k,z2),...,g(k,zm) on the same graph). Similarly, on a separate graph, plot the value functions for every value of z.

(f)      Starting from initial value z = z9, using the Markov matrix you constructed in part b, simulate timeseries for z. Use 10000 time periods. Using this time-series for z, along with initial capital stock equal to the steady-state level you found in part a (find the grid point that is closest to steady-state capital), simulate time-series for capital stock, investment, consumption and output for 10000 time periods. Graph the last 1000 periods of each time-series in log scale (logkt, logct, logIt) on separate plots.

(g)     Calculate the standard deviation (volatility) of each variable and correlation of each variable with output (all in log scale). Report all the results.

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