ECON $6305$: Coding Assignment #$1$

Aaron Betz

George Washington University

$1$ Write your own IRF code

Suppose you have the following solution to a model from Dolo $($Problem $1$ is just a Python exercise, you will not actually be using Dolo$).$ You will generate impulse response functions with these formulas.

$z_t=z_{t-1}+\frac{1-}{b}ar(z)+lo{n}_t$

$k_t=k_{t-1}(1-)+2\mathrm{.}16(z_{t-1}-1)+0\mathrm{.}017(k_{t-1}-9\mathrm{.}354)+9\mathrm{.}354$

$1\mathrm{.}1$ Capital and Z impulses

Where $=0\mathrm{.}9\frac{,}{b}ar(z)=1,$ and $=0\mathrm{.}025.$ You can solve forward to time $t=1$ from time $t=0($steady state$)$ when $z_0\frac{?}{b}=ar(z)=1$ and $k_0\frac{?}{b}=ar(k)=9\mathrm{.}354$ given some initial shock $lo{n}_0.$ To continue to solve forward assume $lo{n}_1=0$ and that every following $lo{n}_t$ is zero. Write a Python function to generate an impulse of length $n,$ given an initial shock, $lo{n}_0.$ Show your code and graph impulses $($for both capital and $z)$ of length $40,$ with an initial shock $lo{n}_0=0\mathrm{.}01.$

$1\mathrm{.}2$ Labor and output impulse

Now that you have a $40$ period series of all the necessary state variables, $k_t$ and $z_t$ solve for $n_t$ and then $y_t$ given the following formulas from Dolo.

$n_t=0\mathrm{.}33+0\mathrm{.}809(z_{t-1}-1)+0\mathrm{.}025(k_{t-1}-9\mathrm{.}354)$

$y_t=z_t{n}_t^{0\mathrm{.}67}{k}_{t-1}^{0\mathrm{.}33}$

Keep in mind you will need to solve for labor, $n_t,$ before you solve for output, $y_t.$ Once you have solved all $40$ periods graph both labor and output over time $($use separate graphs$).$

$1$