<ol>
<li>Start by establishing $ \epsilon _1 = \epsilon , \epsilon_{t&gt;1}=0$ . Now, epsilon will be your impulse.</li>
<li>You can calculate an expression for $A_t,\bar{A}_t\ \forall t$ . From here, you can either:

<ol>
<li>Work recursively with each variable and provide an analytic function for them, for every moment on time.</li>
<li>Calculate a value if you are simulating it, using the policy functions at each step.</li>
</ol></li>
</ol>

<p>Notice that, if you log-linearized your equations, $\epsilon=1$ would mean a 1% deviation from the steady state.</p>

CC BY-SA 3.0
2018-01-04T00:18:18.450Z
2018-01-04T00:18:18.450Z
one_teach_wonder
<p>I have a very basic model about a small open economy. This is a summary: </p>

<blockquote>
  <p>$$
E_0\sum_{t=0}^{\infty}\theta_t U(c_t, h_t)
$$
  utility function:
  $$
d_t = (1 + r_{t-1})d_{t-1} - y_t + c_t + i_t + \Phi(k_{t-1} - k_t)
$$
  evolution of foreign debt:
  $$
y_t = A_t F(k_t, h_t)
$$
  output:
  $$
k_{t+1} = i_t + (1 - \delta)k_t
$$
  stock of capital evolves according to:</p>
  
  <p>The first order conditions of the household's maximization problem are:
  \begin{align}
\lambda_t = \beta(c_t, h_t)(1 + r_t)E_t\lambda_{t+1}\\
\lambda_t = U_c(c_t,h_t) - \eta_t\beta_c(c_t,h_t)\\
\eta_t = -E_t U(c_{t+1}, h_{t+1}) + E_t \eta_{t+1} \beta(c_{t+1}, h_{t+1}) \\
-U_h(c_t,h_t) + \eta_t\beta_h(c_t,h_t) = \lambda_tA_tF_h(k_t,h_t)\\
\lambda_t[1 + \Phi'(k_{t+1} - k_t)] = \beta(c_t,h_t)E_t\lambda_{t+1}[A_{t+1}F_k(k_{t+1},h_{t+1}) + 1 - \delta + \Phi'(k_{t+2} - k_{t+1})]
\end{align}
  Law of motions of the productivity shock:
  $$
\ln A_{t+1} = \rho \ln A_t + \epsilon_{t+1}; ~~~~ \epsilon_{t+1} \sim { NIID}(0,\sigma_\epsilon^2); ~~~~ t\ge 0
$$
  we parametrize this as 
  \begin{align}
U(c,h) = \frac{[c - \omega^{-1}h^\omega]^{1 - \gamma} - 1}{1 - \gamma}\\
\beta(c,h) = [1 + c - \omega^{-1}h^\omega]^{-\psi_1} \\
F(k,h) = k^{\alpha}h^{1-\alpha} \\
\Phi(x) = \frac{\phi}{2}x^2; ~~~~ \phi&gt;0
\end{align}
  $$
\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline
\gamma &amp; \omega &amp;  \psi_1 &amp; \alpha &amp; \phi &amp; r &amp; \delta &amp; \rho &amp; \sigma_{\epsilon} \\\hline
2 &amp; 1.455 &amp; 0.11 &amp; 0.32 &amp; 0.028 &amp; 0.04 &amp; 0.1 &amp; 0.42 &amp; 0.0129 \\ \hline
\end{array}
$$</p>
</blockquote>

<p>I have computed the steady state, but how can I compute the impulse response function? </p>

CC BY-SA 3.0
2018-01-05T09:37:54.480Z
congratulations & thanks to @caverac on an epic edit latex-ifying a giant image
410 gone
<macroeconomics><international-economics>
Impulse response functions?
Federica Brizio
# Impulse response functions?

(Asked by: username_0 on 2018-01-01)

I have a very basic model about a small open economy. This is a summary:

> $$
> E\_0\sum\_{t=0}^{\infty}\theta\_t U(c\_t, h\_t)
> $$
> utility function:
> $$
> d\_t = (1 + r\_{t-1})d\_{t-1} - y\_t + c\_t + i\_t + \Phi(k\_{t-1} - k\_t)
> $$
> evolution of foreign debt:
> $$
> y\_t = A\_t F(k\_t, h\_t)
> $$
> output:
> $$
> k\_{t+1} = i\_t + (1 - \delta)k\_t
> $$
> stock of capital evolves according to:
>
> The first order conditions of the household's maximization problem are:
> \begin{align}
> \lambda\_t = \beta(c\_t, h\_t)(1 + r\_t)E\_t\lambda\_{t+1}\\
> \lambda\_t = U\_c(c\_t,h\_t) - \eta\_t\beta\_c(c\_t,h\_t)\\
> \eta\_t = -E\_t U(c\_{t+1}, h\_{t+1}) + E\_t \eta\_{t+1} \beta(c\_{t+1}, h\_{t+1}) \\
> -U\_h(c\_t,h\_t) + \eta\_t\beta\_h(c\_t,h\_t) = \lambda\_tA\_tF\_h(k\_t,h\_t)\\
> \lambda\_t[1 + \Phi'(k\_{t+1} - k\_t)] = \beta(c\_t,h\_t)E\_t\lambda\_{t+1}[A\_{t+1}F\_k(k\_{t+1},h\_{t+1}) + 1 - \delta + \Phi'(k\_{t+2} - k\_{t+1})]
> \end{align}
> Law of motions of the productivity shock:
> $$
> \ln A\_{t+1} = \rho \ln A\_t + \epsilon\_{t+1}; ~~~~ \epsilon\_{t+1} \sim { NIID}(0,\sigma\_\epsilon^2); ~~~~ t\ge 0
> $$
> we parametrize this as
> \begin{align}
> U(c,h) = \frac{[c - \omega^{-1}h^\omega]^{1 - \gamma} - 1}{1 - \gamma}\\
> \beta(c,h) = [1 + c - \omega^{-1}h^\omega]^{-\psi\_1} \\
> F(k,h) = k^{\alpha}h^{1-\alpha} \\
> \Phi(x) = \frac{\phi}{2}x^2; ~~~~ \phi>0
> \end{align}
> $$
> \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline
> \gamma & \omega & \psi\_1 & \alpha & \phi & r & \delta & \rho & \sigma\_{\epsilon} \\\hline
> 2 & 1.455 & 0.11 & 0.32 & 0.028 & 0.04 & 0.1 & 0.42 & 0.0129 \\ \hline
> \end{array}
> $$

I have computed the steady state, but how can I compute the impulse response function?

**Tags:** macroeconomics, international-economics
> **Comments:**
> **username_1**: congratulations & thanks to @caverac on an epic edit latex-ifying a giant image  

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## 1 answers

### Answer by username_2 on 2018-01-04. Score: 1

1. Start by establishing $ \epsilon \_1 = \epsilon , \epsilon\_{t>1}=0$ . Now, epsilon will be your impulse.
2. You can calculate an expression for $A\_t,\bar{A}\_t\ \forall t$ . From here, you can either:
   1. Work recursively with each variable and provide an analytic function for them, for every moment on time.
   2. Calculate a value if you are simulating it, using the policy functions at each step.

Notice that, if you log-linearized your equations, $\epsilon=1$ would mean a 1% deviation from the steady state.


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