Search and Matching model Endogenous separation Walsh (2003)

General DSGE Modeling Replication Attempts
1

I am trying to replicate the Search and Matching model Walsh (2003) https://escholarship.org/content/qt6tg550dv/qt6tg550dv.pdf

I am struggling with the aggregate output of the wholesale sector (Equation 21):
Q_t=\mathbb{E}[a_t|a\geq\tilde{a_t}]z_t\phi_tN_t \\ = \Bigg[\int_{\tilde{a_t}}^{\infty}a_t(\frac{f(a_t)}{1-F(\tilde{a_t})})da\Bigg]z_t\phi_tN_t

Now to solve the integral analytically, I assumed a logistic distribution and integrating by parts I get:
\int_{\tilde{a_t}}^{\infty}a_tf(a_t)da=\Bigg[a_tF(a_t)\Bigg]_{\tilde{a_t}}^{\infty}-\int_{\tilde{a_t}}^{\infty}F(a_t)da

The problem that arises is that it diverges to infinity
\Bigg[a_tF(a_t)\Bigg]_{\tilde{a_t}}^{\infty}=\Bigg[\infty\frac{1}{1+\exp(\frac{-\infty-a}{b})}\Bigg]-\Bigg[\tilde{a_t}\frac{1}{1+\exp(\frac{-\tilde{a_t}-a}{b})}\Bigg] \\ = \infty*1-\Bigg[\tilde{a_t}\frac{1}{1+\exp(\frac{-\tilde{a_t}-a}{b})}\Bigg]

Now I do not know how to proceed, as I cannot write the Model in Dynare because of \infty.

I would really apreciate any help.
Best

2

Usually, you simply work with the respective integrals in Dynare. After all, the paper assumes a lognormal distribution. See e.g.

3

Thank you for the answer. What do you acctually mean with that? Could you provice me with an example?

4

Have a look at the replication for Carlstrom/Fuerst (1997) at Ambrogio Cesa-Bianchi - Replications

5

Thank you for the anwer. Unfortunately I really cannot manage to get to the right point. I do not understand how I should procede for giving in dynare the following expression:

\int_{\tilde{a_t}}^{\infty}a_t(\frac{f(a_t)}{1-F(\tilde{a_t})})da

6

Usually, with the stated distributional assumption, you can replace the integral with something that can be entered into the computer. An example is Section 7 “7 Adding BGG-type financial frictions as in Christiano, Motto, Rostagno” in https://www.sas.upenn.edu/~schorf/papers/hb%20forecasting%20appendix.pdf