Finite and infinite sums

Dear All,

I am currently working on a New Keynesian model with Dynare. One equation has a particular form. The variable w is forward looking and notably depends on the expected future values for pi in the following way (here I omit arguments that have no impact on my topic):

w_t = E_t [Beta(pi_t+1) + Beta^2(pi_t+1 + pi_t+2) + Beta^3 (pi_t+1 + pi_t+2 + pi_t+3) + Beta^4 (pi_t+1 + pi_t+2 + pi_t+3 + pi_t+4) … up to infinity]

I would be grateful if you could tell me how to write it with Dynare. Especially the way to write the sequence for pi.


See [New Area-Wide Model (NAWM) wage setting) or [How to write a infinite sum FOC into dynare)

Thank you very much for this answer.

In fact, I have already tried to write this equation in a recursive way, as it is suggested in the user guide. However, I was not able to find such a representation, since what happens with pi is not exactly an infinite sum. This is rather an infinity of finite sums (pi_t+1 + pi_t+1 + pi_t+2 + pi_t+1 + pi_t+2 + pi_3 + …).

Note also that the model is already written in a linear form, since pi is the inflation rate.

I think writing the pi in a recursive way would be the solution. But I do not see how to this with this kind of sum.

Thanks by advance for your help.