# Infinite sum recursive form

Hello,

I am trying to rewrite the following equation recursively, but I am not sure if I am doing everything right:

The first result I arrived at is the following (leaving out the expectation operator):

I am not entirely sure that this result is correct: What concerns me most is the variable w_t^*

I would really apreciate your help. Thank you

ps. This is equation of (2) from this work

I am getting

\begin{align} A_t^J\left( {w_t^*} \right) =& \sum\limits_{j = 0}^\infty {{{\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}^j}{E_t}\left[ {\left( {{d_{t + j}} - w_t^*} \right) + \beta \left( {1 - s} \right)A_{t + j + 1}^J\left( {w_{t + j + 1}^*} \right)} \right]} \hfill \\ =& \left( {{d_t} - w_t^*} \right) + \beta \left( {1 - s} \right)E_tA_{t + 1}^J\left( {w_{t + 1}^*} \right) \hfill \\ &+ \sum\limits_{j = 1}^\infty {{{\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}^j}{E_t}\left[ {\left( {{d_{t + j}} - w_t^*} \right) + \beta \left( {1 - s} \right)A_{t + j + 1}^J\left( {w_{t + j + 1}^*} \right)} \right]} \hfill \\ =& \left( {{d_t} - w_t^*} \right) + \beta \left( {1 - s} \right)E_tA_{t + 1}^J\left( {w_{t + 1}^*} \right) \hfill \\ &+ \left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]\sum\limits_{j = 0}^\infty {{{\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}^j}{E_t}\left[ {\left( {{d_{t + 1 + j}} - w_t^*} \right) + \beta \left( {1 - s} \right)A_{t + 1 + j + 1}^J\left( {w_{t + 1 + j + 1}^*} \right)} \right]} \hfill \\ =& \left( {{d_t} - w_t^*} \right) + \beta \left( {1 - s} \right)E_tA_{t + 1}^J\left( {w_{t + 1}^*} \right) + \left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]\sum\limits_{j = 0}^\infty {{{\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}^j}{E_t}\left[ {\left( {w_{t + 1}^* - w_t^*} \right)} \right]} \hfill \\ &+ \left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]\sum\limits_{j = 0}^\infty {{{\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}^j}{E_t}\left[ {\left( {{d_{t + 1 + j}} - w_{t + 1}^*} \right) + \beta \left( {1 - s} \right)A_{t + 1 + j + 1}^J\left( {w_{t + 1 + j + 1}^*} \right)} \right]} \hfill \\ =& \left( {{d_t} - w_t^*} \right) \\ &+ \beta \left( {1 - s} \right)A_{t + 1}^J\left( {w_{t + 1}^*} \right) + \left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]\sum\limits_{j = 0}^\infty {{{\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}^j}{E_t}\left[ {\left( {w_{t + 1}^* - w_t^*} \right)} \right]} \hfill \\ &+ \left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]{E_t}A_{t + 1}^J\left( {w_{t + 1}^*} \right) \hfill \\ =& \left( {{d_t} - w_t^*} \right) + \beta \left( {1 - s} \right)E_tA_{t + 1}^J\left( {w_{t + 1}^*} \right) \hfill \\ &+ {E_t}\left[ {\left( {w_{t + 1}^* - w_t^*} \right)} \right]\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]\sum\limits_{j = 0}^\infty {{{\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}^j}} \hfill \\ &+ \left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]{E_t}A_{t + 1}^J\left( {w_{t + 1}^*} \right) \hfill \\ =& \left( {{d_t} - w_t^*} \right) + \beta \left( {1 - s} \right)E_tA_{t + 1}^J\left( {w_{t + 1}^*} \right) \hfill \\ &+ {E_t}\left[ {\left( {w_{t + 1}^* - w_t^*} \right)} \right]\frac{{\left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}}{{1 - \left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]}} \hfill \\ &+ \left[ {\beta \left( {1 - s} \right)\left( {1 - \gamma } \right)} \right]{E_t}A_{t + 1}^J\left( {w_{t + 1}^*} \right) \hfill \\ \end{align}