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Solving for a difference equation for $s_{t}$
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By applying the above difference equation on $ s_{t}=(1-\beta) E_{t} \sum_{j=0}^{\infty} \beta^{j} f_{t+j}$, we obtain $$s_{t}=(1-\beta) E_{t} \sum_{j=0}^{\infty} \beta^{j} \left [ \bar{P} - u_{t+j} ...

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