I’m studying baseline NK model and wonder why is there the existence of a SDF, for example in Galí (2008 or 2015) since there’s no capital accumulation the price of bonds Q_{t,t+s} \equiv \beta^s\frac{c_{t+s}^{-\sigma}}{c_{t}^{-\sigma}}\frac{1}{(1+\pi_{t+s})} play the role of an inverse yield of bonds i.e. a gross nominal interest rate (correct me if I’m wrong).

Also in Sims (2010) one can find that the SDF is defined as an equivalent manner as \beta^s u'(c_{t+s})/u'(c_{t}) which is equivalent in the model to \frac{(1+\pi_{t+s})}{(1+i_t)} which by Fisher equation would be (sort of) \frac{1}{1+r_{t+s|t}}.

What I understand is that the firm is bringing future net profit flows to present value with respect to the economy’s real or nominal interest rates (depending if the set-up of the firm’s profit is in real or nominal terms), but why aren’t those rates being powered to the time of periods in the future like with \beta? i.e. present value of real profit flow would be:

profits_t+\sum_{s=1}^\infty \prod_{j=1}^s \frac{1}{1+r_{t+s+j|t}}\beta^{s} profits_{t+s|t}

Am I missing the point of the SDF? If so could you please explain me its purpose and intuition? and also why authors do not just put directly the interest rate?

Thank you.