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I think what you are looking for is a standard RBC-model. Because the main step, going from RBC to New Keynesian, is to include Calvo pricing among other things. This is covered in many textbooks like Gali's, where he first introduces the RBC model and then moves on to the New Keynesian setup.


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This simplification of the infinite sum is commonly made by a differencing approach. You can see here an example of this approach in this kind of models. Regarding with the assumptions mentioned by you, for $\hat{mc}_t$ since it's a deviation from the levels variable steady state, in the equilibrium there's no expectations and therefore $mc_t=mc\implies \hat{...


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