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I'm trying to learn about new-keynesian models hw they were derived. However, I found great difficulty in deriving the equations used in many studies and the lack of books and papers that mention their origin. The following equation (see link slide 7/26, https://slideplayer.com/slide/10639387/) represents the aggregate demand block of the canonical gap model (reduced-form new-keynesian model) which is a s follows,

$$\hat{y}_t = a_1 \hat{y}_{t-1} - a_2 mci_t + a_3 \hat{y}_{t+1} + \epsilon^y_t,$$ $$mci_t = a_4 \hat z_t + (1 - a_4) \hat r_t.$$

Where $\hat y_t$ is the output gap, $mci_t$ represents the monetary conditions index, $\hat r_t$ refers to the real interest rate gap, $\hat z_t$ denotes the real exchange rate gap. Can anyone please show how such equation was derived or direct me to its source. Thanks in advance.

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I would recommend first to check Sims' class notes in Graduate Macro Theory. I think it is both very gentle and straightforward introduction to this kind of models. In these notes he derives the model linearized equilibrium conditions starting from the optimization problems.

Once you've read Sims notes, probably you want to check Fenandez-Villaverde's "A Baseline DSGE Model" notes in which he develops essentially the same model but including some extensions, plus some insights into the estimation of the model using bayesian techniques.

If you want to understand these types of models in a more general way and in more depth, I'd recommend the book Recursive Methods in Economic Dynamics by Stokey, Lucas and Prescott, which is the standard reference. This book develops the recursive approach to model economic problems while providing the necessary math background, although you are supposed to have some exposure in proofs and real analysis before going into this reading.

In line with the last recommendation is Recursive Macroeconomic Theory by Ljungqvist and Sargent. This book offers a wider range of topics than SLP, but is also more referencial to my taste since it assumes a some mathematical and rigorous-macro background.

TL;DR. Sims' class notes will give you some working knowledge in NK models; Go into Fenandez-Villaverde's "A Baseline DSGE Model" notes if you want to get a little bit further. Check Recursive Methods in Economic Dynamics and Recursive Macroeconomic Theory if you want to really understand these types of models.

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  • $\begingroup$ Thank you so much @manifold! One thing that confuses me is some papers use economic intuition to augment the baseline model while others are "micro-founded." In the equation above I found a paper that derives the model but without the presence of a real exchange rate gap making it a closed economy new-keynesian model. But others introduce the real exchange rate gap (see Laxton et. al. (2006) page 6 to 12) and a version of the uncovered interest parity condition. In the end, what I'm trying to comprehend is whether the above model is derived explicitly from microeconomic foundations. $\endgroup$
    – Blg Khalil
    Feb 8, 2023 at 0:47

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