# New Keynesian Model in Understanding DSGE

Hi,

I am trying to replicate New Keynesian model from book “Understanding DSGE” by Celso José Costa Junior. Set of non-linear equations for this model are presented as follows:

1. \frac{W_t}{P_t} = L_t^{\phi} C_t^{\sigma}
2. \mathbb{E}_t\left(\frac{C_{t+1}}{C_t}\right)^\sigma = \beta \left[(1-\delta) + \mathbb{E}_t \left(\frac{R_{t+1}}{P_{t+1}}\right)\right]
3. K_{t+1} = (1-\delta)K_t + I_t
4. Y_{t} = A_t K_{t}^\alpha L_{t}^{1-\alpha}
5. K_{t} = \alpha MC_{t} \frac{Y_{t}}{R_t}
6. L_{t} = (1-\alpha) MC_{t} \frac{Y_{t}}{W_t}
7. MC_t = \frac{1}{A_t} \left( \frac{W_t}{1-\alpha} \right)^{1-\alpha} \left( \frac{R_t}{\alpha} \right)^{\alpha}
8. P^*_t = \frac{\psi}{\psi - 1} \mathbb{E}_t \sum_{i=0}^{\infty}\left( \beta \theta \right)^i MC_{t+i}
9. P_t= \left[\theta P_{t-1}^{1-\psi} + (1-\theta) P_{t}^{*\;1-\psi}\right]^{\frac{1}{1-\psi}}
10. \Pi_t = \frac{P_t}{P_{t-1}}
11. Y_t = C_t + I_t
12. \log A_t = (1-\rho)\log A_{ss} + \rho \log A_{t-1} + \epsilon_{A,t}

However I’m having issues with equation 7) as it doesn’t introduce any information into the model as it can be deduced from equations (4,5,6). The book also includes a dynare file corresponding to linearized version of this model.
nk_chapter3.mod (1.4 KB)

But in the included dynare file the model is diffrent from theoretical model with linearized versions of equations 5) and 6) corresponding to:
K(-1) = Y - R
L = Y - W
instead of what the book describes as:
K(-1) = MC + Y - R
L = MC + Y - W

Am i missing something about this model or is it missing an equation? If its missing is there a way to fix it by removing variable/introducing additional equation?

Thanks for any help on this topic.

I have never worked with that book, but whenever people post about it, they report issues. I agree with all your points. The one equation is redundant and the linearization seems wrong.