I am trying to solve following problem regarding the New Keynesian Model:

So, I have calculated following equations by using the Taylor Rule, Phillips Curve and IS Curve: $$\pi=\frac{\kappa\phi_y}{(1-\beta)\phi_y+\kappa(\phi_\pi-1)}(y^*-y^n)\qquad(1)$$

where $y^*$ denotes the efficient level of output and $y^n$ the natural level of output.

Also $$y=y^n+\frac{1-\beta}{\kappa}\frac{\kappa\phi_y}{{1-\beta}\phi_y+\kappa(\phi_\pi-1)}(y^*-y^n).\qquad(2)$$

The parameters were calibrated as follows: $\kappa=0.34,\nu=6,\beta=0.99,\sigma=1,\gamma=1,A_t=1,\phi_y=0.125,\phi_\pi=1.5$ and $$y^*_t-y_t^n=\frac {Y_t^*}{Y}-\frac{Y_t^n}{Y}$$,since $Y=Y^n$ (*).

I tried to solve for $(y^*_t-y_t^n)$ and with (*) and $$w=\frac{\nu-1}{\nu}A_t$$, $$w_t=C^\sigma_tL_t^{\gamma-1}$$, $$C_t=Y_t=A_tL_t$$ I get: $$y_t^*-y_t^n=\frac{1}{(\frac{\nu-1}{\nu})^{\frac{1}{\sigma+\gamma-1}}}-1\qquad(3)$$.

Now, I have a problem solving for (1) and (2). First, the sample solution is $\pi=0.0249$, but if I insert the parameters into (1) I get $\pi=\frac{34}{137}*0.2$ and in order to get the same solution as in the sample solution it should be $\pi=\frac{34}{137}*0.1$. So, did I do something wrong in (3)? Secondly, it seems strange to me that if ( * ) holds, shouldn't (2) equal 0? Also, could someone give me an intuitive explanation for ( * )?

Thank you!


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