How to define the natural rate output

Question

I'm following Woodford's Interest & Prices to derive the microfundations for a New Keynesian model with staggereed prices.
I defined the utility function and disutility function (1.1 at page 144) as
$$ u= frac{xi_t C_t^{1-sigma}}{1-sigma}, v= frac{h_t(j)^{1+eta}}{1+eta}, $$
where $xi$ is a consumption disturbance.
Then I have defined the production function ( 1.7 at page 148) as
$$ y_t(j)= A_t * h_t(j). $$
With this functional forms we can write the real marginal cost (1.10 at page 149) as
$$ s_t(y,Y,xi) = frac{v_h(f^{-1}(y_t(j) /A); xi)}{u_c(Y_t; xi)A} frac{1}{f'(f^{-1}(y_t(j)/A))} = frac{y_t(j)^eta cdot C_t^{1/sigma}}{A_t  cdotxi_t } .$$
Woodford says that with flexible prices we can write the natural rate of output as (1.14 at page 151)
$$s(Y^n, Y^n, xi)= mu^{-1}.$$
Therefore I get that the natural rate of output is equal to
$$ Y_t^n =( mu^{-1} * A_t * xi_t )^{eta/ sigma}. $$
However the problem with this definition is that in the steady state where the shocks($xi$ and $A$) are zero also the natural rate of output will be zero.
Is this formulation correct? Is there another way to express the natural rate of output?

1muflon1 · Answer

That is not correct  conclusion and there are several problems with the formulation you use.

$A$ is not a shock - it is a technology parameter of production function (see Woodford Interest and Prices pp 148). Heck, $A$ cannot even be zero. Following Woodford:

$A_t> 0$ is a time-varying exogenous technology factor

there can be shocks to $A_t$ that increase it or decrease it but $A_t$ itself is strictly positive at all times.

It is actually not $s(Y^n,Y^n,ξ)=μ^{−1}$ but $s(Y^n_t,Y^n_t;ξ)$. This might seem trivial but note $s(.)$ has only 2 arguments ($Y^n_t$ and $Y^n_t;xi$  not 3 arguments $Y^n_t$, $Y^n_t$ and $xi$). In addition, the solution to $s(Y^n,Y^n,ξ)=μ^{−1}$ gives you the natural rate of output not the equation itself. In this context how I read the Woodford semicolon (;)  is used to separate variables and shock parameters. Hence, it is just to indicate that the ξ is not a variable but given by the shock parameters of the economy (i.e. indicate variable conditional on vector of shocks.

Your definition of utility function is extremely non-standard, you define it as:

$$ u= frac{xi_t C_t^{1-sigma}}{1-sigma} $$
but why are the shocks $xi_t$ even multiplied by consumption in the first place? In Woodford the equation (1.1) is given as:
$$E_0 left[ sum^{infty}_{t=0} beta^t left( u_t(C_t, M_t/P_t; xi_t) - int^{1}_{0} v(h_t(i);xi_t)di right) right]$$

first as you can see $C_t$ is not even conditional on shocks $xi_t$, the real money holding is $M_t/P_t; xi_t$ is conditional on vector of shocks.
you can't just assume $u_t(C_t, M_t/P_t; xi_t) = xi_t C_t^{1-sigma}$, I mean that makes no economic sense whatsoever, why should utility be product of vector of shocks and consumption?

I don't think you will get the same result with some more reasonable utility function and take into account that the variables are conditional on the vector of shocks (e.g. $ M_t/P_t; xi_t$ does not mean  $M_t/P_t$ has mean zero even if $xi_t$ is given by Gaussian distribution with mean zero).

How to define the natural rate output

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