Silviase/jgov_v1.4_full_ann huggingface dataset:
where L is the long run level of labor hours.

$$ \left(\widehat{C}_{t}-\eta\frac{\mathcal{A}_{T,t-1}}{\mathcal{A}_{T,t}}\widehat{C}_{t-1}\right)^{-\sigma_{c}}\exp\left\{\varepsilon_{L,t}\frac{\sigma_{c}-1}{1+\sigma_{l}}\left(\frac{L_{t}}{L}\right)^{1+\sigma_{l}}\right\}=\widehat{\lambda}_{t} $$

$$ \widehat{\lambda}_{t}=\beta\varepsilon_{B,t}R_{t}\mathbb{E}_{t}\left[\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{-\sigma_{c}}\frac{\widehat{\lambda}_{t+1}}{\pi_{t+1}}\right] $$

$$ \left[\widehat{C}_{t}-\eta\frac{\mathcal{A}_{T,t-1}}{\mathcal{A}_{T,t}}\widehat{C}_{t-1}\right]^{1-\sigma_{c}}\frac{\varepsilon_{L,t}}{L}\exp\left\{\varepsilon_{L,t}\frac{\sigma_{c}-1}{1+\sigma_{l}}\left(\frac{L_{t}}{L}\right)^{1+\sigma_{l}}\right\}\left(\frac{L_{t}}{L}\right)^{\sigma_{l}}=\widehat{\lambda}_{t}\widehat{w}_{t}^{m} $$

$$ \begin{aligned}\frac{p_{I,t}}{\varepsilon_{I,t}}={}&Q_{t}\left[1-\mathcal{S}\left(\frac{\widehat{I}_{t}}{\widehat{I}_{t-1}}\frac{\mathcal{A}_{T,t}}{\mathcal{A}_{T,t-1}}\right)-\frac{\widehat{I}_{t}}{\widehat{I}_{t-1}}\frac{\mathcal{A}_{T,t}}{\mathcal{A}_{T,t-1}}\mathcal{S}^{\prime}\left(\frac{\widehat{I}_{t}}{\widehat{I}_{t-1}}\frac{\mathcal{A}_{T,t}}{\mathcal{A}_{T,t-1}}\right)\right]\\&+\beta\mathbb{E}_{t}\left[\frac{\widehat{\lambda}_{t+1}}{\widehat{\lambda}_{t}}\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{-\sigma_{c}}Q_{t+1}\frac{\varepsilon_{I,t+1}}{\varepsilon_{I,t}}\left(\frac{\widehat{I}_{t+1}}{\widehat{I}_{t}}\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{2}\mathcal{S}^{\prime}\left(\frac{\widehat{I}_{t+1}}{\widehat{I}_{t}}\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)\right]\end{aligned} $$

$$ Q_{t}\delta^{\prime}\left(z_{t}\right)=r_{t}^{k} $$

$$ Q_{t}=\beta\mathbb{E}_{t}\left[\frac{\widehat{\lambda}_{t+1}}{\widehat{\lambda}_{t}}\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{-\sigma_{c}}\Big(Q_{t+1}\Big(1-\delta(z_{t+1})\Big)+r_{t+1}^{k}z_{t+1}\Big)\right] $$

Notice that Tobin's Q $ Q_{t} $ is a stationary variable. Therefore, by its definition, the Lagrange multiplier $ \mu_{t} $ is trended. We write $ \widehat{\mu}_{t} = \mu_{t} A_{T,t}^{\sigma_{c}} $ .

Let $ K_{t} \equiv \widehat{K}_{t} A_{T,t} $ , and the dynamics for the stock of capital is

$$ \widehat{K}_{t}=(1-\delta(z_{t}))\frac{\mathcal{A}_{T,t-1}}{\mathcal{A}_{T,t}}\widehat{K}_{t-1}+\varepsilon_{I,t}\left(1-\mathcal{S}\left(\frac{\widehat{I}_{t}}{\widehat{I}_{t-1}}\frac{\mathcal{A}_{T,t}}{\mathcal{A}_{T,t-1}}\right)\right)\widehat{I}_{t} $$

On the production side, the factor price frontier becomes

$$ \frac{\widehat{w}_{t}L_{t}^{d}}{r_{t}^{k}\widehat{K}_{t}^{d}}=\frac{1-\alpha}{\alpha}. $$
Marginal cost is written

$$ m c_{t}=A_{C,t}^{\alpha-1}\left(\frac{r_{t}^{k}}{\alpha}\right)^{\alpha}\left(\frac{\widehat{w}_{t}}{1-\alpha}\right)^{1-\alpha} $$

For the price equations, we define $ \mathcal{Z}_{i,t}=\widehat{\mathcal{Z}_{i,t}}\mathcal{A}_{T,t}^{1-\sigma_{c}} $ for i=1,2 and 3, then,

$$ \frac{P_{t}^{\star}}{P_{t}}=\frac{\theta_{f}(1+\psi_{f})}{\theta_{f}(1+\psi_{f})-1}\frac{\widehat{\mathcal{Z}_{1,t}}}{\widehat{\mathcal{Z}_{2,t}}}+\frac{\psi_{f}}{\theta_{f}(1+\psi_{f})-1}\left(\frac{P_{t}^{\star}}{P_{t}}\right)^{1+(1+\psi_{f})\theta_{f}}\frac{\widehat{\mathcal{Z}_{3,t}}}{\widehat{\mathcal{Z}_{2,t}}} $$

and

$$ \widehat{\mathcal{X}}_{1,t}=\widehat{\lambda}_{t}mc_{t}\Theta_{t}^{(1+\psi_{f})\theta_{f}}\widehat{Y}_{t}+\beta\xi_{p}\mathbb{E}_{t}\left[\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{1-\sigma_{c}}\left(\frac{\pi_{t+1}}{\pi_{t-1}^{\gamma_{p}}\bar{\pi}_{t}^{1-\gamma_{p}}}\right)^{(1+\psi_{f})\theta_{f}}\widehat{\mathcal{X}}_{1,t+1}\right] $$

$$ \widehat{\mathcal{X}}_{2,t}=\widehat{\lambda}_{t}\varepsilon_{y,t}\Theta_{t}^{(1+\psi_{f})\theta_{f}}\widehat{Y}_{t}+\beta\xi_{p}\mathbb{E}_{t}\left[\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{1-\sigma_{c}}\left(\frac{\pi_{t+1}}{\pi_{t-1}^{\gamma_{p}}\bar{\pi}_{t}^{1-\gamma_{p}}}\right)^{(1+\psi_{f})\theta_{f}-1}\widehat{\mathcal{X}}_{2,t+1}\right] $$

$$ \widehat{\mathcal{X}}_{3,t}=\widehat{\lambda}_{t}\varepsilon_{y,t}\widehat{Y}_{t}+\beta\xi_{p}\mathbb{E}_{t}\left[\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{1-\sigma_{c}}\left(\frac{\pi_{t-1}^{\gamma_{p}}\bar{\pi}_{t}^{1-\gamma_{p}}}{\pi_{t+1}}\right)\widehat{\mathcal{X}}_{3,t+1}\right] $$

Equations (3.34) and (3.35) are unchanged.

For the unions and the employment agency, we define $ \mathcal{H}_{i,t} = \widehat{\mathcal{H}_{i,t}}\mathcal{A}_{T,t}^{1-\sigma_{c}} $ for i = 1, 2 and 3, $ w_{t}^{\star} = A_{T,t}\widehat{w}_{t}^{\star} $ and $ \varpi_{t} = \widehat{\varpi}_{t}A_{T,t}/A_{T,t-1} $ . The relative wage of a union that has the possibility to reoptimize in period t is

$$ \frac{\widehat{w}_{t}^{\star}}{\widehat{w}_{t}}=\frac{\theta_{s}(1+\psi_{s})}{\theta_{s}(1+\psi_{s})-1}\frac{\widehat{\mathcal{H}_{1,t}}}{\widehat{\mathcal{H}_{2,t}}}+\frac{\psi_{s}}{\theta_{s}(1+\psi_{s})-1}\left(\frac{\widehat{w}_{t}^{\star}}{\widehat{w}_{t}}\right)^{1+(1+\psi_{s})\theta_{s}}\frac{\widehat{\mathcal{H}_{3,t}}}{\widehat{\mathcal{H}_{2,t}}} $$

$$ \widehat{\mathcal{H}_{1,t}}=\widehat{\lambda}_{t}\widehat{w}_{t}^{m}\mathcal{L}_{t}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}+\beta\xi_{w}\mathbb{E}_{t}\left[\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{1-\sigma_{c}}\left(\frac{\widehat{\varpi}_{t+1}\pi_{t+1}}{\pi_{t}^{\gamma_{w}}\bar{\pi}_{t}^{1-\gamma_{w}}}\right)^{(1+\psi_{s})\theta_{s}}\widehat{\mathcal{H}_{1,t+1}}\right] $$
$$ \widehat{\mathcal{H}_{2,t}}=\widehat{\lambda}_{t}\varepsilon_{l,t}\widehat{w}_{t}\mathcal{L}_{t}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}+\beta\xi_{w}\mathbb{E}_{t}\left[\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{1-\sigma_{c}}\left(\frac{\widehat{\varpi}_{t+1}\pi_{t+1}}{\pi_{t}^{\gamma_{w}}\bar{\pi}_{t}^{1-\gamma_{w}}}\right)^{(1+\psi_{s})\theta_{s}-1}\widehat{\mathcal{H}_{2,t+1}}\right] $$

$$ \widehat{\mathcal{H}_{3,t}}=\widehat{\lambda}_{t}\varepsilon_{l,t}\widehat{w}_{t}\mathcal{L}_{t}+\beta\xi_{w}\mathbb{E}_{t}\left[\left(\frac{\mathcal{A}_{T,t+1}}{\mathcal{A}_{T,t}}\right)^{1-\sigma_{c}}\left(\frac{\pi_{t}^{\gamma_{w}}\bar{\pi}_{t}^{1-\gamma_{w}}}{\widehat{\varpi}_{t+1}\pi_{t+1}}\right)\widehat{\mathcal{H}_{3,t+1}}\right] $$

$ \vartheta_{s,t} $ can be expressed as a function of stationarized wages:

$$ \vartheta_{s,t}=(1-\xi_{w})\frac{\widehat{w}_{t}^{\star}}{\widehat{w}_{t}}+\xi_{w}\frac{\bar{\pi}_{t}^{\gamma_{w}}\pi_{t-1}^{1-\gamma_{w}}}{\widehat{\varpi}_{t}\pi_{t}}\vartheta_{s,t-1} $$

Equation (3.53) remains unchanged. Finally, wage distortion is expressed as a function of stationarized wages:

$$ \nabla_{w,t}=\left(1-\xi_{w}\right)\left(\frac{\widehat{w}_{t}^{\star}}{\widehat{w}_{t}}\right)^{-(1+\psi_{s})\theta_{s}}+\xi_{w}\left(\frac{\bar{\pi}_{t}^{\gamma_{w}}\pi_{t-1}^{1-\gamma_{w}}}{\widehat{\varpi}_{t}\pi_{t}}\right)^{-(1+\psi_{s})\theta_{s}}\nabla_{w,t-1} $$

Equilibrium conditions on the good and factor markets are the same for the stationarized variables.

#### 3.6.2 Stationary state

At the stationary state, we have

$$ \left(\widehat{C}-\frac{\eta}{1+g}\widehat{C}\right)^{-\sigma_{c}}\exp\left\{\widetilde{L}\frac{\sigma_{c}-1}{1+\sigma_{l}}\right\}=\widehat{\lambda} $$

$$ \widehat{\lambda}=\beta R(1+g)^{-\sigma_{c}}\frac{\widehat{\lambda}}{\pi^{\star}} $$

$$ \left[\widehat{C}-\frac{\eta}{1+g}\widehat{C}\right]^{1-\sigma_{c}}\frac{\widetilde{L}}{L}\exp\left\{\widetilde{L}\frac{\sigma_{c}-1}{1+\sigma_{l}}\right\}=\widehat{\lambda}\widehat{w}^{m} $$
$$ Q=1 $$

$$ \delta^{\prime}\left(\bar{z}\right)=r^{k} $$

$$ 1=\beta(1+g)^{-\sigma_{c}}\left(1-\bar{\delta}+r^{k}\bar{z}\right) $$

$$ \widehat{K}=\frac{1+g}{g+\bar{\delta}}\widehat{I} $$

$$ \frac{\widehat{w}L^{d}}{r^{k}\widehat{K}^{d}}=\frac{1-\alpha}{\alpha} $$

$$ m c=\left(\frac{r^{k}}{\alpha}\right)^{\alpha}\left(\frac{\widehat{w}}{1-\alpha}\right)^{1-\alpha} $$

$$ \frac{P^{\star}}{P}=\frac{\theta_{f}(1+\psi_{f})}{\theta_{f}(1+\psi_{f})-1}\frac{\widehat{\mathcal{Z}_{1}}}{\widehat{\mathcal{Z}_{2}}}+\frac{\psi_{f}}{\theta_{f}(1+\psi_{f})-1}\left(\frac{P^{\star}}{P}\right)^{1+(1+\psi_{f})\theta_{f}}\frac{\widehat{\mathcal{Z}_{3}}}{\widehat{\mathcal{Z}_{2}}} $$

$$ \widehat{\mathcal{Z}_{1}}=\widehat{\lambda}m c\Theta^{(1+\psi_{f})\theta_{f}}\widehat{Y}+\beta\xi_{p}(1+g)^{1-\sigma_{c}}\widehat{\mathcal{Z}_{1}} $$

$$ \widehat{\mathcal{X}_{2}}=\widehat{\lambda}\Theta^{(1+\psi_{f})\theta_{f}}\widehat{Y}+\beta\xi_{p}(1+g)^{1-\sigma_{c}}\widehat{\mathcal{X}_{2}} $$

$$ \widehat{\mathcal{Z}_{3}}=\widehat{\lambda}\widehat{Y}+\beta\xi_{p}(1+g)^{1-\sigma_{c}}\widehat{\mathcal{Z}_{3}} $$

$$ \vartheta_{f}=\frac{P^{\star}}{P} $$
$$ \frac{\psi_{f}\vartheta_{f}}{1+\psi_{f}}+\frac{\Theta}{1+\psi_{f}}=1 $$

$$ \Delta_{p}=\frac{1}{1+\psi_{f}}\Theta^{(1+\psi_{f})\theta_{f}}\nabla_{p}+\frac{\psi_{f}}{1+\psi_{f}} $$

$$ \nabla_{p}=\left(\frac{P_{t}^{\star}}{P_{t}}\right)^{-(1+\psi_{f})\theta_{f}} $$

$$ \Theta=\frac{P^{\star}}{P} $$

$$ \frac{\widehat{w}^{\star}}{\widehat{w}}=\frac{\theta_{s}(1+\psi_{s})}{\theta_{s}(1+\psi_{s})-1}\frac{\widehat{\mathcal{H}_{1}}}{\widehat{\mathcal{H}_{2}}}+\frac{\psi_{s}}{\theta_{s}(1+\psi_{s})-1}\left(\frac{\widehat{w}^{\star}}{\widehat{w}}\right)^{1+(1+\psi_{s})\theta_{s}}\frac{\widehat{\mathcal{H}_{3}}}{\widehat{\mathcal{H}_{2}}} $$

$$ \widehat{\mathcal{H}_{1}}=\widehat{\lambda}\widehat{w}^{m}\mathcal{L}\Upsilon^{(1+\psi_{s})\theta_{s}}+\beta\xi_{w}(1+g)^{1-\sigma_{c}}\widehat{\mathcal{H}_{1}} $$

$$ \widehat{\mathcal{H}_{2}}=\widehat{\lambda}\widehat{w}\mathcal{L}\Upsilon^{(1+\psi_{s})\theta_{s}}+\beta\xi_{w}(1+g)^{1-\sigma_{c}}\widehat{\mathcal{H}_{2}} $$

$$ \widehat{\mathcal{H}_{3}}=\widehat{\lambda}\widehat{w}\mathcal{L}+\beta\xi_{w}(1+g)^{1-\sigma_{c}}\widehat{\mathcal{H}_{3}} $$

$$ \vartheta_{s}=\frac{\widehat{w}^{\star}}{\widehat{w}} $$

$$ \frac{\psi_{s}\vartheta_{s}}{1+\psi_{s}}+\frac{\Upsilon}{1+\psi_{s}}=1 $$

$$ \Delta_{w}=\frac{1}{1+\psi_{s}}\Upsilon^{(1+\psi_{s})\theta_{s}}\nabla_{w}+\frac{\psi_{s}}{1+\psi_{s}} $$
$$ \nabla_{w}=\left(\frac{\widehat{w}^{\star}}{\widehat{w}}\right)^{-(1+\psi_{s})\theta_{s}} $$

$$ \Upsilon=1 $$

$$ R=R^{\star} $$

$$ \Delta_{p}\widehat{Y}=\left(\widehat{K}^{d}\right)^{\alpha}\left(L^{d}\right)^{1-\alpha} $$

$$ L=\Delta_{w}\mathcal{L} $$

$$ L^{d}=\Delta_{w}\mathcal{L} $$

$$ \frac{\bar{z}}{1+g}\widehat{K}=\widehat{K}^{d} $$

$$ \widehat{C}+\widehat{I}=\left(1-g^{\star}\right)\widehat{Y} $$

We can write equations (3.93a) to (3.93c) equivalently as

$$ \widehat{\mathcal{Z}_{1}}=\frac{\widehat{\lambda}m c\Theta^{(1+\psi_{f})\theta_{f}}\widehat{Y}}{1-\beta\xi_{p}(1+g)^{1-\sigma_{c}}} $$

$$ \widehat{\mathcal{X}_{2}}=\frac{\widehat{\lambda}\Theta^{(1+\psi_{f})\theta_{f}}\widehat{Y}}{1-\beta\xi_{p}(1+g)^{1-\sigma_{c}}} $$

$$ \widehat{\mathcal{X}_{3}}=\frac{\widehat{\lambda}\widehat{Y}}{1-\beta\xi_{p}(1+g)^{1-\sigma_{c}}} $$
Substituting in $ (3.92) $ , we get

$$ \frac{P^{\star}}{P}=\frac{\theta_{f}(1+\psi_{f})}{\theta_{f}(1+\psi_{f})-1}mc+\frac{\psi_{f}}{\theta_{f}(1+\psi_{f})-1}\left(\frac{P^{\star}}{P}\right)^{1+(1+\psi_{f})\theta_{f}}\Theta^{-(1+\psi_{f})\theta_{f}} $$

that implicitly define real marginal cost at the stationary state as a function of optimal relative price $ P^{*}/P $ and the Lagrange multiplier from the optimization program for the firm that produces the homogeneous good. Furthermore, sustitution equation (3.97) in (3.95), (3.94), (3.96b) and (3.96a) we get

$$ \Theta=\vartheta_{f}=\frac{P^{\star}}{P}=\nabla_{p}=\Delta_{p}=1 $$

Marginal cost at the stationary state is then

$$ mc=\frac{\theta_{f}(1+\psi_{f})-1}{\theta_{f}(1+\psi_{f})}-\frac{\psi_{f}}{\theta_{f}(1+\psi_{f})}=\frac{\theta_{f}-1}{\theta_{f}} $$

Finally, substituting (3.97) in equations (3.110a) to (3.110c) we obtain

$$ \widehat{\mathcal{Z}_{1}}=m c\frac{\widehat{\lambda}\widehat{Y}}{1-\beta\xi_{p}(1+g)^{1-\sigma_{c}}} $$

$$ \widehat{\mathcal{L}_{2}}=\widehat{\mathcal{L}_{3}}=\widehat{\mathcal{L}_{1}}/m c $$

where $ \widehat{\lambda} $ , $ \widehat{Y} $ are determined below iwith the other real variables.

We can write equations (3.99a) and (3.99c) in the following manner:

$$ \widehat{\mathcal{H}_{1}}=\frac{\widehat{\lambda}\widehat{w}^{m}\mathcal{L}\Upsilon^{(1+\psi_{s})\theta_{s}}}{1-\beta\xi_{w}(1+g)^{1-\sigma_{c}}} $$

$$ \widehat{\mathcal{H}_{2}}=\frac{\widehat{\lambda}\widehat{w}\mathcal{L}\Upsilon^{(1+\psi_{s})\theta_{s}}}{1-\beta\xi_{w}(1+g)^{1-\sigma_{c}}} $$
$$ \widehat{\mathcal{H}_{3}}=\frac{\widehat{\lambda}\widehat{w}\mathcal{L}}{1-\beta\xi_{w}(1+g)^{1-\sigma_{c}}} $$

Substituting in (3.98) we get

$$ \frac{\widehat{w}^{\star}}{\widehat{w}}=\frac{\theta_{s}(1+\psi_{s})}{\theta_{s}(1+\psi_{s})-1}\frac{\widehat{w}^{m}}{\widehat{w}}+\frac{\psi_{s}}{\theta_{s}(1+\psi_{s})-1}\left(\frac{\widehat{w}^{\star}}{\widehat{w}}\right)^{1+(1+\psi_{s})\theta_{s}}\Upsilon^{-(1+\psi_{s})\theta_{s}} $$

that implicitly defines the ratio of real wage received by the household to the real wage paid to the employment agency as a function of optimal real relative wage, $ \hat{w}^{*}/\hat{w} $ , and the Lagrange multiplier from the optimization program of the employment agency.

Substituting equation (3.103) in equations (3.101), (3.100), (3.102b) and (3.102a) we obtain

$$ \Upsilon=\vartheta_{s}=\frac{\widehat{w}^{\star}}{\widehat{w}}=\nabla_{w}=\Delta_{w}=1 $$

Then, we have

$$ \frac{\widehat{w}^{m}}{\widehat{w}}=\frac{\theta_{s}(1+\psi_{s})-1}{\theta_{s}(1+\psi_{s})}-\frac{\psi_{s}}{\theta_{s}(1+\psi_{s})} $$

Finally, substituting $ (3.103) $ in $ (3.115a) $ to $ (3.115c) $ we get

$$ \widehat{\mathcal{H}_{1}}=\widehat{w}^{m}\frac{\widehat{\lambda}\mathcal{L}}{1-\beta\xi_{w}(1+g)^{1-\sigma_{c}}} $$

$$ \widehat{\mathcal{H}_{2}}=\widehat{\mathcal{H}_{3}}=\widehat{\mathcal{H}_{1}}\frac{\widehat{w}}{\widehat{w}^{m}} $$

Equations (3.104) and (3.84) imply

$$ R^{\star}=\frac{\pi^{\star}}{\beta}(1+g)^{\sigma_{c}} $$

In practice, we are estimating (or calibrating) $ R^{\star} $ and $ \pi^{\star} $ then we deduce $ \beta $ conditionally to the current estimation of the growth rate along the balanced growth path, g. From equation (3.88) we
get the return on capital at the steady state:

$$ r^{k}=\frac{(1+g)^{\sigma_{c}}-\beta(1-\bar{\delta})}{\beta\bar{z}} $$

where $ \bar{\delta} $ and $ \bar{z} $ are estimated (or calibrated). Equation (3.87) establishes the following constraints on the derivative of the depreciation function at the steady state

$$ \delta^{\prime}\left(\bar{z}\right)=\frac{(1+g)^{\sigma_{c}}-\beta(1-\bar{\delta})}{\beta\bar{z}} $$

Conditionally to the long run level of real marginal cost and return on capital, we obtain steady state for the average real wage (relative to labor efficiency trend) using equation (3.91):

$$ \widehat{w}=\left[mc\left(\frac{\alpha}{r^{k}}\right)^{\alpha}\right]^{\frac{1}{1-\alpha}}\left(1-\alpha\right) $$

Along the balanced growth path, real wage growth rate is the same as the growth rate of labor efficiency:

$$ \varpi=1+g $$

Substituting (3.123) in (3.113) we obtain the real wage received by the household (still normalized by labor efficiency trend) at the steady state:

$$ \widehat{w}^{m}=\left(\frac{\theta_{s}(1+\psi_{s})-1}{\theta_{s}(1+\psi_{s})}-\frac{\psi_{s}}{\theta_{s}(1+\psi_{s})}\right)\left[m c\left(\frac{\alpha}{r^{k}}\right)^{\alpha}\right]^{\frac{1}{1-\alpha}}(1-\alpha) $$

As price and wage distortion is zero at the steady state, we get

$$ L^{d}=\mathcal{L}=L $$

and

$$ \widehat{Y}=\widehat{y}=\left(\widehat{K}^{d}\right)^{\alpha}\left(L^{d}\right)^{1-\alpha} $$

In what follows, we write $ \chi_{X,Y} $ the ratio of two variables X and Y. Equation (3.90) and (3.126)
gives the capital labor ratio at the steady state:

$$ \chi_{\widehat{K}^{d},L}=\frac{\widehat{w}}{r^{k}}\frac{\alpha}{1-\alpha} $$

Using equations (3.126), (3.127) and (3.128) it follows that product per capita at the steady state is

$$ \chi_{\widehat{Y},L}=\chi_{\widehat{K}^{d},L}^{\alpha} $$

On writes the equilibrium on the final good market (3.109) as

$$ \chi_{\widehat{C},\widehat{Y}}+\chi_{\widehat{I},\widehat{Y}}=1-g^{\star} $$

Using the equilibrium condition on the market for capital (3.108) as well as equations (3.128), (3.129) and (3.89), one gets the share of investment in output at the steady state:

$$ \chi_{\widehat{I},\widehat{Y}}=\frac{g+\bar{\delta}}{\bar{z}}\chi_{\widehat{K}^{d},L}^{1-\alpha} $$

We can then determine $ \chi_{\widehat{C},\widehat{Y}} $ by complementarity, using equation (3.130). Dividing (3.85) by (3.83) one gets

$$ \left(\widehat{C}-\frac{\eta}{1+g}\widehat{C}\right)\frac{\widetilde{L}}{L}=\widehat{w}^{m} $$

Multiplying and dividing the left handside by $ \hat{Y} $ , one gets

$$ \chi_{\widehat{C},\widehat{Y}}\chi_{\widehat{Y},L}\left(1-\frac{\eta}{1+g}\right)\widetilde{L}=\widehat{w}^{m} $$

and then

$$ \widetilde{L}=\frac{\widehat{w}^{m}}{\chi_{\widehat{C},\widehat{Y}}\chi_{\widehat{Y},L}}\frac{1+g}{1+g-\eta} $$

Then,

$$ \widehat{Y}=\chi_{\widehat{Y},L}L $$
$$ \widehat{C}=\chi_{\widehat{C},\widehat{Y}}\widehat{Y} $$

$$ \widehat{I}=\chi_{\widehat{I},\widehat{Y}}\widehat{Y} $$

Finally, the steady state of the Lagrange multiplier associated with the household budget constraint is obtained from equation (3.83):

$$ \widehat{\lambda}=\left(\widehat{C}-\frac{\eta}{1+g}\widehat{C}\right)^{-\sigma_{c}}\exp\left\{\widetilde{L}\frac{1-\sigma_{c}}{1+\sigma_{l}}\right\} $$

### 3.7 Specifying functional forms

#### 3.7.1 The depreciation function

The rate of depreciation of physical capital depends on the utilization rate of capital. This function must verify $ \delta(0)=0 $ and $ \delta(z)^{\prime}>0 $ for any $ z\in[0,1] $ and we write $ \delta(\bar{z})=\bar{\delta} $ . We use the following functional form:

$$ \delta(z)=\bar{\delta}e^{\frac{\bar{\delta}^{\prime}}{\bar{\delta}}(z-\bar{z})} $$

where $ \bar{\delta}^{\prime}\equiv\delta^{\prime}(\bar{z}) $ is the first derivative of the rate of depreciation evaluated at the steady state. This parameter is linked to parameters $ g,\sigma_{c},\beta,\bar{\delta} $ and $ \bar{z} $ thru equation (3.122):

$$ \bar{\delta}^{\prime}=\frac{(1+g)^{\sigma_{c}}-\beta(1-\bar{\delta})}{\beta\bar{z}}>0 $$

This parameter should therefore be estimated or calibrated. An alternative to calibrate (or estimate) $ \bar{\delta} $ is to calibrate (or estimate) the ratio $ \chi_{\widehat{I},\widehat{Y}} $ . One can show that

$$ \bar{\delta}=\frac{\frac{\theta_{f}\chi_{\hat{I},\hat{Y}}}{\alpha\beta(\theta_{f}-1)}(1+g)^{\sigma_{c}}-\frac{\theta_{f}\chi_{\hat{I},\hat{Y}}}{\alpha(\theta_{f}-1)}-g}{1-\frac{\theta_{f}\chi_{\hat{I},\hat{Y}}}{\alpha(\theta_{f}-1)}} $$

It is possible to examine the data in order to form a prior on $ \chi_{\widehat{I},\widehat{Y}} $ . It is a flexible way of exploiting information about the relation between production and investment in the long run.
#### 3.7.2 The investment adjustment cost function

Adjusting its investment level is costly for the household. This cost is described by function S. One assumes that the cost is zero along the balance growth path, $ \mathcal{S}(1+g)=0 $ , that $ \mathcal{S}(1)^{\prime}=0 $ , and the cost is convex, $ S^{\prime\prime}>0 $ . We choose the following specification form:

$$ \mathcal{S}(x)\equiv\frac{\psi}{2}(x-(1+g))^{2} $$

where $ x \equiv I_{t}/I_{t-1} $ is the growth factor for investment and $ \psi $ a parameter, real and positive, measuring the size of adjustment costs. It needs to be calibrated or estimated. We can then express the stationarized adjustment cost function for investment as

$$ \mathcal{S}\left(\frac{\widehat{I}_{t}}{\widehat{I}_{t-1}}\right)\equiv\frac{\psi(1+g)^{2}}{2}\left(\frac{\widehat{I}_{t}}{\widehat{I}_{t-1}}\frac{A_{T,t}}{A_{T,t-1}}\frac{1}{1+g}-1\right)^{2} $$

#### 3.7.3 Share of labor income in the long run

Parameter $ \alpha $ in the Cobb–Douglas production function used by the firms of the intermediary good sector is linked to the share of labor income in value added in the long run. We could exploit this information to form a prior on $ \alpha $ , or, alternatively, estimate the share of labor income in the long run and deduce from it the value of this technological parameter, as, at the steady state, we have

$$ \frac{wL}{Y}=\frac{\theta_{f}-1}{\theta_{f}}(1-\alpha) $$

or, equivalently,

$$ \alpha=1-\frac{\theta_{f}}{\theta_{f}-1}\frac{wL}{Y} $$

## 4 Estimation

### 4.1 Data

The model is estimated with quarterly data on the period 1994–2008 (3rd quarter). We use the following observed variables: GDP, private consumption expenditures, private investment (sum
of private growth capital formation in residential buildings and in plant and equipment), consumer price index. For the short term nominal interest rate we use the Bank of Japan target rate of unsecured overnight call rate.

### 4.2 Priors

The following parameters were calibrated: the steady value of the depreciation rate ( $ \delta $ ) is set to 0.025, a value usual for quarterly model. The inflation target was set to 0.5%. Given that on the estimation sample, monetary policy was very often constrained by the zero lower bound for nominal interest rate, this parameter is hard to set and hard to interpret. It certainly can't be interpreted as the inflation rate desired by the monetary authority, but rather as a parameter convenient to describe the policy that effectively took place during this crisis period.

The priors used for estimating the other parameters are described in the first three columns of the results tables. The are set so as to reflect the domain of definition of the parameters and their values are often found in the literature or in similar models estimated on US or European data. Given the relatively short estimation sample, they need to be relatively informative.

### 4.3 Estimation results

Following on the arguments developed in the first section of the treatment of trends in DSGE models, we perform two estimation. The first one uses the rate of growth of GDP, private consumption, private investment and CPI inflation. In the second, we use instead the logarithm of the level of these variables. The results are given in Tables 1 and 2. The prior and posterior distribution of the parameters are represented in Figures 1 to 10.

Comparing the estimation results, one observes that estimating in the growth rate or in the level of the variables doesn’t affect greatly the estimation results. For several parameters, the posterior distribution is very close to the prior. This indicates that the parameters are not identified (or that the prior is too tight). This is in part the consequence of a relatively short estimation sample. One could consider using additional observable variables, notably for the behavior of the labor market, but it could also lead to the reformulation of a more parsimonious model.