Silviase/jgov_v1.4_full_ann huggingface dataset:
In the end, inflation dynamics are characterized by equations (3.35), (3.34), (3.31), (3.33a), (3.33b), (3.33c).

### 3.3 Labor

Homogeneous labor $ L_{t} = \int_{0}^{1} L_{t}(h) \, \mathrm{d}h $ provided by the households is differentiated by a continuum of unions, $ \varsigma \in [0,1] $ . We have then $ L_{t} = \int_{0}^{1} l_{t}(\varsigma) \, \mathrm{d}\varsigma $ . Unions sell differentiated labor, $ l_{t}(\varsigma) $ , to an employment agency that aggregate different types of labor to offer it as input to the firms of the intermediary good sector. Unions have monopolistic power and the employment agency operates in a perfectly competitive manner.

#### 3.3.1 Employment agency

It aggregates lagor $ l_{t}(\varsigma) $ provided by unions with an aggregation function as in ?, defined implicitly by

$$ \int_{0}^{1}\mathcal{G}_{s}\left(\frac{l_{t}(\varsigma)}{\mathcal{L}_{t}}\right)\mathrm{d}\varsigma=1 $$

where $ G_{s} $ is a strictly increasing concave function such that $ \mathcal{G}_{s}(1) = 1 $ . We use the following specification:

$$ \mathcal{G}_{s}(x)=\frac{\theta_{s}(1+\psi_{s})}{(1+\psi_{s})(\theta_{s}(1+\psi_{s})-1)}\Big[\big(1+\psi_{s}\big)x-\psi_{s}\Big]^{\frac{(1+\psi_{s})\theta_{s}-1}{(1+\psi_{s})\theta_{s}}}-\left[\frac{\theta_{s}(1+\psi_{s})}{(1+\psi_{s})(\theta_{s}(1+\psi_{s})-1)}-1\right] $$

This function is a generalization of the Dixit–Stiglitz aggregator? that is a particular case when $ \psi_{s}=0 $ . The employment agency chooses the relative quantity of labor of type $ \varsigma $ such as minimizing the cost of production by unit of homogenous labor, $ \frac{W_{t}(\varsigma)}{W_{t}}\frac{l_{t}(\varsigma)}{L_{t}} $ , under the technological constraints (3.36) and (3.37). The first order condition associated to the optimization program of the employment agency determines its demand of differentiated labor $ ^{3} $ $ \varsigma $ :

$$ \frac{l_{t}(\varsigma)}{\mathcal{L}_{t}}=\frac{1}{1+\psi_{s}}\left[\left(\frac{W_{t}(\varsigma)/W_{t}}{\Upsilon_{t}}\right)^{-(1+\psi_{s})\theta_{s}}+\psi_{s}\right] $$
where $ \Upsilon_{t} $ is the Lagrange multiplier associated with the technological constraints (3.36) and (3.37). Substituting (3.38) in the technological constraint, one gets the following expression for the Lagrange multiplier

$$ \Upsilon_{t}=\left(\int_{0}^{1}\left(\frac{W_{t}(\varsigma)}{W_{t}}\right)^{1-\theta_{s}\left(1+\psi_{s}\right)}\mathrm{d}\varsigma\right)^{\frac{1}{1-\theta_{s}\left(1+\psi_{s}\right)}} $$

As the employment agency behaves in a competitive manner, its profit is zero and we get the aggregate wage as

$$ W_{t}=\frac{\psi_{s}}{1+\psi_{s}}\int_{0}^{1}W_{t}(\varsigma)\mathrm{d}\varsigma+\frac{1}{1+\psi_{s}}\left(\int_{0}^{1}W_{t}(\varsigma)^{1-(1+\psi_{s})\theta_{s}}\mathrm{d}\varsigma\right)^{\frac{1}{1-(1+\psi_{s})\theta_{s}}} $$

#### 3.3.2 Unions

Unions supply differentiated labor services from the homogeneous labor supply from the households. Unions have market power because of this differentiation of labor services. We write the profit of a union offering wage $ W_{t}(\varsigma) $ and $ l_{t}(\varsigma) $ units of labor:

$$ \mathcal{S}_{t}\left(W_{t}(\varsigma)\right))=\left(\varepsilon_{l,t}W_{t}(\varsigma)-W_{t}^{m}\right)l_{t}(\varsigma) $$

where $ \log \varepsilon_{l,t} $ , is an exogenous shock on union's gains. It is an AR(1) process with zero mean. For a given demand, when $ \varepsilon_{l,t} = 1 $ , the profit of the union is given by the difference between the wage asked to the employment agency and the wage paid to the household. By substitution in the demand function of the employment agency (3.38):

$$ \mathcal{S}_{t}\left(W_{t}(\varsigma)\right)=\left(\varepsilon_{l,t}W_{t}(\varsigma)-W_{t}^{m}\right)\frac{1}{1+\psi_{s}}\left\lceil\left(\frac{W_{t}(\varsigma)/W_{t}}{\Upsilon_{t}}\right)^{-(1+\psi_{s})\theta_{s}}+\psi_{s}\right\rceil\mathcal{L}_{t} $$

Each union is subject to a Calvo lottery. In each period, a union can adjust the wage $ W_{t}(\varsigma) $ in an optimal manner with probability $ \xi_{w} $ . In this case, the union chooses wage $ W_{t}^{\star} $ that maximizes its profit knowing that in the future it may not have the opportunity to readjust the wage for some periods. When the lottery draw is negative for the union (with probability $ 1-\xi_{w} $ ), it adjusts wages
according to the following ad hoc rule:

$$ W_{t}(\varsigma)=\frac{\mathcal{A}_{T,t}}{\mathcal{A}_{T,t-1}}\bar{\pi}_{t-1}^{\gamma_{w}}\pi_{t-1}^{1-\gamma_{w}}W_{t-1}(\varsigma) $$

We write $ \Omega_{t} = (\mathcal{A}_{T,t}/\mathcal{A}_{T,t-1})\bar{\pi}_{t-1}^{\gamma_{w}}\pi_{t-1}^{1-\gamma_{w}} \equiv \Omega_{t,t-1} $ the growth factor of nominal wage asked by the union $ \varsigma $ at date t when this one doesn't have the opportunity of revising it in an optimal manner. In this case, the union changes the wage by indexing it on (i) a convex mix of the inflation target of the monetary authority and of past inflation and (ii) the efficiency growth in the intermediary goods sector. We write

$$ \Omega_{t+j,t}=\frac{\mathcal{A}_{T,t+j}}{\mathcal{A}_{T,t}}\left(\prod_{h=0}^{j-1}\bar{\pi}_{t+h}\right)^{\gamma_{w}}\left(\prod_{h=0}^{j-1}\pi_{t+h}\right)^{1-\gamma_{w}}=\Omega_{t+1}\Omega_{t+2}\cdots\Omega_{t+j} $$

the growth factor of the wage of a union that gets negative signals during the next j periods (for j = 0, we have $ \Omega_{t,t} = 1 $ , for j = 1, we have $ \Omega_{t+1,t} = \Omega_{t+1} $ ).

Let $ \widetilde{U}_{t} $ be the value of a union that receives a positive signal at date t and $ \mathcal{U}_{t}(W_{t-1}(\varsigma)) $ the value of a union that receives the negative signal. In the latter case, the union follows simply the ad hoc rule (3.42), this explains why its value at date t depends upon $ W_{t-1}(\varsigma) $ . On the opposite, the optimization program of a union that receives a positive signal is purely turn towards the future. As unions have the same expectations about the future, they all choose the same optimal wage $ (W_{t}^{\star}) $ . More formally, the value at date t of a union that receives a positive signal is

$$ \widetilde{\mathcal{U}}_{t}=\max_{\mathbf{W}}\Big\{\mathcal{S}_{t}\Big(\mathbf{W}\Big)+\beta\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\xi_{w})\widetilde{\mathcal{U}}_{t+1}+\xi_{w}\mathcal{U}_{t+1}\big(\mathbf{W}\big)\right)\right]\Big\} $$

where $ \Lambda_{t} $ is the Lagrange multiplier associated to the nominal budget constraint of the representative household.

The value of a union that receives a negative signal is

$$ \mathcal{U}_{t}\big(W_{t-1}(\varsigma)\big)=\mathcal{S}_{t}\Big(\Gamma_{t}W_{t-1}(\varsigma)\Big)+\beta\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\xi_{w})\widetilde{\mathcal{U}}_{t+1}+\xi_{w}\mathcal{U}_{t+1}\big(\Gamma_{t}W_{t-1}(\varsigma)\big)\right)\right] $$
The first order condition and the application of the envelop theorem give

$$ \mathcal{S}_{t}^{\prime}\big(W_{t}^{\star}\big)+\beta\xi_{w}\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\mathcal{U}_{t+1}^{\prime}\big(W_{t}^{\star}\big)\right]=0 $$

$$ \frac{\mathcal{U}_{t}^{\prime}\big(W_{t-1}(\varsigma)\big)}{\Gamma_{t}}=\mathcal{S}_{t}^{\prime}\big(\Gamma_{t}W_{t-1}(\varsigma)\big)+\beta\xi_{p}\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\mathcal{U}_{t+1}^{\prime}\big(\Gamma_{t}W_{t-1}(\varsigma)\big)\right] $$

with the derivative of the union profit at W :

$$ \begin{align*}\mathcal{S}_{t}^{\prime}\big(\mathcal{W}\big)\quad=&\quad\varepsilon_{l,t}\frac{1-\theta_{s}(1+\psi_{s})}{1+\psi_{s}}\left(\frac{\mathcal{W}}{W_{t}}\right)^{-(1+\psi_{s})\theta_{s}}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}\mathcal{L}_{t}\\&+\theta_{s}\left(\frac{\mathcal{W}}{W_{t}}\right)^{-(1+\psi_{s})\theta_{s}-1}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}\frac{W_{t}^{m}}{W_{t}}\mathcal{L}_{t}+\frac{\psi_{s}}{1+\psi_{s}}\varepsilon_{l,t}\mathcal{L}_{t}\end{align*} $$

Let’s write temporarily, in order to simplify notations, W, the price inherited from the past. One can rewrite, one period ahead

$$ \mathcal{U}_{t+1}^{\prime}\big(\mathcal{W}\big)=\Omega_{t+1,t}\mathcal{S}_{t+1}^{\prime}\big(\Omega_{t+1,t}\mathcal{W}\big)+\beta\xi_{p}\Omega_{t+1,t}\mathbb{E}_{t+1}\left[\frac{\Lambda_{t+2}}{\Lambda_{t+1}}\mathcal{U}_{t+2}^{\prime}\big(\Omega_{t+1,t}\mathcal{W}\big)\right] $$

iterating toward the future and applying conditional expectation gives

$$ \mathbb{E}_{t}\left[\mathcal{U}_{t+1}^{\prime}\big(\mathcal{W}\big)\right]=\mathbb{E}_{t}\left[\sum_{j=0}^{\infty}(\beta\xi_{p})^{j}\Omega_{t+1+j,t}\frac{\Lambda_{t+1+j}}{\Lambda_{t+1}}\mathcal{S}_{t+1+j}^{\prime}\big(\Omega_{t+1+j,t}\mathcal{W}\big)\right] $$

Substituting in the first order condition (pour $ W = W_{t}^{\star} $ ), one gets the following condition for an optimal wage choice by a union that receives a positive signal:

$$ \mathbb{E}_{t}\left[\sum_{j=0}^{\infty}(\beta\xi_{p})^{j}\Omega_{t+j,t}\frac{\Lambda_{t+j}}{\Lambda_{t}}\mathcal{S}_{t+j}^{\prime}\big(\Omega_{t+j,t}W_{t}^{\star}\big)\right]=0 $$

The optimal wage $ W_{t}^{\star} $ is the nominal wage that insures that the sum of current and expected discounted marginal profits are zero when the union can only revise nominal wages by using the ad hoc rule (3.42).
It is possible to obtain a recursive expression for multiplier $ \Upsilon_{t} $ that appears in the expression for a union profit. Equation (3.39) can be written equivalently in the form

$$ \Upsilon_{t}^{1-\theta_{s}(1+\psi_{s})}=\int_{0}^{1}\left(\frac{W_{t}(\varsigma)}{W_{t}}\right)^{1-\theta_{s}(1+\psi_{s})}\mathrm{d}\varsigma $$

The wage offered by the union at date t appears under the integral sign. This price has been determined optimally j periods before with probability $ (1-\xi_{w})\xi_{w}^{j} $ . We can then rewrite the integral as

$$ \Upsilon_{t}^{1-\theta_{s}(1+\psi_{s})}=(1-\xi_{w})\sum_{j=0}^{\infty}\xi_{w}^{j}\left(\frac{\Omega_{t,t-j}W_{t-j}^{\star}}{W_{t}}\right)^{1-\theta_{s}(1+\psi_{s})} $$

where $ W_{t-j}^{\star} $ is the optimal wage at date t-j. Finally, one can interpret the infinite sum as the solution of the following recursive equation:

$$ \Upsilon_{t}^{1-\theta_{s}(1+\psi_{s})}=(1-\xi_{w})\left(\frac{W_{t}^{\star}}{W_{t}}\right)^{1-\theta_{s}(1+\psi_{s})}+\xi_{w}\left(\frac{\Omega_{t,t-1}}{W_{t}/W_{t-1}}\right)^{1-\theta_{s}(1+\psi_{s})}\Upsilon_{t-1}^{1-\theta_{s}(1+\psi_{s})} $$

One can get a more explicit expression for the wage that satisfies equation (3.47). Substituting in this equation the expression for marginal profit (3.46) and dividing by $ W_{t}^{\star}-(1+\psi_{s})\theta_{s} $ , one gets

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

where $ w_{t}^{\star} $ is the real wage obtained by the union at date t when it can adjust the nominal wage in an optimal manner and $ w_{t} $ the real nominal wage in the economy, with

$$ \mathcal{H}_{1,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{w})^{j}\lambda_{t+j}w_{t+j}^{m}\left(\frac{\Omega_{t+j}}{\frac{w_{t+j}}{w_{t}}\frac{P_{t+j}}{P_{t}}}\right)^{-(1+\psi_{s})\theta_{s}}\Upsilon_{t+j}^{(1+\psi_{s})\theta_{s}}\mathcal{L}_{t+j} $$

$$ \mathcal{H}_{2,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{w})^{j}\lambda_{t+j}\varepsilon_{l,t+j}w_{t+j}\left(\frac{\Omega_{t+j}}{\frac{w_{t+j}}{w_{t}}\frac{P_{t+j}}{P_{t}}}\right)^{1-(1+\psi_{s})\theta_{s}}\Upsilon_{t+j}^{(1+\psi_{s})\theta_{s}}\mathcal{L}_{t+j} $$

$$ \mathcal{H}_{3,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{w})^{j}\lambda_{t+j}\varepsilon_{l,t+j}w_{t+j}\frac{\Omega_{t+j}}{\frac{w_{t+j}}{w_{t}}\frac{P_{t+j}}{P_{t}}}\mathcal{L}_{t+j} $$
Noticing that $ w_{t+j}/w_{t} $ , the growth factor of the real wage between t and $ t+j $ , can be equivalently written as $ \Pi_{i=1}^{j}\varpi_{t+i}\left(\varpi_{t}\right) $ is the growth factor of the real wage between t and t-1) and that we have

$$ \Omega_{t+j,t}=(1+g)^{j}\left(\prod_{h=1}^{j}\mathcal{E}_{t+h}\right)^{\frac{1}{1-\rho_{x}}}\left(\prod_{h=0}^{j-1}\bar{\pi}_{t+h}\right)^{\gamma_{w}}\left(\prod_{h=0}^{j-1}\pi_{t+h}\right)^{1-\gamma_{w}}, $$

we can finally represent variables $\mathcal{H}_{1,t}$, $\mathcal{H}_{2,t}$ and $\mathcal{H}_{3,t}$ in the recursive form

$$ \mathcal{H}_{1,t}=\lambda_{t}w_{t}^{m}\mathcal{L}_{t}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}+\beta\xi_{w}\mathbb{E}_{t}\left[\left(\frac{\varpi_{t+1}\pi_{t+1}}{(1+g)\mathcal{E}_{t+1}^{\frac{1}{1-\rho_{x}}}\pi_{t-1}^{\gamma_{w}}\bar{\pi}_{t}^{1-\gamma_{w}}}\right)^{(1+\psi_{s})\theta_{s}}\mathcal{H}_{1,t+1}\right] $$

$$ \mathcal{H}_{2,t}=\lambda_{t}\varepsilon_{l,t}w_{t}\mathcal{L}_{t}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}+\beta\xi_{w}\mathbb{E}_{t}\left[\left(\frac{\varpi_{t+1}\pi_{t+1}}{(1+g)\mathcal{E}_{t+1}^{\frac{1}{1-\rho_{x}}}\pi_{t-1}^{\gamma_{w}}\bar{\pi}_{t}^{1-\gamma_{w}}}\right)^{(1+\psi_{s})\theta_{s}-1}\mathcal{H}_{2,t+1}\right] $$

$$ \mathcal{H}_{3,t}=\lambda_{t}\varepsilon_{l,t}w_{t}\mathcal{L}_{t}+\beta\xi_{w}\mathbb{E}_{t}\left[\left(\frac{(1+g)\mathcal{E}_{t+1}^{\frac{1}{1-\rho_{x}}}\pi_{t-1}^{\gamma_{w}}\bar{\pi}_{t}^{1-\gamma_{w}}}{\varpi_{t+1}\pi_{t+1}}\right)\mathcal{H}_{3,t+1}\right] $$

Noticing that $ \vartheta_{s,t} \equiv \int_{0}^{1} \frac{W_{t}(\varsigma)}{W_{t}} \mathrm{d}\varsigma $ can be written in the recursive form

$$ \vartheta_{s,t}=(1-\xi_{w})\frac{w_{t}^{\star}}{w_{t}}+\xi_{w}\frac{(1+g)\mathcal{E}_{t}^{\frac{1}{1-\rho_{x}}}\pi_{t-1}^{\gamma_{w}}\bar{\pi}_{t}^{1-\gamma_{w}}}{\varpi_{t}\pi_{t}}\vartheta_{s,t-1} $$

we can rewrite equation (3.40) as

$$ \frac{\psi_{s}\vartheta_{s,t}}{1+\psi_{s}}+\frac{\Upsilon_{t}}{1+\psi_{s}}=1 $$

In the end, wage dynamics are characterized by equations (3.45), (3.52), (3.48), (3.49), (3.51a), (3.51b), (3.51c).
### 3.4 Government and monetary authority

#### 3.4.1 Fiscal policy

We assume that exogenous government expenditures $ G_{t} = g_{t}Y_{t} $ are exactly financed by lump sum taxes:

$$ T_{t}=P_{t}G_{t}. $$

#### 3.4.2 Central Bank

We assume that the behavior of the central bank is adequately described by a simple Taylor rule:

$$ R_{t}=R_{t-1}^{\rho_{R}}\left[R^{\star}\left(\frac{\pi_{t-1}}{\bar{\pi}_{t}}\right)^{r_{\pi}}\left(\frac{Y_{t}}{\mathcal{Y}_{t}}\right)^{r_{Y}}\right]^{1-\rho_{R}}\varepsilon_{R,t} $$

where $ \bar{\pi}_{t} $ is the inflation target of the central bank, $ Y_{t} $ is a reference output level to be defined later, $ \log \varepsilon_{R,t} $ is an AR(1) stationary process with zero mean.

### 3.5 General equilibrium

#### 3.5.1 Price distortion

Prices in the intermediary good sector are heterogeneous. However, it is possible to show that this heterogeneity doesn’t hinder aggregation. We know that the firms of this sector choose all the same mix of factor of production in the sense that the ratio of capital demand to labor demand is constant across firms (see equation 3.23). Expressing labor demand of firm $ \iota $ as a function of its demand of physical capital, we can write the production of this firm

$$ y_{t}(\iota)=\left(\frac{A_{t}L_{t}^{d}}{K_{t}^{d}}\right)^{1-\alpha}K_{t}^{d}(\iota) $$

When $ K_{t}^{d} \equiv \int_{0}^{1} K_{t}^{d}(\iota) \, \mathrm{d}\iota $ is aggregate demand for physical capital and $ y_{t} \equiv \int_{0}^{1} Y_{t}(\iota) \, \mathrm{d}\iota $ represents the sum of intermediary productions, we can write directly

$$ y_{t}=\left(K_{t}^{d}\right)^{\alpha}\left(A_{t}L_{t}^{d}\right)^{1-\alpha} $$
The sum of intermediary productions is different from $ Y_{t} $ , because aggregation technology isn't linear. Integrating the demand function for good $ \iota $ from the final good producers (3.18) over $ \iota $ , we get

$$ y_{t}=\Delta_{p,t}Y_{t} $$

with

$$ \Delta_{p,t}\equiv\frac{1}{1+\psi_{f}}\int_{0}^{1}\left(\left(\frac{P_{t}(\iota)/P_{t}}{\Theta_{t}}\right)^{-(1+\psi_{f})\theta_{f}}+\psi_{f}\right)\mathrm{d}\iota $$

Price distortion can be written recursively in the following manner:

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

$$ \nabla_{p,t}=\left(1-\xi_{p}\right)\left(\frac{P_{t}^{\star}}{P_{t}}\right)^{-(1+\psi_{f})\theta_{f}}+\xi_{p}\left(\frac{\bar{\pi}_{t}^{\gamma_{p}}\pi_{t-1}^{1-\gamma_{p}}}{\pi_{t}}\right)^{-(1+\psi_{f})\theta_{f}}\nabla_{p,t-1} $$

where the Lagrange multiplier $ \Theta_{t} $ is defined recursively as well. Then, we have

$$ \Delta_{p,t}Y_{t}=\left(K_{t}^{d}\right)^{\alpha}\left(A_{t}L_{t}^{d}\right)^{1-\alpha} $$

#### 3.5.2 Wage distortion

Here, we show how to link aggregate labor supply by the households with aggregated labor supply by the employment agency to the firms of the intermediary good sector. This link is affected by the heterogeneity of wages induced by their nominal rigidity. Integrating labor demand for type $ \varsigma $ by the employment agency (3.38) over $ \varsigma $ , we find directly

$$ L_{t}=\Delta_{w,t}\mathcal{L}_{t} $$

with

$$ \Delta_{w,t}\equiv\frac{1}{1+\psi_{s}}\int_{0}^{1}\left(\left(\frac{W_{t}(\varsigma)/W_{t}}{\Upsilon_{t}}\right)^{-(1+\psi_{s})\theta_{s}}+\psi_{s}\right)\mathrm{d}\varsigma $$
Wage distortion can be written in recursive form:

$$ \Delta_{w,t}=\frac{1}{1+\psi_{s}}\Upsilon_{t}^{(1+\psi_{s})\theta_{s}}\nabla_{w,t}+\frac{\psi_{s}}{1+\psi_{s}} $$

$$ \nabla_{w,t}=\left(1-\xi_{w}\right)\left(\frac{w_{t}^{\star}}{w_{t}}\right)^{-(1+\psi_{s})\theta_{s}}+\xi_{s}\left(\frac{(1+g)\mathcal{E}_{t}^{\frac{1}{1-\rho_{x}}}\bar{\pi}_{t}^{\gamma_{w}}\pi_{t-1}^{1-\gamma_{w}}}{\varpi_{t}\pi_{t}}\right)^{-(1+\psi_{s})\theta_{s}}\nabla_{w,t-1} $$

where the Lagrange multiplier $ \Upsilon_{t} $ is defined recursively in equation (3.48).

#### 3.5.3 Dividends paid by intermediary good firms

Firms in the intermediary good sector interact in monopolistic competition and make profits that are paid to households in the form of dividends. The sum of nominal profits at date t is

$$ \begin{aligned}\Pi_{t}&=\int_{0}^{1}\Pi_{t}(\iota)\mathrm{d}\iota\\&=\int_{0}^{1}\left(\varepsilon_{y,t}\frac{P_{t}(\iota)}{P_{t}}-m c_{t}\right)P_{t}Y_{t}(\iota)\mathrm{d}\iota\\&=\varepsilon_{y,t}\int_{0}^{1}P_{t}(\iota)Y_{t}(\iota)\mathrm{d}\iota-P_{t}m c_{t}y_{t}\\&=\varepsilon_{y,t}P_{t}Y_{t}-\left(P_{t}r_{t}^{k}\int_{0}^{1}K_{t}^{d}(\iota)\mathrm{d}\iota+W_{t}\int_{0}^{1}L_{t}^{d}(\iota)\mathrm{d}\iota\right)\\&=P_{t}\left(\varepsilon_{y,t}Y_{t}-r_{t}^{k}K_{t}^{d}-w_{t}L_{t}^{d}\right)\end{aligned} $$

As households own the firms, the profits are paid to them. The repartition of these profits between the households in undetermined in general equilibrium, but we know that

$$ \int_{0}^{1}\mathcal{D}_{1,t}(h)\mathrm{d}h=P_{t}\left(\varepsilon_{y,t}Y_{t}-r_{t}^{k}K_{t}^{d}-w_{t}L_{t}^{d}\right) $$
#### 3.5.4 Dividends paid by the unions

In the same way, we can compute aggregate nominal profit of the unions at date t. These profits as well are paid to the households. We have

$$ \begin{aligned}\mathcal{S}_{t}&=\int_{0}^{1}\mathcal{S}_{t}(\varsigma)\mathrm{d}\varsigma\\&=\int_{0}^{1}\left(\varepsilon_{l,t}W_{t}(\iota)-W_{t}^{m}\right)l_{t}(\varsigma)\mathrm{d}\varsigma\\&=\varepsilon_{l,t}\int_{0}^{1}W_{t}(\varsigma)l_{t}(\varsigma)\mathrm{d}\varsigma-W_{t}^{m}L_{t}\\&=\varepsilon_{l,t}W_{t}\mathcal{L}_{t}-W_{t}^{m}L_{t}\\ \end{aligned} $$

and, then,

$$ \int_{0}^{1}\mathcal{D}_{2,t}(h)\mathrm{d}h=\varepsilon_{l,t}W_{t}\mathcal{L}_{t}-W_{t}^{m}L_{t} $$

#### 3.5.5 Equilibrium in factor markets and in bond markets

In general equilibrium, labor supply forms the employment agency equals aggregate labor demand by firms of the intermediary good market. In the same way, aggregate supply of physical capital by the households equals aggregate demand from these firms. In formal terms,

$$ \mathcal{L}_{t}\equiv\Delta_{w,t}^{-1}\int_{0}^{1}L_{t}(h)\mathrm{d}h=\int_{0}^{1}L_{t}^{d}(\iota)\mathrm{d}\iota\equiv L_{t}^{d} $$

$$ \widetilde{K}_{t}\equiv\int_{0}^{1}z_{t}(h)K_{t-1}(h)\mathrm{d}h=\int_{0}^{1}K_{t}^{d}(\iota)\mathrm{d}\iota\equiv K_{t}^{d} $$

Finally, aggregate demand for bonds must be zero, as we assume a close economy and no government debt.

$$ \int_{0}^{1}B_{t}(h)\mathrm{d}h=0 $$

#### 3.5.6 Equilibrium on the good market

By summing the budget constraints of the households (3.2) over $ h \in [0, 1] $ and by substituting the equilibrium conditions on the bond market, the definition of aggregate dividends and the budget
constraint of the government, we get

$$ \begin{aligned}&P_{t}G_{t}+P_{t}C_{t}+p_{I,t}P_{t}I_{t}=W_{t}^{m}L_{t}+P_{t}r_{t}^{K}z_{t}K_{t-1}+P_{t}\left(\varepsilon_{y,t}Y_{t}-r_{t}^{k}K_{t}^{d}-w_{t}L_{t}^{d}\right)\\ &\\&\quad+\varepsilon_{lt}W_{t}\mathcal{L}_{t}-W_{t}^{m}L_{t}\\ \end{aligned} $$

After simplification and knowing that the factor markets are in equilibrium, we obtain

$$ G_{t}+C_{t}+p_{I,t}I_{t}=\varepsilon_{y,t}Y_{t}+\left(\varepsilon_{l,t}-1\right)w_{t}\mathcal{L}_{t} $$

or equivalently

$$ G_{t}+C_{t}+p_{I,t}I_{t}=\varepsilon_{y,t}\Delta_{p,t}^{-1}y_{t}+\left(\varepsilon_{l,t}-1\right)\Delta_{w,t}^{-1}w_{t}L_{t} $$

### 3.6 Long run

The economy described in the model is growing at a constant rate in the long run, in absence of shocks. In order to compute a local approximation in order to solve it and later to estimate it, we need first to make it stationary.

In this model, there is a single source of growth in the long run: the Harrodian technical change that is characterized by a stochastic trend (with a deterministic component) and affects most of the real variables. In order to stationarize the model, we divide all non-stationary variables by $ A_{T,t} $ or a power of this variable. In this section we present the stationarized model and then its stationary state.

#### 3.6.1 Stationary version of the model

On the household side, we write $ C_{t} \equiv \widehat{C}_{t} A_{T,t}, \lambda_{t} = \widehat{\lambda}_{t} A_{T,t}^{-\sigma_{c}}, w_{t}^{m} = \widehat{w}_{t}^{m} A_{T,t}, I_{t} \equiv \widehat{I}_{t} A_{T,t} $ . In order to make estimation (or calibration) of the long run level of $ \varepsilon_{L,t} $ (ie the scale factor that lets obtain the desired long run level for labor hours), we redefine the utility function, replacing $ L_{t} $ by $ L_{t}/L $ .