Silviase/jgov_v1.4_full_ann huggingface dataset:
with Z = MW. Note that in this formulation of the state space representation, only the state variables are transformed, structural shocks and measurement errors stay the same as in the original formulation.

The diffuse initialization of the filter is now as follows. The initial values for the state variables are $ a_{0}=0 $ . This is the unconditional mean of the stationary elements in $ a_{t} $ and has no effects for the non-stationary ones.

Following Durbin and Koopman, we set

$$ \begin{aligned}P_{0}\quad&=\quad P_{0}^{\infty}+P_{0}^{\star}\\&=\quad\left[\begin{array}{cc}{{{I}}}&{{{0}}} \\ \\{{{0}}}&{{{0}}} \\\end{array}\right]+\left[\begin{array}{cc}{{{0}}}&{{{0}}} \\ \\{{{0}}}&{{{\Sigma_{\tilde{a}}}}} \\\end{array}\right]\end{aligned} $$

where I is an identity matrix of the same dimensions as $ T_{11} $ . It corresponds to the diffuse prior on the initial values of the stochastic trends. $ \Sigma_{\tilde{a}} $ is the covariance matrix of the stationary part of $ a_{t} $ . $ \Sigma_{\tilde{a}} $ is the covariance matrix of $ \tilde{a}_{t} $ with dynamics

$$ \tilde{a}_{t}=T_{22}\tilde{a}_{t-1}+\tilde{R}\eta_{t} $$

or

$$ \Sigma_{\tilde{a}}=T_{22}\Sigma_{\tilde{a}}T_{22}^{\prime}+\tilde{R}Q\tilde{R}^{\prime} $$

where $ \tilde{R} $ is the conforming sub-matrix of R. As $ T_{22} $ is already quasi upper-triangular, it is only necessary to use part of the usual algorithm for the Lyapunov equation.

While $ P_{t}^{\infty} $ is different from zero, the filter (and smoother) is in a diffuse step. When t > d, where d is the maximum integration order, the procedure falls back on standard recursions.

$$ \mathbf{A}\mathbf{t}t=0 $$

$$ E\left(a_{1|0}\right)=P_{1|0}=P_{1|0}^{\infty}+P_{1|0}^{\star} $$
Then,

$$ \begin{array}{r c l}{{F_{t}^{\infty}}}&{{=}}&{{Z P_{t|t-1}^{\infty}Z^{\prime}}}\end{array} $$

$$ \begin{array}{r l r}{F_{t}^{\star}}&{{}=}&{Z P_{t}^{\star}Z^{\prime}+H}\end{array} $$

$$ \begin{array}{r c l}{K_{t}^{\infty}}&{=}&{T P_{t|t-1}^{\infty}Z^{\prime}\left(F_{t}^{\infty}\right)^{-1}}\end{array} $$

$$ \begin{array}{r c l}{K_{t}^{\star}}&{=}&{T\left(P_{t|t-1}^{\star}Z^{\prime}\left(F_{t}^{\infty}\right)^{-1}-P_{t|t-1}^{\infty}Z^{\prime}\left(F_{t}^{\infty}\right)^{-1}F_{t}^{\star}\left(F_{t}^{\infty}\right)^{-1}\right)}\end{array} $$

$$ \begin{array}{r c l}{v_{t}}&{=}&{\tilde{y}_{t}^{\star}-Z a_{t|t-1}}\end{array} $$

$$ \begin{aligned}a_{t+1|t}\quad&=\quad T a_{t|t-1}+K_{t}^{\infty}v_{t}\end{aligned} $$

$$ \begin{array}{r c l}{P_{t+1|t}^{\infty}}&{=}&{T P_{t|t-1}^{\infty}\left(T^{\prime}-Z^{\prime}K_{t}^{\infty\prime}\right)}\end{array} $$

$$ \begin{array}{r l r}{P_{t+1|t}^{\star}}&{=}&{-T P_{t|t-1}^{\infty}Z^{\prime}K_{t\;t|t-1}^{\star\prime\star}\left(T^{\prime}-Z^{\prime}K_{t}^{\infty\prime}\right)+R Q R^{\prime}}\end{array} $$

where $ a_{t|t-1}=E_{t-1}a_{t} $

Finally, the log-likelihood is given by

$$ -\frac{n T}{2}\ln2\pi-\frac{1}{2}\sum_{t=1}^{T}\ln\left|F_{t}\right|-\frac{1}{2}\sum_{t=1}^{T}v_{t}^{\prime}F_{t}^{-1}v_{t} $$

## 3 A DSGE model

What follows describes a DSGE model whose structure is close to ?.

### 3.1 Households

The economy is populated with a continuum of households $ h \in [0,1] $ . Each household values consumption of a composite good. We write $ C_{t}(h) $ the demand of this good by household h in period t. A household offers as well labor hours. We write $ L_{t}(h) $ the labor supply of household h in period t. Welfare is defined as:

$$ \begin{aligned}\mathcal{W}_{t}(h)&=u_{t}(h)+\beta\mathbb{E}[\mathcal{W}_{t+1}(h)]\\u_{t}(h)&=\frac{\left[C_{t}(h)-\eta\overline{C}_{t-1}\right]^{1-\sigma_{c}}}{1-\sigma_{c}}\exp\left\{\varepsilon_{L,t}\frac{\sigma_{c}-1}{1+\sigma_{l}}L_{t}(h)^{1+\sigma_{l}}\right\}\end{aligned} $$
where $ \varepsilon_{L,t} $ is a shock to labor supply. $ \log \varepsilon_{L,t} $ is an AR exogenous shock process with mean $ \log \tilde{L} $ (this parameter gives us an extra degree of freedom for adjusting the stationary level of hours). We choose this form for the utility function in order to build a model compatible with balanced growth.

We assume that utility obtained by household h in period t depends not only on its own consumption but as well on aggregate consumption in previous period, $ \overline{C}_{t-1} = \int_{0}^{1} C_{t-1}(h) \, \mathrm{d}h $ . This is a mechanism of external habits.

The budget constraint of household h, in period t, in real terms, is the following:

$$ \begin{aligned}C_{t}(h)&+p_{I,t}I_{t}(h)=\Biggl\{\frac{B_{t-1}(h)}{P_{t}}-\frac{B_{t}(h)}{P_{t}\varepsilon_{B,t}R_{t}}+\left(1-\tau_{W,t}\right)\frac{W_{t}^{m}}{P_{t}}L_{t}(h)\\&+r_{t}^{k}z_{t}(h)K_{t-1}(h)+\frac{\mathcal{D}_{1,t}(h)+\mathcal{D}_{2,t}(h)}{P_{t}}\Biggr\}+T_{t}\end{aligned} $$

where $ P_{t} $ is the aggregate price index; $ R_{t} = 1 + i_{t} $ , corresponds to the rate of interest plus one, $ B_{t}(h) $ the nominal value of bonds detained by household h at the end of period t, $ \varepsilon_{t}^{B} $ is the risk premium requested by households in order to detain the bond; $ \log \varepsilon_{t}^{B} $ is an AR process with zero mean; $ I_{t}(h) $ is investment of h during period t; $ \log p_{I,t} $ is an exogenous shock on the relative price of investment and follows an AR process with zero mean; $ W_{t}^{m} $ is the hourly wage rate received by household h in period t; $ T_{t} $ represents net transfers received by the household during the period; $ \mathcal{D}_{1,t}(h) $ and $ \mathcal{D}_{2,t}(h) $ are the dividends received from firms and from the unions that differentiate household labor supply.

On the resource side, return on physical capital, $ r_{t} $ , is given by

$$ r_{t}^{k}=z_{t}(h)K_{t-1}(h) $$

where the stock of physical capital at date t is

$$ K_{t}(h)=(1-\delta(z_{t}(h)))K_{t-1}(h)+\varepsilon_{I,t}\left(1-\mathcal{S}\left(\frac{I_{t}(h)}{I_{t-1}(h)}\right)\right)I_{t}(h) $$
$ z_{t}(h) \in [0,1] $ is the rate of utilization of physical capital with steady value $ z^{\star} $ ; the depreciation rate, $ \delta $ , is a function of the rate of utilization that verifies $ \delta(0) = 0 $ , $ \delta(1) = 1 $ , $ \delta(z)^{\prime} > 0 $ for all $ z \in [0,1] $ and we write $ \delta(z^{\star}) = \delta^{\star} $ ; $ \varepsilon_{I,t} $ is a random shock to the efficiency of capital accumulation $ \log \varepsilon_{I,t} $ is an AR process with zero mean; function S describes adjustment costs on investment, we assume $ \mathcal{S}(1 + g) = 0 $ , where g is the rate of growth of investment on the balanced growth path, furthermore $ \mathcal{S}(1 + g)^{\prime} = 0 $ and $ S^{\prime\prime} > 0 $ .

Each household h chooses its consumption, labor supply, bond holdings, investment, and capital utilization rate so as to maximize its inter-temporal utility (3.1) under the budget constraint (3.2) and the law of evolution of physical capital (3.3), taking as given evolution of prices and exogenous variables.

The first order optimality conditions are given by:

$$ \left(C_{t}(h)-\eta\overline{C}_{t-1}\right)^{-\sigma_{c}}\exp\left\{\varepsilon_{L,t}\frac{\sigma_{c}-1}{1+\sigma_{l}}L_{t}(h)^{1+\sigma_{l}}\right\}=\lambda_{t}(h) $$

where $ \lambda_{t}(h) $ is the Lagrange multiplier associated to the real budget constraint,

$$ \lambda_{t}(h)=\beta\varepsilon_{B,t}R_{t}\mathbb{E}_{t}\left[\frac{\lambda_{t+1}(h)}{\pi_{t+1}}\right] $$

where $ \pi_{t+1} \equiv P_{t+1}/P_{t} $ is the inflation rate between period t and $ t+1 $ ,

$$ u_{t}(h)\left(\sigma_{c}-1\right)\varepsilon_{L,t}L_{t}(h)^{\sigma_{l}}=-\lambda_{t}(h)\frac{W_{t}^{m}}{P_{t}}. $$

Writing $ \mu_{t}(h) $ the Lagrange multiplier associated to the capital accumulation function one gets:

$$ \begin{aligned}p_{I,t}\lambda_{t}(h)&=\mu_{t}(h)\varepsilon_{I,t}\left[1-\mathcal{S}\left(\frac{I_{t}(h)}{I_{t-1}(h)}\right)-\frac{I_{t}(h)}{I_{t-1}(h)}\mathcal{S}^{\prime}\left(\frac{I_{t}(h)}{I_{t-1}(h)}\right)\right]\\&+\beta\mathbb{E}_{t}\left[\mu_{t+1}(h)\varepsilon_{I,t+1}\left(\frac{I_{t+1}(h)}{I_{t}(h)}\right)^{2}\mathcal{S}^{\prime}\left(\frac{I_{t+1}(h)}{I_{t}(h)}\right)\right]\end{aligned} $$

$$ \mu_{t}(h)\delta^{\prime}\left(z_{t}(h)\right)=\lambda_{t}(h)r_{t}^{k} $$
and at last:

$$ \mu_{t}(h)=\beta\mathbb{E}_{t}\Big[\mu_{t+1}(h)\Big(1-\delta(z_{t+1}(h))\Big)+\lambda_{t+1}(h)r_{t+1}^{k}z_{t+1}(h)\Big] $$

Given the symmetrical nature of the solution for the household’s problem, we get the following aggregated relationships:

$$ \left(C_{t}-\eta C_{t-1}\right)^{-\sigma_{c}}\exp\left\{\varepsilon_{L,t}\frac{\sigma_{c}-1}{1+\sigma_{l}}L_{t}^{1+\sigma_{l}}\right\}=\lambda_{t} $$

$$ \lambda_{t}=\beta\varepsilon_{B,t}R_{t}\mathbb{E}_{t}\left[\frac{\lambda_{t+1}}{\pi_{t+1}}\right] $$

$$ u_{t}\varepsilon_{L,t}\left(\sigma_{c}-1\right)L_{t}^{\sigma_{l}}=-\lambda_{t}\frac{W_{t}^{m}}{P_{t}} $$

$$ \frac{p_{I,t}}{\varepsilon_{I,t}}=Q_{t}\left[1-\mathcal{S}\left(\frac{I_{t}}{I_{t-1}}\right)-\frac{I_{t}}{I_{t-1}}\mathcal{S}^{\prime}\left(\frac{I_{t}}{I_{t-1}}\right)\right]+\beta\mathbb{E}_{t}\left[\frac{\lambda_{t+1}}{\lambda_{t}}Q_{t+1}\frac{\varepsilon_{I,t+1}}{\varepsilon_{I,t}}\left(\frac{I_{t+1}}{I_{t}}\right)^{2}\mathcal{S}^{\prime}\left(\frac{I_{t+1}}{I_{t}}\right)\right] $$

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

$$ Q_{t}=\beta\mathbb{E}_{t}\left[\frac{\lambda_{t+1}}{\lambda_{t}}\left(Q_{t+1}\Big(1-\delta(z_{t+1})\Big)+r_{t+1}^{k}z_{t+1}\right)\right] $$

Here, $ Q_{t} \equiv \mu_{t}/\lambda_{t} $ is Tobin's Q.

### 3.2 Production

#### 3.2.1 Final good producers

Producers of final good, $ Y_{t} $ , operate in a perfectly competitive environment, assembling a continuum of diversified intermediary goods written $ Y_{t}(\iota) $ with $ \iota \in [0,1] $ . They have access to a unique
constant return aggregation technology as in ?, implicitly defined by

$$ \int_{0}^{1}\mathcal{G}_{f}\left(\frac{Y_{t}(\iota)}{Y_{t}}\right)\mathrm{d}\iota=1 $$

where $ G_{f} $ is a strictly increasing concave function such that $ \mathcal{G}_{f}(1)=1 $ . We follow ? or ? and adopt the following functional form for this aggregation function:

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

Parameter $ \psi_{f} $ characterize the curvature of the demand function.

The producer of final good chooses the quantity of intermediary goods $ \iota $ so as to maximize her real profit:

$$ Y_{t}-\int_{0}^{1}\frac{P_{t}(\iota)}{P_{t}}Y_{t}(\iota)\mathrm{d}\iota $$

under the technological constraints (3.16) and (3.17). As the aggregation function is homogeneous of degree one, it is equivalent to minimize cost per unit with respect to the relative demand of intermediary good $ \iota $ under the technological constraint.

The first order condition of optimality determines the demand for intermediary good v:

$$ \frac{Y_{t}\big(\iota\big)}{Y_{t}}=\frac{1}{1+\psi_{f}}\left\lceil\left(\frac{P_{t}\left(\iota\right)/P_{t}}{\Theta_{t}}\right)^{-\left(1+\psi_{f}\right)\theta_{f}}+\psi_{f}\right\rceil $$

where $ \Theta_{t} $ is the Lagrange multiplier associated with the technological constraints (3.16) and (3.17) for the representative firm. Substituting (3.18) in the technological constraints, one gets the following expression for the Lagrange multiplier:

$$ \Theta_{t}=\left(\int_{0}^{1}\left(\frac{P_{t}(\iota)}{P_{t}}\right)^{1-\theta_{f}(1+\psi_{f})}\mathrm{d}\iota\right)^{\frac{1}{1-\theta_{f}(1+\psi_{f})}} $$
The price elasticity of demand is given by:

$$ \epsilon\left(\widetilde{Y}_{t}(\iota)\right)=-\frac{\mathcal{G}^{\prime}\left(\widetilde{Y}_{t}(\iota)\right)}{\widetilde{Y}_{t}(\iota)\mathcal{G}^{\prime \prime}\left(\widetilde{Y}_{t}(\iota)\right)} $$

and, with particular aggregation function adopted in this study,

$$ \epsilon\left(\widetilde{Y}_{t}(\iota)\right)=\theta_{f}\left[1+\psi_{f}-\frac{\psi_{f}}{\widetilde{Y}_{t}(\iota)}\right] $$

When $ \psi_{f} $ is equal to zero, we get back to the more usual case of the CES aggregator of Dixit-Stiglitz? with a price elasticity of demand equal to $ \theta_{f} $ . More generally, one remarks that demand is more sensitive to price when the level of demand is important if and only if parameter $ \psi_{f} $ is positive. We expect therefore to obtain a negative value for this parameter.

Finally, as the final good sector is perfectly competitive, profit for the representative firm must be zero and we derive the aggregate price index:

$$ P_{t}=\frac{\psi_{f}}{1+\psi_{f}}\int_{0}^{1}P_{t}(\iota)\mathrm{d}\iota+\frac{1}{1+\psi_{f}}\left(\int_{0}^{1}P_{t}(\iota)^{1-(1+\psi_{f})\theta_{f}}\mathrm{d}\iota\right)^{\frac{1}{1-(1+\psi_{f})\theta_{f}}} $$

#### 3.2.2 Intermediary goods producers

A continuum of firms $ \iota \in [0,1] $ in monolistic competition produce intermediary goods for the producers of the final good. These firms have all access to the same Cobb–Douglas technology in to transform physical capital and labor in differentiated intermediary goods:

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

where $ K_{t}^{d}(\iota) $ and $ L_{t}^{d}(\iota) $ are demands of intermediary good firm $ \iota $ for physical capital, and labor, respectively; $ A_{t} $ is technical progress, neutral in Harrod sense. The latter term is further decom-
posed in a trend component $ A_{T,t} $ and a cyclical on $ A_{C,t} $ . We have then,

$$ \Delta\log\mathcal{A}_{T,t}\sim\mathrm{AR}(1)\mathrm{stationary with mean}\log(1+g) $$

$$ \log\mathcal{A}_{C,t}\sim\mathrm{AR}(1)\mathrm{stationary with zero mean}. $$

Each intermediary firm $ \iota\in[0,1] $ buys freely its production factors on competitive markets taking their price as given. The firm $ \iota\in[0,1] $ decides upon the mix of physical capital $ (K_{t}^{d}(\iota)) $ and labor $ (L_{t}^{d}(\iota)) $ so as to minimize its cost, $ r_{t}^{k}K_{t}^{d}(\iota)+w_{t}L_{t}^{d}(\iota) $ , under the technological constraint (3.21). The firm optimal behavior on the factor markets is summarized by the following factor prices frontier:

$$ \frac{w_{t}L_{t}^{d}(\iota)}{r_{t}^{k}K_{t}^{d}(\iota)}=\frac{1-\alpha}{\alpha} $$

where $ w_{t} \equiv W_{t}/P_{t} $ is the real wage. The ratio of capital to labor is invariant across firms. Using the factor prices frontier, we rewrite the total cost of firm $ \iota $ as a function of the stock of capital:

$$ C T_{t}(\iota)=\frac{r_{t}^{k}K_{t}^{d}(\iota)}{\alpha} $$

On the other hand, as the returns of scale is constant, we know that the total cost can also be written as

$$ C T_{t}(\iota)=m c_{t}(\iota)Y_{t}(\iota) $$

where $ mc_{t}(\iota) $ is the real marginal cost. We derive then the following expression for the marginal cost of firm $ \iota $ :

$$ m c_{t}(\iota)=A_{t}^{\alpha-1}\left(\frac{r_{t}^{k}}{\alpha}\right)^{\alpha}\left(\frac{w_{t}}{1-\alpha}\right)^{1-\alpha}\equiv m c_{t} $$

Again, marginal cost doesn’t depend on the size of the firm and is constant across firms.
The nominal profit of a firm that offers price P at date t is given by:

$$ \Pi_{t}(\mathcal{P})=\left(\varepsilon_{y,t}\frac{\mathcal{P}}{P_{t}}-m c_{t}\right)\left[\left(\frac{\mathcal{P}}{P_{t}}\right)^{-(1+\psi_{f})\theta_{f}}\left(\int_{0}^{1}\left(\frac{P_{t}(\iota)}{P_{t}}\right)^{1-(1+\psi_{f})\theta_{f}}\mathrm{d}f\right)^{\frac{(1+\psi_{f})\theta_{f}}{1-(1+\psi_{f})\theta_{f}}}+\psi_{f}\right]\frac{P_{t}Y_{t}}{1+\psi_{f}} $$

where $ \log \varepsilon_{y,t} $ , an zero mean AR stationary process, is a shock to the sales of the firm.

Firm $ \iota $ has market power but can't decide of the its optimal price in each period. Following a Calvo scheme, at each date, the firm receives a signal telling it whether it can revise its price $ P_{t}(\iota) $ in an optimal manner or not. There is a probability $ \xi_{p} $ that the firm can't revise its price in a given period. In such a case, the firm follows the following rule:

$$ P_{t}(\iota)=[\bar{\pi}_{t}]^{\gamma_{p}}\left[\frac{P_{t-1}}{P_{t-2}}\right]^{1-\gamma_{p}}P_{t-1}(\iota)=\Gamma_{t}P_{t-1}(\iota) $$

where $ \bar{\pi}_{t} $ is the inflation target of the monetary authorities. More generally, we write

$$ \Gamma_{t+j,t}=\left(\prod_{h=0}^{j-1}\bar{\pi}_{t+h}\right)^{\gamma_{p}}\left(\prod_{h=0}^{j-1}\pi_{t+h}\right)^{1-\gamma_{p}}=\Gamma_{t+1}\Gamma_{t+2}\cdots\Gamma_{t+j} $$

the growth factor of the price of a firm that doesn’t receive a favorable signal during j successive periods(for j = 0 we have $ \Gamma_{t,t} = 1 $ ; for j = 1, we have $ \Gamma_{t+1,t} = \Gamma_{t+1} $ ). When the firm $ \iota $ receives a positive signal (with probability $ 1 - \xi_{p} $ ), it chooses price $ P_{t}(\iota) $ that maximizes its profit.

Let $ \widetilde{\mathcal{V}}_{t} $ be the value of a firm that receives a positive signal in period t and $ \mathcal{V}_{t}(P_{t-1}(\iota)) $ the value of a firm that receives a negative signal. As a firm that receives a negative signal follows simply the ad hoc pricing rule (3.25), its value at time t depends only on $ P_{t-1}(\iota) $ . For a firm that receives a positive signal, its value at period t is

$$ \widetilde{\mathcal{V}}_{t}=\max_{\mathbf{P}}\Big\{\Pi_{t}\Big(\mathbf{P}\Big)+\beta\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\xi_{p})\widetilde{\mathcal{V}}_{t+1}+\xi_{p}\mathcal{V}_{t+1}\big(\mathbf{P}\big)\right)\right]\Big\} $$

where $ \Lambda_{t} $ is the Lagrange multiplier of the budget constraint of the representative household and $ P_{t}\Lambda_{t}=\lambda_{t} $ . Let $ P^{\star} $ be the optimal price chosen by the firm that can re-optimize.
The value of a firm that can’t re-optimize is

$$ \mathcal{V}_{t}\big(P_{t-1}(\iota)\big)=\Pi_{t}\Big(\Gamma_{t}P_{t-1}(\iota)\Big)+\beta\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\xi_{p})\widetilde{\mathcal{V}}_{t+1}+\xi_{p}\mathcal{V}_{t+1}\big(\Gamma_{t}P_{t-1}(\iota)\big)\right)\right] $$

The first order condition and the envelope theorem give

$$ \Pi_{t}^{\prime}\big(P^{\star}\big)+\beta\xi_{p}\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\mathcal{V}_{t+1}^{\prime}\big(P^{\star}\big)\right]=0 $$

$$ \frac{\mathcal{V}_{t}^{\prime}\big(P_{t-1}(\iota)\big)}{\Gamma_{t}}=\Pi_{t}^{\prime}\big(\Gamma_{t}P_{t-1}(\iota)\big)+\beta\xi_{p}\mathbb{E}_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\mathcal{V}_{t+1}^{\prime}\big(\Gamma_{t}P_{t-1}(\iota)\big)\right] $$

with the derivative of profit at P :

$$ \begin{align*}\Pi_{t}^{\prime}\big(\mathcal{P}\big)\quad=&\quad\varepsilon_{y,t}\frac{1-\theta_{f}\big(1+\psi_{f}\big)}{1+\psi_{f}}\left(\frac{\mathcal{P}}{P_{t}}\right)^{-(1+\psi_{f})\theta_{f}}\Theta_{t}^{(1+\psi_{f})\theta_{f}}Y_{t}\\&+\theta_{f}\left(\frac{\mathcal{P}}{P_{t}}\right)^{-(1+\psi_{f})\theta_{f}-1}\Theta_{t}^{(1+\psi_{f})\theta_{f}}mc_{t}Y_{t}+\frac{\psi_{f}}{1+\psi_{f}}\varepsilon_{y,t}Y_{t}\end{align*} $$

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

$$ \mathcal{V}_{t+1}^{\prime}\big(\mathcal{P}\big)=\Gamma_{t+1,t}\Pi_{t+1}^{\prime}\big(\Gamma_{t+1,t}\mathcal{P}\big)+\beta\xi_{p}\Gamma_{t+1,t}\mathbb{E}_{t+1}\left[\frac{\Lambda_{t+2}}{\Lambda_{t+1}}\mathcal{V}_{t+2}^{\prime}\big(\Gamma_{t+1,t}\mathcal{P}\big)\right] $$

Iterating toward the future and applying conditional expectation at time t, one gets

$$ \mathbb{E}_{t}\left[\mathcal{V}_{t+1}^{\prime}\big(\mathcal{P}\big)\right]=\mathbb{E}_{t}\left[\sum_{j=0}^{\infty}(\beta\xi_{p})^{j}\Gamma_{t+1+j,t}\frac{\Lambda_{t+1+j}}{\Lambda_{t+1}}\Pi_{t+1+j}^{\prime}\big(\Gamma_{t+1+j,t}\mathcal{P}\big)\right] $$

By substitution $ (P = P^{\star}) $ inf the first order condition, one gets the following condition for the price chosen by the firm that gets a positive signal:

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

One can get a more explicit expression for the price that satisfies equation (3.30). Substituting in
this equation the expression of marginal profit (3.29) and dividing by $ P_{t}^{\star} - (1 + \psi_{f}) \theta_{f} $ one gets:

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

with

$$ \mathcal{Z}_{1,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{p})^{j}\lambda_{t+j}\left(\frac{\Gamma_{t+j}}{P_{t+j}/P_{t}}\right)^{-(1+\psi_{f})\theta_{f}}\Theta_{t+j}^{(1+\psi_{f})\theta_{f}}m c_{t+j}Y_{t+j} $$

$$ \mathcal{Z}_{2,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{p})^{j}\lambda_{t+j}\varepsilon_{y,t+j}\left(\frac{\Gamma_{t+j}}{P_{t+j}/P_{t}}\right)^{1-(1+\psi_{f})\theta_{f}}\Theta_{t+j}^{(1+\psi_{f})\theta_{f}}Y_{t+j} $$

$$ \mathcal{Z}_{3,t}=\mathbb{E}_{t}\sum_{i=0}^{\infty}(\beta\xi_{p})^{j}\lambda_{t+j}\varepsilon_{y,t+j}\frac{\Gamma_{t+j}}{P_{t+j}\big/P_{t}}Y_{t+j} $$

writing $ P_{t+j}/P_{t} $ , the inflation factor between t and $ t+j $ , can be written equivalently $ \Pi_{i=1}^{j}\pi_{t+i} $ , and we can represent variables $ X_{1,t} $ , $ X_{2,t} $ and $ X_{3,t} $ in recursive form:

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

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

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

Writing $ \vartheta_{f,t} \equiv \int_{0}^{1} \frac{P_{t}(\iota)}{P_{t}} \mathrm{d}\iota $ can be written in recursive form:

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

we can finally write the equation $ (3.20) $ equivalently as:

$$ \frac{\psi_{f}\vartheta_{f,t}}{1+\psi_{f}}+\frac{\Theta_{t}}{1+\psi_{f}}=1 $$