*2.1. Households*

We assume a continuum of infinitely-lived households [0, 1] that are divided in two fractions: asset holders and non-asset holders. Asset holders are denoted with the fraction 1 − *λ*. They trade a risk-less one period bond and hold shares in firms. The non-asset holders are denoted by *λ*. They do not participate in asset markets and simply consume their disposable income.

#### 2.1.1. Asset Holders

These households face the following intertemporal problem:

$$\max\_{\{\mathbf{C}\_{A,t}, L\_{A,t}, B\_{A,t+1}\}} E\_t \sum\_{t=o}^{\infty} \beta^t \frac{\left(\mathbf{C}\_{A,t} L\_{A,t}^{\vartheta}\right)^{1-\sigma}}{1-\sigma} \tag{1}$$

where *β* ∈ (0, 1) denotes the discount factor, *ϕ* indicates the inverse of the Frish elasticity and *σ* is the inverse of the intertemporal elasticity of substitution. Moreover, *CA*,*t*, *LA*,*<sup>t</sup>* and *BA*,*t*+<sup>1</sup> denote, respectively, consumption, leisure and nominal bond holdings for each asset holder.

The asset holder intertemporal budget constraint is expressed by:

$$R\_t^{-1}B\_{A,t+1} + P\_l C\_{A,t} + P\_l T\_l = B\_{A,t} + (1 - \tau) \left( \mathcal{W}\_l N\_{A,t} + P\_l D\_{A,t} \right) \tag{2}$$

where *τ* is the income tax rate that is assumed to be constant and ( *Tt*) denotes the real lump-sum taxes that are adjusted to a rule specified below. Moreover, we indicate by *Rt* the gross nominal return on bonds purchased in period *t*, whereas *Pt* is the price level, *Wt* the nominal wage and *DA*,*<sup>t</sup>* the real dividend payments to households who own shares in the monopolistically-competitive firms. Finally, *NA*,*<sup>t</sup>* indicates the hours worked by the asset holder. If we assume that time endowment is normalized to one, then we have: *NA*,*<sup>t</sup>* = 1− *LA*,*t*.

#### 2.1.2. Non-Asset Holders

In each period *t*, these households solve the following intratemporal problem:

$$\max\_{\{C\_{N,t}, L\_{N,t}\}} \frac{\left(C\_{N,t} L\_{N,t}^{\mathcal{P}}\right)^{1-\sigma}}{1-\sigma} \tag{3}$$

subject to the following budget constraint:

$$P\_t \mathbb{C}\_{N,t} = (1 - \tau) \,\,\mathcal{W}\_t \mathcal{N}\_{N,t} - P\_t T\_t \tag{4}$$

where *C N*,*t* and *NN*,*<sup>t</sup>* denote consumption and hours worked by non-asset holders, respectively. Equation (4) implies that non-asset holder consumption equals their net income.

<sup>2</sup> Appendices A–C report the full derivation of the model.

*2.2. Firms*

> Firms in the final goods market are competitive. They use the following aggregation technology:

$$Y\_t = \left(\int\_0^1 Y\_t\left(i\right)^{\frac{t-1}{t}} \, di\right)^{\frac{t}{t-1}}\tag{5}$$

where *Yt* (*i*) denotes the quantity of intermediate goods *i* ∈ [0, 1], at time *t*, used as input. Moreover, *ε* is the constant elasticity of substitution.

Firms in the final goods market have the following profit maximization problem:

$$\max\_{\{Y\_{l}(i)\}} P\_{l}Y\_{l} - \int\_{0}^{1} P\_{l}\left(i\right) \mathcal{Y}\_{l}\left(i\right) di\tag{6}$$

where *Pt* is the price index for the final goods and *Pt* (*i*) denotes the price of the intermediate goods *i*. From the first order condition for *Yt* (*i*), we obtain the downward sloping demand for each intermediate input:

$$\mathcal{Y}\_t\left(i\right) = \left(\frac{P\_t\left(i\right)}{P\_t}\right)^{-\varepsilon}\mathcal{Y}\_t\tag{7}$$

This implies a price index equal to:

$$P\_t = \left[\int\_0^1 \left(P\_t\left(\dot{t}\right)\right)^{1-\varepsilon} d\dot{t}\right]^{\frac{1}{1-\varepsilon}}\tag{8}$$

The intermediate goods, *Yt* (*i*), are produced by monopolistically-competitive producers that face a production function that is linear in labour and subject to a fixed cost *F*:

$$N\_l\left(i\right) = N\_l\left(i\right) - F,\text{ if }N\_l\left(i\right) > F,\text{ otherwise, }\,\,\,\chi\_l\left(i\right) = 0\tag{9}$$

Thus, real profits for these firms correspond to:

$$O\_t\left(i\right) \equiv \left[\frac{P\_t\left(i\right)}{P\_t}\right] \mathcal{Y}\_t\left(i\right) - \left[\frac{\mathcal{W}\_t}{P\_t}\right] \mathcal{N}\_t\left(i\right)$$

We assume that intermediate goods firms face Calvo-style price-setting frictions (Calvo 1983). This implies that intermediate firms can reoptimize their prices with probability (1 − *<sup>α</sup>*), whereas with probability *α*, they keep their prices constant as in a given period. A firm *i*, resetting its price in period *t*, solves the following maximization problem:

$$\max\_{\{P\_t^\*(i)\}} E\_t \sum\_{s=0}^{\infty} \mathfrak{a}^s \Lambda\_{t, t+s} \left[ P\_t^\* \left( i \right) Y\_{t, t+s} \left( i \right) - \mathcal{W}\_{t+s} Y\_{t, t+s} \left( i \right) \right] \tag{10}$$

subject to the demand function:

$$Y\_{t+s}\left(i\right) = \left(\frac{P\_t^\*\left(i\right)}{P\_{t+s}}\right)^{-\varepsilon} Y\_{t+s} \tag{11}$$

where *P*∗*t* (*i*) is the optimal price chosen by firms resetting prices at time *t*. Finally, the expression for the price law of motion is equal to:

$$P\_t = \left[ \alpha \left( P\_{t-1} \right)^{1-\varepsilon} + \left( 1 - \alpha \right) \left( P\_t^\* \right)^{1-\varepsilon} \right]^{\frac{1}{1-\varepsilon}} \tag{12}$$
