Notice that $ p_{t}^{k}=r_{t}^{k} $ , if $ \alpha_{g}=0 $ . When public capital is wasteful, i.e. $ \alpha_{g}=0 $ , there are no entrants to capital-producing business since their profits become negative due to the fixed cost $ \Theta^{k} $ .

#### A.1.2 Households

Following the GSW framework, we consider a large household with a continuum of members represented by the unit square and indexed by a pair $(j,h)\in[0,1]\times[0,1]$. The first dimension indexed by $j\in[0,1]$ represents a differentiated skill in which a given household member is specialized. The second dimension indexed by $h\in[0,1]$ indicates member's labor disutility. Households have market power due to a differentiated skills indexed by $j\in[0,1]$, but they are assumed to face nominal wage rigidities a la Calvo, in line with Erceg et al. (2000).

Members within each household have different labor disutilities with uniformly distributed as $ h \in [0,1] $ . Given the nominal wage determined by each household, members decide to work or not taking their labor disutilities into consideration. In addition, we assume full risk sharing of consumption across members: Thanks to sharing labor income equally, members can enjoy consuming with the same level.

Then, the preference of a member h (who has a disutility h) in any household j at period t can be written by

$$ \zeta_{t}^{c}\ln\left(\tilde{C}_{j,t}-\theta\tilde{C}_{t-1}\right)-1_{t}(j,h)\zeta_{t}^{h}\chi_{t}^{h}A_{H}h^{\sigma_{h}} $$

$ \tilde{C}_{j,t} $ stands for consumption of member h in union j and $ \tilde{C}_{t} \equiv \int_{0}^{1} \tilde{C}_{j,t} d j $ stands for aggregate consumption. The term $ \theta \tilde{C}_{t-1} $ indicates (external) habits on consumption and the parameter $ \theta \in (0,1) $ depicts the importance of habit formation. $ 1_{t}(j,h) $ is the indicator function which takes a value equal to one if the member h is employed at period t, and zero otherwise. It is worth noting that the indicator function means members decide to work with fixed hours (normalized as unity) or not. $ \chi_{t}^{h} $ stands the endogenous preference shifter defined as the following equation:

$$ \begin{aligned}&\chi_{t}^{h}\equiv\frac{Z_{\chi,t}}{\tilde{C}_{t}-\theta\tilde{C}_{t-1}},\\&Z_{\chi,t}=Z_{\chi,t-1}^{1-v}\left(\tilde{C}_{t}-\theta\tilde{C}_{t-1}\right)^{v},\\ \end{aligned} $$

This preference specification leads marginal labor disutility decreases during (aggregate) consumption booms. Two structural shocks are embedded: $ \zeta_{t}^{c} $ is the preference shock and $ \zeta_{t}^{h} $ is the labor supply shock. $ A_{H} $ is the scale parameter and $ \sigma_{h} $ is the inverse Frisch elasticity.

Let $ H_{j,t} $ be defined as employment of household j. Then, aggregating the member's utility regarding h, we derive utility of union j at period t as follows:

$$ \begin{align*}\zeta_{t}^{c}\ln\left(\tilde{C}_{j,t}-\theta\tilde{C}_{t-1}\right)-\zeta_{t}^{h}\chi_{t}^{h}A_{H}\int_{0}^{H_{j,t}}h^{\sigma_{h}}dh\\=\zeta_{t}^{c}\ln\left(\tilde{C}_{j,t}-\theta\tilde{C}_{t-1}\right)-\zeta_{t}^{h}\chi_{t}^{h}A_{H}\frac{H_{j,t}^{1+\sigma_{h}}}{1+\sigma_{h}}\end{align*} $$

Thus, the preference of household j falls into the standard functional form.