j. Consequently, because $ R_{t}^{E} = R_{t}^{E}(j) $ and $ s_{t} = s_{t}(j) $ for all j, the expected risk-adjusted nominal return for banker m becomes equal across all bankers, i.e.:

$$ E_{t}\left[R_{t+1}^{F}(m)\right]=\frac{R_{t}^{E}}{s_{t}}\;\mathrm{f o r}\;\mathrm{a l l}\;m. $$

Next, we derived the law of motion of the aggregate net worth of the corporate sector. As for notation, the aggregate variable is expressed by suppressing the argument j. Aggregating over the income statement (3.3) and taking into account the entrance and exit of entrepreneurs from period t to $ t + 1 $ , we obtained the following aggregate net worth transition equation:

$$ n_{t+1}^{E}=\gamma_{t+1}^{E}\left[r_{t+1}^{k}q_{t}k_{t+1}-\frac{R_{t}^{E}}{\pi_{t+1}}b_{t}^{E}\right]+\xi^{E}n_{t}^{E} $$

where $ r_{t+1}^{k} $ is the realized gross return from capital investment at period $ t+1 $ and is defined as

$$ r_{t+1}^{k}\equiv\frac{\alpha p_{t+1}^{m c}\overline{{y}}_{t+1}/k_{t+1}+(1-\delta)q_{t+1}}{q_{t}}. $$

Here, $ \overline{y}_{t+1} $ is the average of project outcomes, $ y_{t+1}(j) $ , for all entrepreneurs. Thus, the idiosyncratic factor stemming from $ \omega_{t}(j) $ is averaged away, and $ r_{t+1}^{k} $ only reflects the aggregate factors in the economy. Using the entrepreneur's balance sheet ([balance_sheet]), the aggregate net worth transition (3.10) can be rearranged as follows:

$$ n_{t+1}^{E}=\gamma_{t+1}^{E}\left[\left(r_{t+1}^{k}-\frac{R_{t}^{E}}{\pi_{t+1}}\right)q_{t}k_{t+1}+\frac{R_{t}^{E}}{\pi_{t+1}}n_{t}^{E}\right]+\xi^{E}n_{t}^{E}. $$

Notice how the realization of $ r_{t+1}^{k} $ can affect the aggregate net worth in the next period. Ex-ante, by the rational expectation equilibrium condition (3.6), the expected return from the capital investment and the borrowing cost are equalized. As Eq.(3.11) ex-post, however, the realized return from the capital investment can exceed or fall below the borrowing cost, depending on the realizations of the aggregate shocks, and the ex-post return $ r_{t+1}^{k} $ affects the evolution of the aggregate net worth as well as the ex-ante one as described in Eq.(3.12). This is a case where the forecast error has an actual effect on the economy. Another factor that affects the evolution of the aggregate net worth is the realization of the stochastic survival rate $ \gamma_{t+1}^{E} $ . At the micro-level, $ \gamma_{t+1}^{E} $ has an interpretation of the stochastic survival rate of entrepreneur j from period t to $ t+1 $ . At the aggregate level, $ \gamma_{t+1}^{E} $ is interpreted as an exogenous shock to the aggregate net worth in the corporate sector. In our paper, we interpret it as an aggregate corporate net-worth shock.

### 3.2 Financial Friction in Banking Sector

#### 3.2.1 Entrance and Exit of Bankers

According to Gertler and Karadi (2010; 2011), there is a continuum of bankers indexed by $ m \in [0,1] $ where each banker is risk neutral and has a finite horizon. We assume that each banker faces an exogenous time-varying stochastic survival rate of $ \gamma_{t+1}^{F} $ from period t to $ t+1 $ , which is common to all bankers. By the same token, as in the corporate sector, the stochastic process of $ \gamma_{t}^{F} $ is uncorrelated with any other shocks in the economy, and its mean is equal to $ \gamma^{F} $ , i.e., $ E[\gamma_{t}^{F}] = \gamma^{F} $ .

After $ 1 - \gamma_{t+1}^{F} $ , a fraction of the bankers exit; between period t and $ t + 1 $ , exactly the same number of new bankers will enter the banking business from the household. Each banker entering the baking business will receive a ‘start-up’ transfer from the household,