while each banker exiting the business will transfer his net worth back to the household. In aggregate, the “start-up” transfer is assumed to be the constant fraction $ \xi^{F} $ of the aggregate net worth available in the banking sector, $ n_{t}^{F} $ , i.e., $ \xi^{F}n_{t}^{F} $ , and the aggregate transfer from the exiting bankers is equal to $ \gamma_{t+1}^{F}n_{t}^{F} $ . Thus, the net transfer from the banking sector to the household, $ \Xi_{t}^{F} $ , is equal to $ (1-\gamma_{t+1}^{F}-\xi^{F})n_{t}^{F} $ .

#### 3.2.2 Individual Banker's Problem

We will now describe the individual banker’s problem. The treatment here basically follows that of Gertler and Karadi (2011) and the perfect inter-bank market version of Gertler and Kiyotaki (2010). The balance sheet equation of the individual banker m is given by the following:

$$ b_{t}^{E}(m)=n_{t}^{F}(m)+b_{t}^{F}(m) $$

where $ b_{t}^{E}(m) $ is the asset of banker m, which is lent to an arbitrarily chosen entrepreneur j at period t; $ n_{t}^{F}(m) $ is the net worth of banker m; and $ b_{t}^{F}(m) $ is the liability of banker m, which is also a deposit made by the household at period t.

By receiving deposits $ b_{t}^{F}(m) $ from the household at period t, banker m pledges to pay the deposit rate of $ R_{t}/\pi_{t+1} $ in real terms during the next period. As a result of the banking business, the net worth transition for banker m at period $ t+1 $ is given by the following: $ n_{t+1}^{F}(m) = r_{t+1}^{F}(m)b_{t}^{E}(m) - r_{t+1}b_{t}^{F}(m) $ , where $ r_{t+1}^{F}(m) \equiv R_{t+1}^{F}(m)/\pi_{t+1} $ and $ r_{t+1} \equiv R_{t}/\pi_{t+1} $ . Using the balance sheet equation ([banker's balance sheet]), the net worth transition equation can be reformulated as follows:

$$ n_{t+1}^{F}(m)=\left(r_{t+1}^{F}(m)-r_{t+1}\right)b_{t}^{E}(m)+r_{t+1}n_{t}^{F}(m). $$

As shown by Gertler and Kiyotaki (2010), with the agency cost present between banker m and the depositor, the expected spread between $ r_{t+1}^{F}(m) $ and the real deposit rate $ r_{t+1} $ becomes strictly positive, i.e., $ E_{t}\left[r_{t+1}^{F}(m)-r_{t+1}\right]>0 $ . However, of course, whether the net worth of banker m increases or decreases in the next period depends on the realization of $ r_{t+1}^{F}(m) $ .

Given the above net-worth transition equation, risk-neutral banker m will maximize the net worth accumulation by maximizing the following objective function with respect to bank lending, $ b_{t}^{E}(m) $ :

$$ V_{t}^{F}(m)=E_{t}\sum_{i=0}^{\infty}\beta^{i}(1-\gamma_{t+1}^{F})\gamma_{t+1,t+1+i}^{F}\left[\left(r_{t+1+i}^{F}(m)-r_{t+1+i}\right)b_{t+i}^{E}(m)+r_{t+1+i}n_{t+i}^{F}(m)\right] $$

where $ \gamma_{t+1,t+1+i}^{F}\equiv\prod_{j=0}^{i}\gamma_{t+1+j}^{F} $ . Now, since the expected spread between the risk-adjusted bank lending rate and the deposit rate is strictly positive, it is in the interest of banker m to lend an infinite amount to an entrepreneur by accepting an infinite number of deposits from the depositor.

In order to avoid the infinite risk-taking by the banker, Gertler and Karadi (2011) and Gertler and Kiyotaki (2010) imposed a moral hazard/costly enforcement problem between the banker and the depositor. In each period, the banker has the technology to divert the fraction $ \lambda $ of his asset holding to the household and exit from the banking business. However, by doing so, the banker is forced to file bankruptcy, and a fraction $ (1 - \lambda) $ of his asset will be seized by the depositors. Thus, in order for the banker to continue business and for the depositors to safely deposit their funds to the banker, the following incentive constraint must be met each period: