3.1 Households

Households provide labor to goods producers, consume, and save. Following Gertler, et al. (Reference Gertler, Kiyotaki, Prestipino, Gertler, Kiyotaki and Prestipino2016), we assume that households provide project management services for regular banks and charge them a competitive price $f_{t\text{ }}$ per unit of project, where $f_{t\text{ }}$ is set to be equal to the marginal cost of providing the service. Households maximize the expected present value of lifetime utility as given by:

(1) \begin{equation} E_{0}\sum _{t=0}^{\infty }\beta ^{t}\left \{ \ln C_{t}-\eta \frac {L_{t}^{1+\phi }}{1+\phi }\right \} , \end{equation}

where $C_{t}$ is consumption and $L_{t}$ is labor supply. The parameter $\beta$ is the discount factor, $\eta$ denotes the relative importance of labor in utility, and $\phi$ denotes the inverse of the elasticity of labor supply.

In every period, households face the following constraints

(2) \begin{equation} C_{t}+D_{t}\leq W_{t}L_{t}+R_{t-1}D_{t-1}+\Pi _{t}+\Delta _{t}-\Upsilon _{t}, \end{equation}

where $D_{t}$ denotes deposits, $W_{t}$ stands for the wage rate, $R_{t}$ is the deposit rate, $\Pi _{t}$ denotes the dividends transferred from regular banks and capital producers, $\Delta _{t}$ is the profit obtained from project management services, and $\Upsilon _{t}$ represents a lump-sum tax. We assume that households are only able to deposit funds in regular banks since shadow banks do not have the capacity to directly raise funds from households.

3.2 Capital producers

The capital producers procure final goods as inputs, convert them into capital-producing projects and subsequently sell these projects to banks. There exist two distinct types of capital-producing technologies. The first is a risk-free method, through which one unit of project can deterministically generate one unit of capital. The second is a risky method, wherein one unit of project has a probability of $q_{t}$ of generating $\Theta _{t}(q_{t})$ units of capital and a probability of $1-q_{t}$ of yielding nothing. We assume that regular banks exclusively invest in risk-free projects, while shadow banks solely invest in risky projects.Footnote ⁴ Hence,

(3) \begin{equation} k_{r,t}^{i}=o_{r,t}^{i}; \end{equation}

and

(4) \begin{align} k_{s,t}^{\, j} &=o_{s,t}^{\, j}\Theta _{t}(q_{t}^{\, j})=o_{s,t}^{\, j}\left ( \omega _{1}-\frac {\omega _{2}}{2}q_{t}^{\, j}\right ) \text{ with probability of }q_{t}^{\, j}; \end{align}

(5) \begin{align} k_{s,t}^{\, j} &=0 \text{with a probability of }1-q_{t}^{\, j} \end{align}

where $o_{r,t}^{i}$ represents the size of risk-free project acquired by regular bank $i$ , and $o_{s,t}^{\, j}$ and $q_{t}^{\, j}$ , respectively, denote the size and risk level of risky project purchased by shadow bank $j$ ; $k_{r,t}^{i}$ stands for the capital invested by regular banks, hereafter denoted as regular capital and $k_{s,t}^{\, j}$ represents the capital invested by shadow banks, hereafter denoted as shadow capital. $\Theta _{t}(q_{t})$ is inversely related to the success probability $q_{t}$ , indicating a higher return for taking greater risks.

At each period, capital producers purchase $I_{x,t}$ ( $\{x\in r,s\}$ , where $r$ denotes the risk-free project and $s$ denotes the risky project) units of final goods and transform them into $O_{x,t}^{new}$ units of new project, incurring a capital adjustment cost $\Gamma$ ,

(6) \begin{equation} O_{x,t}^{new}=(1-\Gamma (I_{x,t}/I_{x,t-1}))I_{x,t} \end{equation}

where $\Gamma (I_{x,t}/I_{x,t-1})=\kappa \lbrack (I_{x,t}/I_{x,t-1})-1]^{2}$ , the parameter $\kappa$ governs the magnitude of capital adjustment cost. Capital producers maximize their expected discounted profits as follows:

(7) \begin{equation} \max _{I_{x,t+i}}E_{t}\sum _{i=0}^{\infty }\beta ^{i}\frac {C_{t}}{C_{t+i}}\left [ Q_{x,t+i}(1-\Gamma (I_{x,t+i}/I_{x,t+i-1}))I_{x,t+i}-I_{x,t+i}\right ] \end{equation}

where $Q_{x,t}$ represents the price of projects. Additionally, at the beginning of each period, capital producers repurchase the existing projects after depreciation, $O_{x,t}^{old}=(1-\delta )K_{x,t-1}$ , from banks at the current market price $Q_{x,t}$ and then resell them in the market. Here $K_{r,t}$ and $K_{s,t}$ denote the aggregate amounts of $k_{r,t}^{i}$ and $k_{s,t}^{\, j}$ , respectively. Thus, the total quantity of projects available for sale in period $t$ is $O_{x,t}=$ $O_{x,t}^{new}+$ $O_{x,t}^{old}$ .

3.3 Regular banks

We assume that there is a continuum of regular banks indexed by $i\in \lbrack 0,1]$ . Each regular bank combines its own net worth, $n_{t}^{i}$ , and funds raised from households, $d_{t}^{i}$ , to make investments in both regular loans, $s_{t}^{i}$ , and shadow banking products, $b_{t}^{i}$ . The regular loans are utilized to acquire risk-free capital-producing projects, $o_{r,t}^{i}$ . We assume that regular banks purchase project management services from households at a price $f_{t}$ per unit of project, as in Gertler, et al. (Reference Gertler, Kiyotaki, Prestipino, Gertler, Kiyotaki and Prestipino2016). The budget constraint for regular banks is presented as follows:

(8) \begin{equation} s_{t}^{i}+b_{t}^{i}=[Q_{r,t}+f_{t}]o_{r,t}^{i}+b_{t}^{i}=n_{t}^{i}+d_{t}^{i}, \end{equation}

Following Gertler, et al. (Reference Gertler, Kiyotaki, Prestipino, Gertler, Kiyotaki and Prestipino2016), we assume that the capacity of regular banks to raise funds from households is constrained by minimum capital requirements. One interpretation of this financial friction is that regular bankers can divert assets for personal gain rather than operating with integrity. To incentivize regular banks’ investments in shadow banking products, we assume that diverting investments in shadow banking products is more difficult than diverting investments in regular loans. Specifically, we assume that depositors believe banks can divert $\theta$ percent of their regular loans, but only $\gamma$ percent of their investments in shadow banking products, where $\gamma \lt \theta$ . To rationalize this assumption, Gertler, et al. (Reference Gertler, Kiyotaki, Prestipino, Gertler, Kiyotaki and Prestipino2016) argue that interbank loans (an instrument for shadow banking) are less idiosyncratic in nature than non-financial loans and thus more amenable to monitoring by outside depositors. However, we interpret that banking regulatory authorities have their own evaluations of shadow banking products and can align depositors’ beliefs with their own. In this regard, $\gamma$ serves as a policy instrument: regulatory authorities can set the values of $\gamma$ to adjust the amount of regulatory arbitrage allowance enjoyed by shadow banks.

With the above assumptions, any financial agreement between regular banks and depositors should satisfy the following incentive constraint:

(9) \begin{equation} v_{t}^{i}\geq \theta (Q_{r,t}+f_{t})o_{r,t}^{i}+\gamma b_{t}^{i}, \end{equation}

which indicates that the franchise value of the bank, $v_{t}^{i}$ (to be defined later), should not be less than the gain from diverting funds. Given that the franchise value of a bank, $v_{t}^{i}$ , is positively correlated with its net worth, $n_{t}^{i}$ , Eq. (9) can be regarded as a minimum capital requirement for regular banks. This requirement stipulates that $\theta$ units of regular banks’ capital (net value) are needed for each unit of investment in regular loans and $\gamma$ units of regular banks’ capital are required for each unit of investment in shadow banking. When $\gamma \lt \theta$ , meaning that investments in shadow banking enjoy regulatory privilege, regular banks can alleviate the constraint imposed by the minimum capital requirement by reallocating investments from regular loans to shadow banking products. This assumption can also be reconciled with the fact that in China, some investments in shadow banking are recorded as interbank loans on commercial banks’ balance sheets and are considered less risky assets than regular loans when calculating the risk-based capital requirement.

According to Eq. (9), the limit on the total assets that a regular bank can hold is contingent upon its net value. If regular banks accumulate sufficient net value such that they no longer require external financing, the constraint ceases to be binding. To preclude this scenario, we assume that regular banks have a probability $\sigma$ of surviving until the next period and a probability of $1-\sigma$ of exiting the market. To maintain the total number of regular banks, new regular banks will enter each period with an initial endowment $w$ , financed by a lump-sum tax. As long as there is a positive probability that the regular banks’ capacity to raise funds may be constrained at some point in the future, it will be optimal for regular banks to defer dividends and accumulate net worth until exit. We assume that regular banks transfer net worth back to households and employ a stochastic discount factor $\Lambda _{t+m}=\beta ^{m}(C_{t}/C_{t+m})$ . Consequently, the expected utility of a continuing regular bank at the end of period $t$ is given by:

(10) \begin{equation} v_{t}^{i}=E_{t}\left [ \sum _{m=1}^{\infty }\Lambda _{t+m}(1-\sigma )\sigma ^{m-1}n_{t+m}^{i}\right ] , \end{equation}

where $(1-\sigma )\sigma ^{m-1}$ is the probability of exiting at date $t+m$ . Equation (10) can be rewritten as a recursive form:

(11) \begin{equation} v_{t}^{i}=E_{t}\left \{ \Lambda _{t+1}\left [ (1-\sigma )n_{t+1}^{i}+\sigma v_{t+1}^{i}\right ] \right \} =E_{t}\left [ \Omega _{t+1}^{i}n_{t+1}^{i}\right ] , \end{equation}

where $\Omega _{t+1}^{i}=\Lambda _{t+1}\left ( 1-\sigma +\sigma \frac {v_{t+1}^{i}}{n_{t+1}^{i}}\right )$ .

At each period, regular banks earn a return rate, $R_{k,t}^{r}$ on regular loans and a return rate, $R_{b,t}$ , on shadow banking investments. Thus, the net worth of a regular bank at period $t+1$ is:

(12) \begin{equation} n_{t+1}^{i}=R_{k,t+1}^{r}(Q_{r,t}+f_{t})o_{r,t}^{i}+R_{b,t+1}b_{t}^{i}-R_{t+1}d_{t}^{i}, \end{equation}

where $R_{k,t+1}^{r}=\frac {Q_{r,t+1}(1-\delta )+Z_{t+1}}{Q_{r,t}+f_{t}}$ . At period $t$ , each unit of regular loans can be utilized to acquire $1/(Q_{r,t}+f_{t})$ units of risk-free capital-producing projects. At period $t+1$ , each unit of capital yields a rental rate of capital $Z_{t+1}$ and a resale value $Q_{r,t+1}(1-\delta )$ , where $\delta$ represents the depreciation rate.

Therefore, the optimization problem for each regular bank is to choose $(o_{r,t}^{i},d_{t}^{i},b_{t}^{i})$ in order to maximize the bank value as given by Eq. (11), subject to the budget constraints Eqs. (8) and (12), as well as the incentive constraint specified in Eq. (9).Footnote ⁵

3.4 Shadow banks

There is a continuum of shadow banks, indexed by $j\in \lbrack 0,1]$ . They do not have the ability to directly obtain financing from households. Instead, they can acquire funds from regular banks either through liability funds or equity funds. Specifically, following Abbate and Thaler (Reference Abbate and Thaler2019), we assume that each shadow bank determines its capital structure by raising liability funds, denoted as $LF_{t}$ , and equity funds, denoted as $EF_{t}$ , from regular banks. The equity ratio of shadow banks is defined as $e_{t}^{\, j}=EF_{t}^{\, j}/(LF_{t}^{\, j}+EF_{t}^{\, j})$ . This assumption regarding the capital structure is made to facilitate the explanation of the inefficient risk-taking behavior of shadow banks. Based on this assumption, we further introduce a moral hazard problem associated with debts. We assume that the risk of capital-producing projects cannot be predetermined in the debt contract. Consequently, shadow banks always have an incentive to take on more risks. This is because they can reap all the benefits arising from higher risks while only partially bearing the associated costs. We will show that the implicit guarantees provided by the government have an impact on the optimal capital structure and, thus, affect the risk-taking behavior of shadow banks.

Each shadow bank is only permitted to invest in one risky capital-producing project with a size of $o_{s,t}^{\, j}$ at a unit price of $Q_{s,t}$ . They are confronted with investment risk: with a probability of $q_{t}^{\, j}$ , the project succeeds, and the shadow bank repays the liability funds at a promised return rate of $R_{l,t}$ and pays the equity funds a state-contingent return rate of $R_{e,t}$ . In the case where the probability is $1-q_{t}^{\, j}$ , the equity funds receive nothing, while the liability funds can still obtain some compensation from the government, which is financed by a lump-sum tax. We assume that this compensation amounts to $\Psi$ percent of the total assets managed by shadow banks, $LF_{t}^{\, j}+EF_{t}^{\, j}$ , indicating that each unit of liability funds can receive an amount of compensation equal to $\Psi /(1-e_{t}^{\, j})$ . Assuming that regular banks can fully diversify the idiosyncratic risks through investment in numerous different shadow banks, the no-arbitrage condition requires that:

(13) \begin{equation} E_{t}\left ( \Lambda _{t+1}R_{b,t}\right ) =E_{t}\left \{ \Lambda _{t+1}\left [ q_{t}^{\, j}R_{l,t}+(1-q_{t}^{\, j})\frac {\Psi }{1-e_{t}^{\, j}}\right ] \right \} =E_{t}\left ( \Lambda _{t+1}q_{t}^{\, j}R_{e,t+1}\right ) , \end{equation}

which implies that the expected payoffs, when discounted by the stochastic discount factor, ought to be identical. The compensation provided to liability funds functions as a subsidy for these liability funds. This, in turn, reduces the required rate of return for liability funds, thereby rendering equity funds relatively more expensive. We refer to $R_{b,t}$ as shadow banks’ financing cost thereafter because it determines regular banks’ expected rate of return when they channel funds to shadow banks.

Now, we introduce the moral hazard problem associated with risk-taking. We assume that shadow banks determine the risk level of projects by setting $q_{t}^{\, j}$ . However, this risk level cannot be prespecified in the debt contract with liability funds providers. In other words, even if shadow banks were to promise to take a certain risk level, they could easily renege on their promise and choose a different risk level to maximize the profits for equity holders. Shadow banks can do so because liability funds providers lack the ability to observe the actual project risk. Since both liability funds and equity funds are sourced from regular banks, to preserve the information friction, we assume that each regular bank has one department responsible for equity funds and another department responsible for liability funds, with the information sharing between these two departments prohibited. Anticipating this moral hazard problem, liability funds providers must first put themselves in the shoes of shadow banks and conjecture their optimal choice of risk level, and then accordingly determine the required return rate of debts, $R_{l,t}$ .

Specifically, with this timing structure, the optimization problem of shadow banks can be characterized as a two-stage problem. At the first stage, shadow banks choose the optimal capital structure, $e_{t}^{\, j}$ , while liability funds providers determine the required return rate of debts, $R_{l,t}$ , anticipating shadow banks’ risk choice. At the second stage, shadow banks select the optimal risk level, $q_{t}^{\, j}$ , given the capital structure $e_{t}^{\, j}$ and the return rate of debts, $R_{l,t}$ , specified in the first stage.

As a result, the optimization problem of shadow banks can be solved in a backward manner. At the second stage, given the required return rate of debts, $R_{l,t}$ , and the capital structure, $e_{t}$ , shadow banks select the project risk level, $q_{t}^{\, j}$ , in order to maximize the expected profits of equity funds providers:

(14) \begin{equation} \max _{q_{t}^{\, j}}Q_{s,t}o_{s,t}^{\, j}E_{t}\left \{ \Lambda _{t+1}\left [ q_{t}^{\, j}\left ( \omega _{1}-\frac {\omega _{2}}{2}q_{t}^{\, j}\right ) R_{k,t+1}^{s}-q_{t}^{\, j}R_{l,t}(1-e_{t}^{\, j})\right ] \right \} , \end{equation}

where $R_{k,t+1}^{s}=\frac {Z_{t+1}+(1-\delta )Q_{s,t+1}}{Q_{s,t}}$ is the rate of return on shadow capital. The first order condition with respect to $q_{t}^{\, j}$ is:

(15) \begin{equation} q_{t}^{\, j}=\frac {\omega _{1}E_{t}(\Lambda _{t+1}R_{k,t+1}^{s})-E_{t}(R_{l,t}\Lambda _{t+1})(1-e_{t}^{\, j})}{\omega _{2}E_{t}(\Lambda _{t+1}R_{k,t+1}^{s})}. \end{equation}

It is straightforward to demonstrate that $\frac {\partial q_{t}^{\, j}}{\partial e_{t}^{\, j}}\gt 0$ and $\frac {\partial q_{t}^{\, j}}{\partial R_{k,t+1}^{s}}\gt 0$ . This indicates that a higher leverage (lower $e_{t}^{\, j}$ ) and a lower rate of return on shadow capital induce shadow banks to take greater risks (lower $q_{t}^{\, j}$ ).

At the first stage, shadow banks determine the capital structure, $e_{t}^{\, j}$ , with the aim of maximizing the expected excess profits (profits net of the opportunity costs of equity), anticipating that $q_{t}^{\, j}$ will be selected at the second stage as a function of $e_{t}^{\, j}$ . Note that regular banks also anticipate the second-stage risk choice and price equity funds and liability funds accordingly. The shadow banks’ second-stage problem is therefore:

(16) \begin{equation} \max _{e_{t}^{\, j}}Q_{s,t}o_{s,t}^{\, j}E_{t}\left \{ \Lambda _{t+1}\left [ q_{t}^{\, j}\left ( \omega _{1}-\frac {\omega _{2}}{2}q_{t}^{\, j}\right ) R_{k,t+1}^{s}-q_{t}^{\, j}R_{l,t}(1-e_{t}^{\, j})-q_{t}^{\, j}R_{e,t+1}e_{t}^{\, j}\right ] \right \} , \end{equation}

subject to the constraints implied in Eq. (13),

(17) \begin{equation} E_{t}(\Lambda _{t+1}R_{l,t})=\frac {E_{t}\left [ \Lambda _{t+1}\left ( R_{b,t}-\frac {1-q_{t}^{\, j}}{1-e_{t}^{\, j}}\Psi \right ) \right ] }{q_{t}^{\, j}}\text{ and }E_{t}(\Lambda _{t+1}R_{e,t+1})=\frac {E_{t}\left ( \Lambda _{t+1}R_{b,t}\right ) }{q_{t}^{\, j}}. \end{equation}

We can substitute Eq. (17) into Eq. (16) and reformulate the shadow banks’ first stage problem as:

(18) \begin{equation} \max _{e_{t}^{\, j}}Q_{s,t}o_{s,t}^{\, j}E_{t}\left \{ \Lambda _{t+1}\left [ q_{t}^{\, j}\left ( \omega _{1}-\frac {\omega _{2}}{2}q_{t}^{\, j}\right ) R_{k,t+1}^{s}-R_{b,t}+(1-q_{t}^{\, j})\Psi \right ] \right \} . \end{equation}

The first order condition with respect to $e_{t}^{\, j}$ is given by

(19) \begin{equation} E_{t}\left [ \Lambda _{t+1}(\omega _{1}R_{k,t+1}^{s}-\omega _{2}q_{tt}^{\, j}R_{k,t+1}^{s}-\Psi )\right ] \frac {\partial q_{t}^{\, j}}{\partial e_{t}^{\, j}}=0. \end{equation}

Equation (19) demonstrates that shadow banks are faced with the following trade-off when determining the capital structure: increasing the equity ratio could incentivize shadow banks to choose projects with higher probability of success (higher $q_{t}^{\, j}$ ); a higher probability of success leads to greater expected return from successful projects as long as $q_{t}^{\, j}\lt$ $\omega _{1}/\omega _{2}$ , but results in less compensation from governments once projects fail. The presence of implicit guarantees from governments induces shadow banks to choose higher leverage (lower $e_{t}^{\, j}$ ) even though they anticipate that a lower equity ratio leads to greater risk taking. Without the implicit guarantees from governments ( $\Psi =0$ ), the optimal condition regarding the choice of equity ratio Eq. (19) requires that $q_{t}^{\, j}=$ $\omega _{1}/\omega _{2}$ , which can be achieved when shadow banks finance only through equity funds ( $e_{t}^{\, j}=1$ , as shown by Eq. (15)).

Finally, the assumption of free entry into the shadow banking market ensures that the expected excess profit is zero:

(20) \begin{equation} E_{t}\left \{ \Lambda _{t+1}\left [ q_{t}^{\, j}\left ( \omega _{1}-\frac {\omega _{2}}{2}q_{t}^{\, j}\right ) R_{k,t+1}^{s}-R_{b,t}+(1-q_{t}^{\, j})\Psi \right ] \right \} =0. \end{equation}

Equation (20) indicates that a decline in the shadow banking financing cost $R_{b,t}$ or an increase in the level of government guarantees $\Psi$ attracts more funds to shadow banking, leading to an increase in the shadow capital and therefore a decrease in the rate of return on shadow capital $R_{k,t}^{s}$ .

We then combine Eq. (20) with Eqs. (15) and (19) to solve for the equity ratio $e_{t}^{\, j}$ and the project risk level $q_{t}^{\, j}$ :

(21) \begin{align} e_{t}^{\, j} &=\frac {\widetilde {R}_{b,t}-\widetilde {\Psi }}{\widetilde {R}_{b,t}}, \end{align}

(22) \begin{align} q_{t}^{\, j} &=1-\frac {\widetilde {R}_{b,t}}{\widetilde {\Psi }}+\frac {\sqrt {\omega _{2}(\widetilde {R}_{b,t}-\widetilde {\Psi })\left [ 2\omega _{1}\widetilde {\Psi }+\omega _{2}(\widetilde {R}_{b,t}-\widetilde {\Psi })\right ] }}{\omega _{2}\widetilde {\Psi }}, \end{align}

where $\widetilde {R}_{b,t}=E_{t}(\Lambda _{t+1}R_{b,t})$ and $\widetilde {\Psi }=E_{t}(\Lambda _{t+1}\Psi )$ .

With all the aforementioned results, we demonstrate that a higher ratio of government guarantees in the event of project failure reduces the required rate of return for liability funds, $R_{l,t}$ , making liability funds less costly than equity funds. This encourages the use of higher leverage, exacerbating the moral hazard problem and prompting shadow banks to take on greater risks. In addition, we show that both the equity ratio $e_{t}^{\, j}$ and the probability of success $q_{t}^{\, j}$ decrease as shadow banks’ financing cost $R_{b,t}$ declines. According to the zero-profit condition Eq. (20), a lower financing cost for shadow banks leads to more funds flowing to shadow banks and a decline in the rate of return on shadow capital, $R_{k,t}^{s}$ , thereby reducing the benefits of safer projects. This, in turn, induces shadow banks to choose riskier projects. Furthermore, as shown by the no-arbitrage condition Eq. (13), a decline in $R_{b,t}$ causes $R_{l,t}$ to decrease more than $R_{e,t}$ , making liability funds even less expensive relative to equity funds, which also induces a higher level of leverage.

3.5 Goods producers

Goods producers employ labor from households and rent capital from both regular banks and shadow banks. The production function is as follows:

(23) \begin{equation} Y_{t}=A_{t}L_{t}^{1-\alpha }K_{t-1}^{\alpha }. \end{equation}

where $K_{t}=K_{r,t}+K_{s,t}$ , $K_{r,t}$ represents the aggregate amount of regular capital and $K_{s,t}$ stands for the aggregate amount of shadow capital. We assume that regular capital and shadow capital are perfect substitutes in the process of goods production. Given the capital rental rate $Z_{t}$ and the wage rate $W_{t}$ , goods producers determine the demand for capital and labor so as to maximize their profits.

3.6 Aggregation and equilibrium

It is straightforward to show that $o_{r,t}^{i}/n_{t}^{i}$ , $d_{t}^{i}/n_{t}^{i}$ , and $b_{t}^{i}/n_{t}^{i}$ are identical for all regular banks provided that the marginal management cost of the risk-free project, $f_{t}$ , is independent of individual bank-specific factors. Here, in line with Gertler, et al. (Reference Gertler, Kiyotaki, Prestipino, Gertler, Kiyotaki and Prestipino2016), we assume that the marginal cost of project management service increases with the total project size, namely, $f_{t}=\varepsilon O_{r,t}$ ( $\varepsilon \gt 0$ ), where $O_{r,t}=\int _{0}^{1}o_{r,t}^{i}\,\text{d}i$ represents the total size of risk-free projects. Consequently, the total management cost is given by $F_{t}=\frac {1}{2}\varepsilon O_{r,t}^{2}$ . Under this condition, we can aggregate across banks in order to derive relations among the total net worth of all regular banks, denoted as $N_{t}=\int _{0}^{1}n_{t}^{i}\,\text{d}i$ , the sum of all household deposits, $D_{t}=\int _{0}^{1}d_{t}^{i}\,\text{d}i$ , the total investments in shadow banks, $B_{t}=\int _{0}^{1}b_{t}^{i}\,\text{d}i$ , and the total investments in risk-free projects $O_{r,t}$ :

(24) \begin{equation} (Q_{r,t}+f_{t})O_{r,t}^{{}}+B_{t}^{{}}=N_{t}^{{}}+D_{t}^{{}} \end{equation}

Meanwhile, the effective leverage $\Phi _{t}^{i}$ and the franchise value per unit of net worth $v_{t}^{i}/n_{t}^{i}$ are the same across all regular banks:

(25) \begin{align} \Phi _{t}^{i} &=\Phi _{t}=\frac {(Q_{r,t}+f_{t})O_{r,t}^{{}}+(\gamma /\theta )B_{t}^{{}}}{N_{t}^{{}}}, \end{align}

(26) \begin{align} \frac {v_{t}^{i}}{n_{t}^{i}} &=\frac {v_{t}}{n_{t}}=E_{t}\left \{ \Omega _{t+1}\left [ (R_{k,t+1}^{r}-R_{t+1})\Phi _{t}^{i}+R_{t+1}\right ] \right \} . \end{align}

The aggregate net worth of regular banks, denoted as $N_{t}$ , comprises two components. The first component is the accumulated net worth of banks that operated at time $t-1$ and survive until time $t$ . These surviving banks account for $\sigma$ percent of the total number of regular banks. Each of these surviving banks carries forward a net worth of $\left [ (R_{k,t}^{r}-R_{t})\Phi _{t-1}^{i}+R_{t}\right ] n_{t-1}^{i}$ . The second component is the endowment of entering banks, which account for $1-\sigma$ percent of total number of regular banks. Each entering bank brings forward an initial endowment of $w$ . Thus, the evolution of $N_{t}$ given by:

(27) \begin{equation} N_{t}=\sigma \left [ (R_{k,t}^{r}-R_{t})\Phi _{t-1}^{{}}+R_{t}\right ] N_{t-1}^{{}}+(1-\sigma )w \end{equation}

The aggregate capital supplied by regular banks is simply equal to the total size of all risk-free projects: $K_{r,t}=O_{r,t}.$ In equilibrium, all shadow banks choose the same capital structure and the same risk level, namely, $e_{t}^{\, j}=e_{t}$ and $q_{t}^{\, j}=q_{t}$ . Let $O_{s,t}=\int _{0}^{1}o_{s,t}^{\, j}dj$ denote the total size of risk projects. Then the aggregate shadow capital is given by $K_{s,t}^{{}}=\int _{0}^{1}o_{s,t}^{\, j}q_{t}^{\, j}\left ( \omega _{1}-\frac {\omega _{2}}{2}q_{t}^{\, j}\right )dj = O_{s,t}q_{t}^{{}}\left ( \omega _{1}-\frac {\omega _{2}}{2}q_{t}^{{}}\right ).$

Finally, the goods market clears when the output is equal to the sum of the consumption of households, the investments in risk-free and risky projects, and the management costs of regular bank projects:

(28) \begin{equation} Y_{t}=C_{t}+I_{r,t}+I_{s,t}+F_{t}. \end{equation}

The inter-bank funds market clears when the total demand for liability funds and equity funds by shadow banks is equal to the supply of funds from regular banks:

\begin{equation*} \int _{0}^{1}(LF_{t}^{\, j}+EF_{t}^{\, j})dj=LF_{t}+EF_{t}=B_{t}. \end{equation*}

6.1 The optimum with social planner

We assume that the social planner can optimally allocate resources without being concerned about the financial constraints. The planner maximizes social welfare by optimally choosing consumptions, hours worked, and investments in capital. There are two ways to produce capital: the first corresponds to the technique of regular banks, which involves management costs, while the second corresponds to the technique of shadow banks, which is characterized by a project-to-capital conversion rate of $t$ . We assume that the social planner always chooses the optimal risk level $q^{\ast }$ $=0.99$ to maximize the expected project-to-capital conversion rate, and thus, $t^{\ast }=q_{{}}^{\ast }\left ( \omega _{1}-\frac {\omega _{2}}{2}q_{{}}^{\ast }\right ) =0.953$ . Additionally, to simplify the analysis, we ignore the capital adjustment cost.

Thus, the social planner chooses $C_{t}$ , $K_{t}$ , $K_{s,t}$ , and $L_{t}$ to maximize the household’s expected present value of lifetime utility:

(29) \begin{equation} E_{0}\sum _{t=0}^{\infty }\beta ^{t}\left \{ \ln C_{t}-\eta \frac {L_{t}^{1+\phi }}{1+\phi }\right \} , \end{equation}

subject to the following resource constraints:

\begin{eqnarray*} C_{t}+I_{r,t}+I_{s,t}+F_{t} &\leq &A_{t}L_{t}^{1-\alpha }K_{t-1}^{\alpha }\text{,} \\ K_{r,t} &=&I_{r,t}+(1-\delta )K_{r,t-1}\text{,} \\ K_{s,t} &=&t^{\ast }\left [ I_{s,t}+(1-\delta )K_{s,t-1}\right ] \text{,} \\ K_{t} &=&K_{r,t}+K_{s,t}, \end{eqnarray*}

where $K_{r,t}$ and $K_{s,t}$ are capital produced using the techniques of regular banks and shadow banks, respectively. $F_{t}=\frac {1}{2}\varepsilon K_{r,t}^{2}$ is the management cost of producing regular capital.

The first order conditions with respect to $L_{t}$ , $K_{t}$ , and $K_{s,t}$ are as follows:

(30) \begin{align} (1-\alpha )A_{t}L_{t}^{-\alpha }K_{t-1}^{\alpha }/C_{t} &=\eta L_{t}^{\phi }, \end{align}

(31) \begin{align} 1+\varepsilon K_{r,t} &=\beta E_{t}\left \{ \frac {C_{t}}{C_{t+1}}\left [ \alpha A_{t+1}L_{t+1}^{1-\alpha }K_{t}^{\alpha -1}+1-\delta \right ] \right \} , \end{align}

(32) \begin{align} 1/t^{\ast } &=1+\varepsilon K_{r,t}. \end{align}

Equation (30) is the optimality condition for labor. Equation (31) describes the optimality conditions for the total capital, equating the marginal benefit and cost of acquiring an additional unit of capital. Equation (32) indicates that the marginal cost of producing additional regular capital and shadow capital should be the same. Note that the marginal cost of producing regular capital depends on the project management cost, while the marginal cost of producing shadow capital depends on the project-to-capital conversion rate $t$ .

6.2 Obtaining the first best in decentralized economy

We now examine whether the first-best equilibrium can be achieved in the decentralized economy. Specifically, the first-best equilibrium necessarily requires that Eqs. (30)–(32) are satisfied. Since there are no frictions in the labor market, Eq. (30) can always be achieved in the decentralized economy. Thus, we focus on the two optimality conditions governing capital accumulation, namely, Eqs. (31) and (32). The optimality condition Eq. (31) requires that the total capital reach a level such that the marginal benefit of investing an additional unit of capital equals the marginal cost. The optimality condition Eq. (32) requires that resources are appropriately allocated to regular banks and shadow banks so that the marginal cost of acquiring an additional unit of capital is equal regardless of whether it is financed by regular banks or shadow banks. In the online Appendix C, we prove that Eqs. (31) and (32) can only be achieved when the constraint on regular banks’ capacity for asset holding is slack and the shadow banking insurance coverage ratio is zero. In other words, as long as the constraint on regular banks’ capacity for asset holding is binding, it is impossible to satisfy the two optimality conditions simultaneously in the decentralized economy. This is because the capital requirements impose a constraint on regular banks’ capacity to absorb deposits and thus prevents the total capital from reaching the optimal level required by Eq. (31).

To further illustrate the role of the capital requirements, we consider a special case of the baseline model with no regulatory arbitrage ( $\gamma$ equals $\theta$ ) and no implicit government guarantees ( $\Psi$ equals zero). In this special case, according to Eqs. (21) and (15), shadow banks will only use equity funds and choose the optimal level of risk, $q=q^{\ast }=0.99$ . Under this circumstance, the only friction preventing us from reaching the first-best resource allocation is the constraint imposed on the regular banks’ capacity to absorb deposits and expand asset holdings, measured by $\theta$ . Figure 5 shows how the key variables depend upon $\theta$ in the steady state. First of all, as shown by the figure, when $\theta$ approaches to zero, the constraint limiting the regular bank’s capacity to absorb deposits and expand asset holdings is relaxed. All the key variables and household utility converges to the first-best level. Meanwhile, as shown in Figure 5(f), both the rate of return on regular capital $R_{k}^{r}$ and the rate of return on shadow banking investments $R_{b}$ approach to the deposit rate $R$ ( $R=1/\beta =1.01$ ),Footnote ⁸ which is a criterion of obtaining the first best. As $\theta$ increases, the constraint on regular banks’ capacity to absorb deposits and expand asset holdings is tightened, causing a decline in the total capital (Figure 5(d)) and an increase in the rate of return $R_{k}^{r}$ and $R_{b}$ (Figure 5(f)), leading to a drop of the household’s utility (Figure 5(a)).

Particularly, we can identify a critical value $\theta ^{\ast }$ denoted by triangles in Figure 5. When $\theta \geq \theta ^{\ast }$ , the marginal cost of producing regular capital is lower than that of shadow capital (Figure 5(e)). Consequently, all capital will be intermediated by regular banks. As $\theta$ decreases, the increasing amount of regular capital drives up the management cost, eroding the regular banks’ cost advantage. When $\theta \lt$ $\theta ^{\ast }$ , the most cost-effective approach is to maintain regular capital at $K_{r}(\theta ^{\ast })$ and let shadow banks finance the remainder, so that the marginal costs of producing regular capital and shadow capital are equal.

Figure 5.

The impact of $ \theta$ on key variables and social welfare.

6.3 Finding the second best

Taking as given the constraint on regular banks’ capacity to absorb deposits and expand asset holdings, we are left with two policy instruments to utilize. The first policy instrument pertains to the degree of regulatory arbitrage allowance, governed by $\gamma$ , and the second instrument pertains to the percentage of shadow banking losses covered by government implicit guarantees, governed by $\Psi$ . In the following experiments, we focus on identifying the second best in the decentralized economy by varying these policy instruments, considering that the regular banks’ capacity to absorb deposits and expand asset holdings is always constrained by their net worth. We commence by examining the impact of $\gamma$ on resource allocations and welfare, holding $\Psi$ constant, and then proceed to explore the impact of $\Psi$ , holding $\gamma$ constant. In this manner, we investigate under the assumption that different governmental authorities make decisions independently, taking the choices made by others as given. For instance, in the context of China, we could interpret this as the banking regulatory authority governs $\gamma$ , while local governments determine the value of $\Psi$ . The coordination of these two policy instruments will be studied in the next section.

6.3.1 The impact of $ \gamma$ given $\Psi$

We first explore the impact of the degree of regulatory arbitrage allowance, $\gamma$ , on resource allocation and social welfare. In doing so, we assume that the value of $\gamma$ is less than $\theta$ so that investments in shadow banking enjoy regulatory privilege. Figure 6 shows how the key variables depend on $\gamma$ in the steady state. A drop in $\gamma$ means that investments in shadow banking can relax the constraint on regular banks’ capacity for asset holdings, thereby boosting the incentive to supply funds to shadow banks.

The solid lines in Figure 6 represent the scenario where there are no government guarantees, i.e., $\Psi =0$ . As shown in Figure 6(f), a decrease in $\gamma$ reduces the required rate of return on shadow banking investments for regular banks, $R_{b}$ , and consequently lowers the financing cost of shadow banks. However, there exists a critical level of $\gamma ^{\ast \text{ }}=0.39$ (triangles in Figure 6(b)); beyond this point, the profit of shadow banks becomes negative, and thus no capital will be intermediated by shadow banks. Only when $\gamma \lt$ $\gamma ^{\ast \text{ }}$ and the financing cost of shadow banks drops to a sufficiently low level can shadow banks break even and begin to finance shadow capital. As $\gamma$ continues to decline, more capital is intermediated by shadow banks, partially substituting for the capital originally financed by regular banks. Meanwhile, regular banks attract more deposits and expand their total assets, resulting in an increase in the total capital.

Figure 6.

The impact of $ \gamma$ on key variables and social welfare.

The household’s steady-state utility continuously increases as $\gamma$ declines, attaining its highest level when $\gamma$ approaches zero. However, even at this peak level, the household’s steady-state utility remains 0.4% lower than the first-best level. To understand how the household’s utility changes with respect to $\gamma$ , we derive the total derivative of the steady-state utility $U$ with respect to $\gamma$ as follows (see online Appendix D for details):

(33) \begin{equation} \frac {dU}{d\gamma }=\frac {1}{C}\left \{ \Omega _{K}\frac {\partial K}{\partial \gamma }-\Omega _{K_{s}}\frac {\partial K_{s}}{\partial \gamma }+\Omega _{q}\frac {\partial q}{\partial \gamma }\right \} \end{equation}

where $\Omega _{K}=(Z+1-\delta )-(1+\varepsilon K_{r})$ , $\Omega _{K_{s}}=1/t-(1+\varepsilon K_{r})$ , and $\Omega _{q}=\frac {K_{s}}{t^{2}}\left ( \omega _{1}-\omega _{2}q\right )$ . The term $\Omega _{K}$ represents the marginal net benefit of an additional unit of total capital when financed by regular banks. $\Omega _{K}$ is positive provided that $R_{k}^{r}=\frac {Z+1-\delta }{1+\varepsilon K_{r}}\gt 1$ . The term $\Omega _{K_{s}}$ represents the marginal extra cost of financing through shadow banks. This cost is positive when the marginal cost of shadow capital $1/t$ exceeds that of regular capital $1+\varepsilon K_{r}$ . Meanwhile, $\Omega _{q}$ represents the marginal benefit of reducing the shadow banks’ project risk level, and it is always positive. Consequently, we can decompose the impact of $\gamma$ on the steady-state household utility $U$ into three components. The first component, $\Omega _{K}\frac {\partial K}{\partial \gamma }$ , is termed the total capital effect. The second component, $\left ( -\Omega _{K_{s}}\frac {\partial K_{s}}{\partial \gamma }\right )$ , is labeled the financing structure effect. And the third component, $\Omega _{q}\frac {\partial q}{\partial \gamma }$ , is referred to as the risk-taking effect. As shown in Figure 6(d), an increase in $\gamma$ leads to a decline in the total capital, and thus the total capital effect of a rising $\gamma$ is negative. Figure 6(b) shows that an increase in $\gamma$ results in a decrease in the capital intermediated by shadow banks, and so $\frac {\partial K_{s}}{\partial \gamma }\lt 0$ . This means that the financing structure effect of a rising $\gamma$ is positive. As shown in Figure 6(g), changes in $\gamma$ cause negligible variations in the shadow bank project risk, and consequently, the risk-taking effect is close to zero. Overall, in Figure 6(h), we demonstrate that the marginal utility gain $\frac {dU}{d\gamma }$ is negative for $\gamma \lt \gamma ^{\ast }$ . On one hand, a decline in $\gamma$ increases the total capital amount by relaxing the constraint on regular banks’ capacity for asset holdings, which improves social welfare. On the other hand, as $\gamma$ declines, more capital is financed through the more costly shadow banking, having a negative impact on social welfare. When the total capital effect outweighs the financing structure effect, social welfare improves as $\gamma$ decreases. This explains why a decline in $\gamma$ (for $\gamma \lt \gamma ^{\ast }$ ) leads to an increase in the household’s steady-state utility level.

We then examine scenarios where the government covers part of the losses of shadow banking products. The dashed lines in Figure 6 illustrate the resource allocation and welfare changes for various values of $\gamma$ when $\Psi$ is set to the calibrated value of 0.122, while the dash-dotted lines represent the situation where $\Psi$ is set to 0.14. Government compensation leads to an increase in shadow capital, a decrease in regular capital, and an increase in total capital for any level of $\gamma$ . Meanwhile, it prompts shadow banks to take on more risks, thereby reducing the project-to-capital conversion rate and resulting in a higher extra cost of shadow capital, $\Omega _{K_{s}}$ . Regarding the welfare implications, as shown by the dashed line in Figure 6(a), the utility reaches its maximum at $\gamma =0.5$ when $\Psi = 0.122$ . However, for $\Psi = 0.14$ , the utility gain from the financing structure effect consistently outweighs the utility loss due to the total capital effect as $\gamma$ increases. The dash-dotted line in Figure 6(a) indicates that the utility is maximized only when $\gamma$ equals $\theta$ , at which point regulatory arbitrage disappears. Overall, these experimental results suggest that the presence of government compensation modifies the trade-off between the total capital effect and the financing structure effect.

6.3.2 The impact of $\Psi$ given $ \gamma$

Next, we analyze the impact of implicit government guarantees, denoted as $\Psi$ , on social welfare, while holding the value of $\gamma$ constant. The government guarantees serve as insurance for liability funds invested in shadow banking projects, facilitating the shadow banks’ ability to break even.

Figure 7.

The impact of $\Psi$ on key variables and social welfare.

The solid lines in Figure 7 illustrate how the key variables vary with $\Psi$ in the steady state when $\gamma$ is set to the baseline value of 0.4663. As depicted in Figure 7(b), when the insurance coverage ratio exceeds a critical value $\Psi ^{\ast }$ (marked with triangles in Figure 7), shadow banking projects start to generate non-negative profits, thereby enabling financing through shadow banks. As $\Psi$ continues to rise, not only does a larger proportion of the total capital get financed by shadow banks, but also the total capital also increases. More significantly, an increase in $\Psi$ leads to a substantial decline in the success probability of shadow banking projects (Figure 7(g)), which rapidly drives up the marginal cost of shadow capital (Figure 7(e)).

Regarding welfare, the solid line in Figure 7(a) shows that the utility attains its maximum level at $\Psi ^{\ast \ast }=0.105$ , which remains 0.8% lower than the first-best level. To understand this phenomenon, we derive the total derivative of the steady-state utility $U$ with respect to $\Psi$ (the derivation is similar to that of $\frac {dU}{d\gamma }$ ):

(34) \begin{equation} \frac {dU}{d\Psi }=\frac {1}{C}\left \{ \Omega _{K}\frac {\partial K}{\partial \Psi }-\Omega _{K_{s}}\frac {\partial K_{s}}{\partial \Psi }+\Omega _{q}\frac {\partial q}{\partial \Psi }\right \} \end{equation}

where the definitions of $\Omega _{K}$ , $\Omega _{K_{s}}$ , and $\Omega _{q}$ are the same as in Eq. (33). The impact of $\Psi$ on the steady-state utility is decomposed into three parts once again: the total capital effect, $\Omega _{K}\frac {\partial K}{\partial \Psi }$ ; the financing structure effect, $\left ( -\Omega _{K_{s}}\frac {\partial K_{s}}{\partial \Psi }\right )$ ; and the risk-taking effect, $\Omega _{q}\frac {\partial q}{\partial \Psi }$ . First, $\Omega _{K}$ is positive and when $\Psi \gt \Psi ^{\ast }$ , an increase in $\Psi$ results in an increase in the total capital. Consequently, the total capital effect of a rising $\Psi$ is positive. Second, as shown in Figure 7(e), the marginal extra cost of shadow capital, $\Omega _{K_{s}}$ , is positive. Meanwhile, Figure 7(e) also shows that an increase in $\Psi$ leads to a rise in the capital intermediated by shadow banks. As a result, the financing structure effect of a rising $\Psi$ is negative. Third, $\Omega _{q}$ is always positive. As indicated by Figure 7(g), an increase in $\Psi$ reduces the success probability of shadow banking projects. Thus, the risk-taking effect is negative. The optimal policy choice of the insurance coverage ratio, $\Psi$ , hinges on the trade-off among the three aforementioned effects. With regard to the total capital effect, an increase in $\Psi$ offers greater incentives to finance capital through shadow banking. This, in turn, drives the total capital closer to the socially optimal level, thereby enhancing social welfare. However, an increase in $\Psi$ also exerts negative impacts on social welfare via the risk-taking effect and the financing structure effect. As $\Psi$ rises, shadow banks exploit the lack of commitment regarding their risk choices and thus take on excessive risks. Consequently, the capital intermediated by shadow banks becomes even more costly.

For the baseline value of $\gamma =0.4663$ , Figure 7(a) indicates that social welfare reaches its maximum at $\Psi ^{\ast \ast }=0.105$ . This can be explained by referring to Figure 6(h). The marginal utility gain, $\frac {dU}{d\Psi }$ , is positive when $\Psi \lt \Psi ^{\ast \ast }$ , implying that the total capital effect dominates the financing structure effect and the risk-taking effect. Conversely, when $\Psi \gt \Psi ^{\ast \ast }$ , the marginal utility gain, $\frac {dU}{d\Psi }$ , becomes negative, implying that the financing structure effect and the risk-taking effect assume greater significance than the total capital effect. Notably and intriguingly, the value of $\Psi =0.122$ in the baseline model surpasses this optimal level. Although a higher $\Psi$ yields a total capital amount closer to the first-best level, this advantage cannot compensate for the cost associated with the greater risk-taking of shadow banks and the higher marginal cost of capital intermediated by them. Reducing the degree of implicit guarantees, $\Psi$ , from 0.122 to the optimal level would decrease the ratio of shadow capital to regular capital from 24.2% to 15.5%, resulting in a 0.1% improvement in welfare.

Finally, we conduct experiments to demonstrate that the optimal choices of $\Psi$ are dependent on the value of $\gamma$ . As indicated by the dashed line in Figure 6(a), when $\gamma =0$ , the utility reaches its maximum only when the government compensation rate is zero. In this scenario ( $\gamma =0$ ), with regular banks already having a strong incentive to channel funds to shadow banking investments through regulatory arbitrage, total capital is not scarce. The adverse effect of inducing shadow banks to take more risks due to government guarantees dominates, and thus any provision of such insurance would worsen social welfare. In another extreme case, when we eliminate the possibility of regulatory arbitrage by setting $\gamma =\theta$ , as shown by the dash-dotted lines in Figure 6, regular banks only have sufficient incentive to invest in shadow banking projects when the government provides a higher loss guarantee compared to the baseline case.

6.4 Coordination of two policy instruments

The above experiments imply that the optimal choices of $\gamma$ and $\Psi$ are interdependent. In this section, we further investigate the coordination between these two policy instruments by presenting all possible outcomes across different levels of $\gamma$ and $\Psi$ . Figure 8 presents these results regarding social welfare and total capital. Specifically, in the upper panel of Figure 8, we graph the utility values as elevations above a grid in the $\gamma -\Psi$ plane; whereas in the middle and lower panels, we depict the levels of utility and total capital under various combinations of $\gamma$ and $\Psi$ using distinct colors, with warmer hues indicating higher levels.

The plots in Figure 8 show that the maximum welfare level can be obtained when shadow banks have the largest amount of regulatory arbitrage ( $\gamma =0$ ) and simultaneously, the government provides no guarantees for shadow banking products ( $\Psi =0$ ). However, it is realistic to suspect whether this cooperative outcome could be achieved in the context of China. For instance, if, as interpreted, the banking regulation authority sets the capital requirement, and on the other hand, local governments provide guarantees for shadow banking finance, it is legitimate to expect that local governments have strong incentives to stimulate capital investment, which is a key engine for economic growth. In this case, as shown by the lower panel of Figure 8, local governments always have incentives to increase $\Psi$ given any value of $\gamma$ , since doing so can expand total capital. With local governments’ distorted incentives, in order to maximize social welfare, the optimal response of the banking regulation authority is to adjust the level of regulatory arbitrage allowance by setting $\gamma$ . This non-cooperative equilibrium leads to a lower welfare level than the cooperative equilibrium. For instance, when the level of government guarantees is capped at 15% due to a fiscal constraint, the non-cooperative equilibrium (northeast corner in the middle panel of Figure 8) generates a level of welfare 0.83% lower than that at the cooperative equilibrium.

Figure 8.

Welfare and total capital under various combination of $ \gamma$ and $\Psi$ .

6.5 The optimal policies under different $ \theta$

In the model, the parameter $\theta$ measures the constraint that limits the capacity of regular banks to absorb deposits and expand asset holdings. In reality, the level of this constraint depends on the development of the financial sector and government intervention. Thus, it is reasonable to assume that different countries and/or economies have different values of $\theta$ . In order to generalize some implications for cross-country comparisons and investigate the sensitivity of optimal policies to $\theta$ , in this section, we conduct experiments to explore the impacts of the parameter $\theta$ .

In Figure 9, we plot resource allocations and corresponding welfare levels for a country with a relatively lower bank capital requirement ( $\theta =0.5$ ) and a country with a relatively higher bank capital requirement ( $\theta =0.8$ ), along with the baseline case. As shown in the figure, a higher bank capital requirement for regular loans provides banks with greater incentives to channel funds through shadow banking (Figure 9-b) and results in a lower required return for shadow banking investments (Figure 9-b). Specifically, the lower financing cost makes it easier for shadow banks to break even. Consequently, shadow banking emerges in countries with higher capital requirements for regular loans, even when the government provides no guarantees for losses ( $\Psi =0$ ) from shadow banking investments, as indicated by the solid line in Figure 9(b).

Regarding the roles of government guarantees, the figure illustrates that they can narrow the capital gap (Figure 9-d) and the welfare gap (Figure 9-a) between countries with a high level of constraint and those with a low level of constraint. It is interesting to note that the total capital in a country with a higher banking capital requirement can exceed that in a country with a lower requirement when the level of government guarantees exceeds a threshold, as shown in Figure 9(d). However, even in this case, its social welfare remains lower, as shown in Figure 9(a), because more capital is financed through costly and risky shadow banking. Moreover, we also demonstrate that countries facing more constraints can benefit more from government guarantees. In particular, compared with the case without any support, government guarantees can enhance welfare by up to 0.34% in a country with a higher banking capital requirement ( $\theta =0.8$ ), whereas they only increase welfare by up to 0.26% in a country with a lower banking capital requirement ( $\theta =0.5$ ). This is because, in countries with fewer constraints, regular banks have an incentive to invest in shadow banking projects only when governments provide sufficiently high loss guarantees. Consequently, the welfare gain from shadow banking is limited as the guarantees encourage excessive risk-taking and their negative effects quickly predominate.

Figure 9.

The optimal policies under various values of $ \theta$ .

Figure 10.

The dynamics of welfare during the transition path.

6.6 Dynamic welfare analysis

We have examined the welfare changes at the steady states in the previous experiments. In this section, we further investigate the welfare dynamics associated with policy changes along the entire transition path between the initial steady state and the terminal state.Footnote ⁹ Figure 10 plots the transition paths of household utility, consumption, hours worked, and total capital when the degree of implicit government guarantees, $\Psi$ , is reduced from 0.122 to 0.105 and other parameters are held constant at their baseline values. The vertical axis reports the percentage deviations from the initial steady state, and the horizontal axis shows the time elapsed. As shown in Figure 10, a reduction in implicit guarantees leads to a decline in the capital stock, making more resources available for consumption during the first 100 periods after the policy change. Meanwhile, households work fewer hours and thus enjoy a higher utility. At the peak level, households can enjoy a utility that is 0.38% higher than that in the initial steady state. Nevertheless, households still achieve a gain of about 0.1% in utility at the terminal steady state, even though the consumption level decreases by 0.3%, because the benefit of having more leisure outweighs the cost of reduced consumption. Overall, the results in Figure 10 indicate that households experience an increase in utility throughout the entire transition path following the policy change.

It would be interesting to further quantify the long run benefit of the above policy change by comparing the welfare effect of transition and the welfare effect of staying at the initial steady state. Specifically, we measure the effect of transition by calculating the weighted sum of households’ instantaneous utility over 5000 periods after the policy change:

\begin{equation*} \text{Wel}_{trans}=\sum _{j=0}^{5000}\beta ^{\, j}\left \{ \ln C_{j}-\eta \frac {L_{j}^{1+\phi }}{1+\phi }\right \} , \end{equation*}

where $\left \{ C_{j},L_{j}\right \} _{j=0}^{5000}$ denotes the transition path of variables, and we measure the welfare effect of maintaining status quo as:

\begin{equation*} \text{Wel}_{ss}=\sum _{j=0}^{5000}\beta ^{\, j}\left \{ \ln C_{0}-\eta \frac {L_{0}^{1+\phi }}{1+\phi }\right \} , \end{equation*}

where $C_{0}$ and $L_{0\text{ }}$ denote the initial steady state. We find that $\text{Wel}_{trans}$ is 0.12% higher than $\text{Wel}_{ss}$ , indicating that the policy change of lowering implicit government guarantees from the current level 0.122 to the optimal level 0.105 suggested by our study could benefit the economy in the long run.