2.1 Data

Our sample comprises countries listed by Cantú et al. (Reference Cantú, Cavallino, De Fiore and Yetman2021), who provide detailed information on quantitative easing programs implemented by central banks during the COVID-19 pandemic.

The variables considered for the study are equity return, asset purchases, investment growth, and consumer price index (CPI) inflation. “Asset purchases” is measured as the ratio of the central bank’s net claims on the central government to the total assets of the central bank. “Investment growth” is the annual change in real gross fixed capital formation. “Equity return” is the year-on-year percentage change of the stock market index of the economy. All these data are collected from the IMF’s International Financial Statistics database. We use quarterly data because our key variable—private gross fixed capital formation—is at a quarterly frequency.

We wish to classify economies based on their overall credit market regulations. To this end, we use the credit market regulation index from the economic freedom index provided by the Fraser Institute. We classify countries as high- and low-credit-regulated based on the median credit regulation of the entire sample between 2013 (post-GFC) and 2021. We classify a country as highly regulated if the mean credit regulation index (over time) for the country is greater than the sample median value. The empirical analysis is carried out for these two categories of economies from 2019Q1 to 2022Q3.

Tables 1 and 2 in online appendix present the various characteristics of the public QE announcements made in both low- and high-credit regulation countries based on data provided by Cantú et al. (Reference Cantú, Cavallino, De Fiore and Yetman2021). While data on the end date of the QE exercise is not always available, all countries in our sample commenced their QE programs in 2020. It is also clear from the tables that the QE exercise mainly involved purchasing long-term bonds.

To address any apprehensions about the results being impacted by the varying levels of quantitative easing in different economies, we define “Asset Purchase” in our VAR model as the ratio of the central bank’s net claims on the central government to the total assets of the central bank. For robustness, we also demonstrate that the findings remain consistent when we define “Asset purchases” in line with Weale and Wieladek (Reference Weale and Wieladek2016), that is, as the ratio of the central bank’s net claims on central government to nominal GDP.

2.2 Methodology

We use the PVAR method to estimate the impact of QE. It combines the conventional VAR method which considers all system variables endogenous, with the panel data method that allows for unobserved individual heterogeneity. This enables us to utilize cross-sectional dimensions to control for heterogeneity and to increase the number of observations for small sample sizes.

A $k$ -variate panel VAR of order $p$ is specified as follows:

(1) \begin{equation} Y_{it} = A_{1} Y_{it-1} + A_{2} Y_{it-2} + \ldots + A_{p} Y_{it-p} + u_i + \eta _{it} \end{equation}

where $i$ denotes an economy and $t ={1, 2, \ldots, T}$ is the time period. $Y_{it}$ represents the (1 x k) vector of variables used for estimation, $A_l$ represents the (k x k) matrix of reduced form coefficients, and $u_i$ and $\eta _{it}$ represent the (1 x k) vector of fixed effect and residuals, respectively. The orthogonal impulse response is obtained using the Cholesky decomposition. It requires all variables to be arranged in such a way that the variable that comes first has both a contemporaneous and a lagging effect on the variable that comes next. Specifically, we follow an approach similar to Weale and Wieladek (Reference Weale and Wieladek2016) and arrange variables in the following order - equity return, asset purchases, investment growth, and CPI inflation - with the equity return being the most endogenous. In Appendix A, we show that the result remains robust to an alternate identification scheme with investment growth being the most exogenous variable.

To avoid any endogeneity concerns and to obtain consistent and efficient estimates, we estimate our PVAR model using the system-generalized method of moment (system-GMM) developed by Blundell and Bond (1998). This methodology employs the lagged differences of the dependent variable as instruments for level equations and the lagged levels of the dependent variable as instruments for first difference equations. The system is identified using the same number of instruments as the endogenous variables. Furthermore, the GMM style estimator is advantageous in situations where the sample size is large and the time period is short, as it avoids the Nickell bias described in Nickell (Reference Nickell1981). We have chosen the VAR model with lag one based on the moment model selection criteria (MMSC) proposed by Andrews and Lu (Reference Andrews and Lu2001). These criteria include the MMSC-Bayesian information criterion, MMSC-Akaike information criterion, MMSC-Hannan and Quinn information criterion and coefficient of determination (CD). Additionally, the first-order panel VAR models do not reject Hansen’s over-identification restriction, indicating that there is no possible misspecification in the model. The results of the Hansen J statistics along with moment model selection criteria for both the PVAR models can be found in Online Appendix A.

2.3 Results

Figure 2 illustrates the estimated impulse response for the two sets of economies derived from the two PVARs. Confidence bands are calculated by approximating a Gaussian distribution, which is based on 1000 Monte Carlo simulations from the fitted panel VAR model. We find that, in economies with low credit market regulation, an asset purchase equal to 1% of the size of the central bank’s balance sheet leads to a statistically significant cumulative increase of 20 basis (0.20%) points in private investment. However, in economies with high-credit market regulation, the impact of a similar asset purchase is negligible and insignificant on private investment.

Figure 2. Impact of hundred basis point increase in central bank’s claim on central government as a percentage of central bank assets on fixed capital formation in the economy.Note: The impulse response is generated using 1000 Monte Carlo simulation of system-GMM estimation of Equation (1).

For robustness, we use an alternative definition of the variable “Asset Purchase” in line with Weale and Wieladek (Reference Weale and Wieladek2016). The estimated impulse response function for both groups of economies is shown in Figure 2 of Appendix A. We find that the results remain robust as a 1% increase in asset purchase relative to GDP leads to a statistically significant cumulative increase of 0.20% in private investment only in those economies that have low credit market regulations. The results also remain robust to the alternative identification scheme where the investment growth is the most exogenous (see Figure 3 of Appendix A.)

The literature on the impact of QE policies on the real economy has largely been centered around the US and Europe (see Fabo et al. (Reference Fabo, Jančoková, Kempf and Pástor2021) for a survey). The results of this survey are mixed. Weale and Wieladek (Reference Weale and Wieladek2016) for instance, finds that an announcement of a purchase of 1% of GDP leads to a statistically significant increase of 0.58% in GDP in the US. These results are broadly in the ballpark of our findings.

To address the concern that our results are not driven by the short data series, we identify eight central banks from Tables 1 and 2 of Appendix A (five in the low-regulation economies and three in the high-regulation economies) that had an active asset purchase program even before the pandemic. We now run PVARs only for these eight economies from 2014 to 2022. The results are shown in Figure 4 of Appendix A. Even with longer time period data series, the results show that the public QE program or “Asset Purchase” has a much higher and significant impact on private investment in economies with low credit regulation compared to economies with higher credit regulation.

3.1 Households

There is a representative household that consumes the final good, supplies labor to wholesale firm, and makes deposits with financial intermediaries. Its lifetime discounted utility takes the following form:

\begin{equation*} U_t = \mathbb {E}_t \sum _{i=0}^{\infty } \beta ^i \left \{ ln(C_{t+i} - hC_{t+i-1}) - \frac {L_{t+i}^{1+\eta }}{1+\eta } \right \} \end{equation*}

where $0\lt \beta \lt 1$ is the household’s utility discounting factor, $0\lt h\lt 1$ is the habit persistence parameter, and $\eta \gt 0$ is inverse of Frisch elasticity of labor supply. The deposits are made in the form of one-period nominal debt $D_t$ so that the household has the option of deciding whether to roll over the deposits or not. These bonds earn a gross return of $R_t^D$ in the next period $(t+1)$ .

The household earns labor income $W_t$ , dividends $DIV_t$ from their ownership in production firms and intermediaries, and pays a lump-sum tax $T_t$ to the government. Formally, the budget constraint of the household can be written as:

\begin{equation*} P_t C_t + D_t = W_t L_t + R_{t-1}^D D_{t-1} + DIV_t - P_t X - P_t T_t \end{equation*}

Here, $P_t$ is the price of the final output good and $X$ is the amount of fixed real equity (in terms of consumption units) infused by household to start up new intermediaries each period.

The first-order conditions with respect to $C_t, L_t,$ and $D_t$ , respectively, are:

(2) \begin{equation} \mu _t = \frac{1}{C_t - hC_{t-1}} - \beta h \mathbb{E}_t \frac{1}{C_{t+1} - h C_t} \end{equation}

(3) \begin{equation} \chi L_t^\eta = \mu _t w_t \end{equation}

(4) \begin{equation} 1=R_t^D \mathbb{E}_t\left[\Lambda _{t,t+1} \Pi _{t+1}^{-1}\right] \end{equation}

Here, $\mu _t = U^{'}(C_t)$ is the marginal utility of consumption, $w_t$ is the real wage, $\Lambda _{t,t+1}=\beta \frac{\mu _{t+1}}{\mu _t}$ is the real stochastic discount factor of the household, and $\Pi _{t+1} = \frac{P_{t+1}}{P_{t}}$ is the gross inflation rate. Together, Equations (2) and (3) give us the households labor supply curve while Equation (4) is the familiar Fisher equation.

3.2 Financial Intermediaries

There is a unit mass of intermediaries (indexed by $i$ ) in the economy. Each period, a fraction $1 - \sigma$ of total intermediaries stochastically exit and return their net worth to their owner household. They are replaced by an equal number of new intermediaries with a startup equity of $X$ given to them by the household.

On its asset side, each intermediary holds perpetual government debt ( $B_{i,t}$ ) and wholesale firm’s debt ( $F_{i,t}$ ) along with the reserves $RE_{i,t}$ (issued by the Central Bank).Footnote ⁶ It finances them using one-period deposits ( $D_{i,t}$ ) from households and its own net worth ( $N_{i,t}$ ). The balance-sheet equation of an intermediary $i$ is therefore given by:

(5) \begin{equation} Q_t F_{i,t} + Q_{B,t} B_{i,t} + RE_{i,t} = D_{i,t} + N_{i,t} \end{equation}

Until an intermediary stochastically exits, it accumulates its net worth instead of paying it out as dividends to the households. Therefore, its net worth evolves according to:

\begin{equation*} N_{i,t} = \big(R_t^F \big) Q_{t-1} F_{i,t-1} + \big(R_t^B \big) Q_{B,t-1} B_{i,t-1} + \big(R_{t-1}^{re} \big) RE_{i,t-1} - R_{t-1}^D D_{i,t-1} \end{equation*}

where $R_t^F = \frac{1+\kappa Q_{t}}{Q_{t-1}}$ , $R_t^B = \frac{1+\kappa Q_{B,t}}{Q_{B,t-1}}$ are the realized holding period returns on private and government debt, respectively, and $(R_{t-1}^{re} )$ is the gross interest rate on reserves set by the central bank which is known at time $t-1$ . Combining the above equation with (5), we get:

(6) \begin{equation} N_{i,t} = \big(R_t^F - R_{t-1}^D\big) Q_{t-1} F_{i,t-1} + \big(R_t^B - R_{t-1}^D\big) Q_{B,t-1} B_{i,t-1} + \big(R_{t-1}^{re} - R_{t-1}^D\big) RE_{i,t-1} + R_{t-1}^D N_{i,t-1} \end{equation}

The interpretation of the above equation is standard. The first three terms represent the excess returns over the deposit rate of holding private debt, government debt, and reserves. The last term reflects the savings made from financing assets using net worth as opposed to deposits. The stochastic exit assumption prevents an intermediary from accumulating enough net worth to make the limited enforcement constraint, that we describe below, redundant.

It is important to note that the net worth of a bank can change in two ways: (i) through its balance-sheet Equation (5), and (ii) through the net worth accumulation Equation (6). The unconventional policy works through the balance-sheet equation as the bank’s reserves holdings change, while the conventional policy works through the net worth accumulation equation as the excess returns earned by the bank change.

3.2.1 Intermediary’s problem

The presence of excess returns implies that the objective of a financial intermediary is to maximize the expected terminal value of its net worth. The expected continuation value of remaining an intermediary at the end of period $t$ is given by:

(7) \begin{equation} V_{i,t} = (1-\sigma ) \mathbb{E}_t [\Lambda _{t,t+1} n_{i,t+1}] + \sigma \mathbb{E}_t [V_{i,t+1} \Lambda _{t,t+1}] \end{equation}

where $V_{i,t}$ is the maximized expected value of the intermediary’s terminal net worth, $n_{i,t+j}=\frac{N_{i,t+j}}{P_{t+j}}$ is the real net worth at $t+j$ , and $\Lambda _{t,t+j}$ is the real stochastic discount factor of households.

3.2.2 Leverage constraint

The financial intermediary faces two constraints. The first, as modeled in Gertler and Karadi (2011), arises due to an agency problem under which an intermediary can divert a fraction $\theta _t$ of the total value of private debt and $\Delta \theta _t$ fraction of the government debt, where $\theta _t \geq 0$ and $\Delta \leq 1$ . This implies that it is easier for the intermediary to divert private debt than government debt. Creditors can recover the rest of the intermediary’s assets including reserves which are held with the central bank. As a consequence of this agency problem, the following endogenous leverage constraint must be satisfied for depositors to be willing to lend to intermediaries:

(8) \begin{equation} V_{it} \geq \theta _t (Q_t f_{i,t} + \Delta Q_{B,t} b_{i,t}) \end{equation}

where $f_{i,t}$ and $b_{i,t}$ are the real values of private and government debt, respectively. Here, the left-hand side denotes the expected value of continuing as an intermediary after time $t$ , and the right-hand side denotes the gain if it decides to divert assets. The constraint ensures that it is never optimal for the intermediary to abscond with assets in equilibrium. It also implies that net worth limits the intermediary’s ability to attract deposits.

3.2.3 Regulatory constraint

The second constraint faced by the intermediaries is a regulatory constraint. Following Chari et al. (Reference Chari, Dovis and Kehoe2020) and Kriwolutzy et al. (Reference Kriwolutzy, Muller and Scheer2018), financial intermediaries face an additional constraint wherein they have to hold a certain minimum fraction ( $\Gamma$ ) of their bond assets in the form of government bonds. Formally, the constraint is given byFootnote ⁷

\begin{equation*} Q_{B,t} b_{i,t} \geq \Gamma \left(Q_{B,t} b_{i,t} + Q_t f_{i,t}\right) \end{equation*}

which can be rewritten as

(9) \begin{equation} Q_{B,t} b_{i,t} \geq \gamma Q_{t} f_{i,t} \end{equation}

where $\gamma = \frac{\Gamma }{1-\Gamma }$ .

3.2.4 Discussion

The objective of an intermediary is to maximize Equation (7) subject to Equations (6), (8), and (9). The Lagrangian optimization yields the following first-order conditions:

(10) \begin{equation} \mathbb{E}_t \left[\Lambda _{t,t+1} \Pi _{t+1}^{-1} \Omega _{i,t+1} \big(R_{t}^{re} - R_t^D\big)\right] = 0 \end{equation}

(11) \begin{equation} \mathbb{E}_t \left[\Lambda _{t,t+1} \Pi _{t+1}^{-1} \Omega _{i,t+1} \big(R_{t+1}^F - R_t^D\big)\right] = \frac{\lambda _{it}}{1+\lambda _{it}} \theta _t + \frac{\zeta _{it}}{1+\lambda _{it}} \gamma \end{equation}

(12) \begin{equation} \mathbb{E}_t \left[\Lambda _{t,t+1} \Pi _{t+1}^{-1} \Omega _{i,t+1} \big(R_{t+1}^B - R_t^D\big)\right] = \frac{\lambda _{it}}{1+\lambda _{it}} \theta _t \Delta - \frac{\zeta _{it}}{1+\lambda _{it}} \end{equation}

where $\left(\Omega _{i,t+1} = 1-\sigma + \sigma \frac{\partial V_{it+1}}{\partial n_{i,t+1}}\right)$ . The Lagrange multipliers $\lambda _{it}$ and $\zeta _{it}$ represent the tightness of leverage and regulatory constraints, respectively. Equation (10) implies intermediaries do not earn any excess returns on the reserves, i.e., $R_{t}^{re}= R_{t}^{D}$ . Equations (11) and (12) imply that if the leverage constraint is binding ( $\lambda _{it}\gt 0$ ), then excess returns earned on both private and government debt persist in equilibrium. We note that the presence of the regulatory constraint drives a wedge between the expected returns on private and government bonds. Intuitively, such a constraint raises the demand for government bonds and depresses their yield. To see this, consider the case when there are no regulations. Under this scenario, the regulatory constraint is slack i.e. $Q_{B,t} b_{i,t} \gt \gamma Q_t f_{i,t}$ and $\zeta _{it}=0$ . Following Kriwolutzy et al. (Reference Kriwolutzy, Muller and Scheer2018), Equation (12) can therefore be rewritten as

(13) \begin{equation} \mathbb{E}_t \left[\Lambda _{t,t+1} \Pi _{t+1}^{-1} \Omega _{i,t+1} \big(\tilde{R}_{t+1}^B - R_t^D\big)\right] = \frac{\lambda _{it}}{1+\lambda _{it}} \theta _t \Delta \end{equation}

where, Kriwolutzy et al. (Reference Kriwolutzy, Muller and Scheer2018) term $\tilde{R}_{t+1}^B$ as the laissez-faire return on government debt. To see the impact of credit regulation, we combine Equations (12) and (13) to get

\begin{equation*} \mathbb {E}_t \left[\Lambda _{t,t+1} \Pi _{t+1}^{-1} \Omega _{i,t+1} \left(R_{t+1}^B - \tilde {R}_{t+1}^B\right) \right] = - \frac {\zeta _{it}}{1+\lambda _{it}} \end{equation*}

It follows from the above equation that whenever the regulatory constraint binds $(\zeta _{it}\gt 0)$ , the return on government debt is lesser than the laissez-faire case, i.e., $R_{t+1}^B \lt \tilde{R}_{t+1}^B$ . Throughout the paper, we assume that the regulatory constraint binds. Although the extent of credit regulation does not change over time, the tightness of regulatory constraints does change as a result of changing economic conditions that impact term premiums.

We also assume that the leverage constraint always binds. Following Sims and Wu (Reference Sims and Wu2021), we assume that the value function is linear in net worth, i.e., $V_{it} = a_t n_{it}$ . Defining $\phi _{it} = \frac{Q_t f_{it} + \Delta Q_{B,t} b_{it}}{n_{it}}$ as the modified leverage ratio for an intermediary and solving for $\phi _t$ (see Appendix D for derivation), we get

(14) \begin{equation} \phi _t = \frac{ (1+\Delta \gamma ) \mathbb{E}_t \left[\tilde{\Lambda }_{t,t+1} R_t^D\right] }{\theta _t + \theta _t \Delta \gamma - \mathbb{E}_t \left[\tilde{\Lambda }_{t,t+1} \left\{ \left(R_{t+1}^F - R_t^D\right) + \gamma \left(R_{t+1}^B - R_t^D\right) \right\} \right] } \end{equation}

where $\tilde{\Lambda }_{t,t+1} = \Lambda _{t,t+1} \Pi _{t+1}^{-1} \Omega _{t+1}$ . Note that the leverage ratio decreases with increasing $\theta _t$ . Intuitively, a higher $\theta _t$ means the intermediary can divert a larger fraction of the assets and hence the depositor will require the intermediary to put in more equity. On the other hand, higher expected excess returns $ \left(R_{t+1}^F - R_t^D\right)$ or $\left(R_{t+1}^B - R_t^D\right)$ results in a higher leverage ratio. Intuitively, higher expected excess returns on assets increase the expected net worth of the intermediary and reduce the risk of default. Similarly, the discounted deposit rate $\mathbb{E}_t \left[\tilde{\Lambda }_{t,t+1} R_t^D\right]$ increases the net worth and hence raises the continuation value of staying a banker. Thus, it also impacts the leverage ratio positively.

Lastly, the leverage ratio decreases with an increase in the credit regulation parameter $\Gamma$ . Essentially, higher regulation reduces the excess returns on corporate and government bonds received by the intermediary. The consequent lower expected net worth increases the risk of default and results in a lower leverage ratio. This is displayed in Figure 3.

Figure 3. Steady-state leverage ratio $(\phi )$ as a function of regulatory constraint parameter $(\Gamma )$ .Note: The range of regulatory constraint parameter $\Gamma$ , as shown in the above Figure, is the only possible set of its values given our model parameters. The lower and upper bounds for $\Gamma$ correspond to the limiting cases of $\Delta$ and $\theta$ approaching unity in the steady state, respectively.

3.2.5 Term premium

The presence of both leverage and regulatory constraints results in a term premium in the model. Following Carlstrom et al. (Reference Carlstrom, Fuerst and Paustian2017), we define (log) term premium as the difference between the observed (log) yield on a long-term bondFootnote ⁸ and the implied (log) yield on a short-term bond. This implied yield is obtained by applying the expectation hypothesis (EH) of the term structure to the series of short rates. The price of such a hypothetical (EH) bond satisfies

\begin{equation*} R_t^D = \frac {1+\kappa Q_{t+1}^{EH}}{Q_t^{EH}} \end{equation*}

and its yield is given by

\begin{equation*} R_t^{EH} = \frac {1}{Q_t^{EH}}+\kappa \end{equation*}

The yield of a long bond is given by

\begin{equation*} RL^i_t = \frac {1}{Q_{i,t}} + \kappa \ ; \ i \in \{B,F\} \end{equation*}

We define term-premium as the ratio of these two yields, i.e,

(15) \begin{equation} TP_{i,t} = \frac{RL^i_t}{R_t^{EH}} \ ; \ i \in \{B,F \} \end{equation}

It follows from Equations (11), (12), and (15) that the term premium on private and government bonds is a function of asset market segmentation ( $\lambda _{it}$ ), credit regulation ( $\zeta _{it}$ ) and credit shocks (shock to $\theta _t$ ). This insight turns out to be crucial during the analysis of monetary policy transmission.

3.3 Production

Our production section is similar to Sims and Wu (Reference Sims and Wu2021) which consists of four different types of production firms: a competitive final good producer, monopolistic retailers, wholesale firm, and investment good firm. A representative investment good firm produces new physical capital from final output subject to a convex adjustment cost. The key departure from a standard framework is that the wholesale firm produces output using its own capital, which is accumulated via the purchase of new capital from the investment goods firm and labor hired competitively from the households. A continuum of retail firms then repackages this wholesale output and sells to the final good firm. These retail firms behave as monopolistic competitors and are subject to price stickiness. In the text, we focus on the wholesale firm, while the rest of the production section is described in the Appendix B.

The wholesale firm produces output using labor input $L_{d,t}$ and the capital that it accumulates using a Cobb-Douglas technology:

\begin{equation*} Y_{w,t} = A_t {(u_t K_t)}^\alpha {L^{1-\alpha }_{d,t}} \end{equation*}

where $A_{t}$ is an exogenous productivity variable that obeys an exogenous stochastic process, $K_{t}$ is the stock of physical capital that the firm owns and $u_t$ is the capital utilization factor. In addition, $\alpha \in (0,1)$ is the exponent on capital services in production.

Physical capital accumulates according to the following law of motion, which generates faster depreciation as a cost of utilization:

(16) \begin{equation} K_{t+1} = \hat{I_t} + (1-\delta (u_t))K_t \end{equation}

Here, $\delta (u_t) = \delta _0 + \delta _1 (u_t -1) + \frac{\delta _2}{2}{(u_t -1)}^2$ is the utilization adjustment cost, which maps utilization into depreciation.

We assume that the wholesale firm purchases at least a constant fraction $\psi$ of its new physical capital $\hat{I_t}$ using fresh issues of perpetual debt or bonds. This results in a “loan in advance” constraint of the form:

(17) \begin{equation} \psi P_t^K \hat{I_t} \leq Q_t CF_{w,t} = Q_t( F_{w,t} - \kappa F_{w,t-1}) \end{equation}

where $P_t^K$ is the price of the new physical capital. $CF_{w,t}$ is the number of new bonds issued (see Appendix B for details), while $Q_{t}$ is the nominal price of a bond. The labor is hired in a competitive market at the nominal wage $W_{t}$ . The firm maximizes the present discounted value of the real dividend. The nominal dividend of the firm is given by:

\begin{equation*} DIV_{w,t} = P_{w,t} A_t {(u_t K_t)}^\alpha {L^{1-\alpha }_{d,t}} - W_t L_{d,t} - P_t^K \hat {I_t} - F_{w,t-1} + Q_t (F_{w,t} - \kappa F_{w,t-1}) \end{equation*}

The first-order conditions are:

(18) \begin{equation} w_t = (1-\alpha ) p_{w,t} A_t{(u_t K_t)}^\alpha L^{-\alpha }_{d,t} \end{equation}

(19) \begin{equation} \nu _{1,t} \delta ^{'}(u_t) = \alpha p_{w,t} A_t{(u_t K_t)}^{\alpha -1} L^{1-\alpha }_{d,t} \end{equation}

(20) \begin{equation} \nu _{1,t} = (1+\psi \nu _{2,t}) p_t^k \end{equation}

(21) \begin{equation} p_t^k M_{1,t} = \mathbb{E}_t \Lambda _{t,t+1} \left [ \alpha p_{w,t+1} A_{t+1}{(u_{t+1} K_{t+1})}^{\alpha -1} u_{t+1} L^{1-\alpha }_{d,t+1} + (1-\delta (u_{t+1})) p^k_{t+1} M_{1,t+1} \right ] \end{equation}

(22) \begin{equation} Q_t M_{2,t} = \mathbb{E}_t \Lambda _{t,t+1} \Pi ^{-1}_{t+1}[1+\kappa Q_{t+1} M_{2,t+1}] \end{equation}

Here, $\nu _{1,t}$ and $\nu _{2,t}$ are the Lagrange multipliers for (16) and (17) respectively, and $M_{1,t}=1+\psi \nu _{2,t}$ and $M_{2,t} = 1+ \nu _{2,t}$ are two auxiliary variables. Variables $w_t$ , $p_{m,t}$ , and $p_{t}^{k}$ are real wages, the relative price of wholesale product, and the relative price of new capital, respectively. Equation (18) is the standard labor demand equation. As discussed in Sims and Wu (Reference Sims and Wu2021), the uniqueness lies with $M_{1,t}$ and $M_{2,t}$ which serve as endogenous “investment wedge” and “financial wedge,” respectively. The fluctuations in these wedges are key channels through which any monetary action, either conventional or unconventional, has an impact on the real economy.

3.4 Fiscal authority

There is a government that consumes a constant $\overline{G}$ amount of real output of final goods and makes the interest payments on its outstanding debt with the help of lump-sum tax raised from households, transfers from the central bank, and fresh issue of perpetual government bonds. The budget constraint is given by:

(23) \begin{equation} P_t \overline{G} + B_{G,t-1} = P_t T_t + P_t T_{cb,t} + Q_{B,t} \big(B_{G,t} - \kappa B_{G,t-1}\big) \end{equation}

where $B_{G,t-1}$ is the total liability (payable at $t$ ) on outstanding bonds issued till $t-1$ , $T_{cb,t}$ is the transfer from the central bank, and the last term denotes the value of bonds freshly issued at $t$ .

3.5 Central bank

The central bank sets the interest rate on reserves according to the following Taylor Rule:

\begin{equation*}ln R_t^{re} = (1-\rho _r) ln R^{re} + \rho _r ln R_{t-1}^{re} + (1-\rho _r)[\phi _\pi (ln \Pi _t - ln \Pi ) + \phi _y (ln Y_t - ln Y_{t-1}) ] + s_r \epsilon _{r,t} \end{equation*}

where $R^{re}$ and $\Pi$ denote the steady state values of the policy rate and inflation respectively, with $0\lt \rho _r \lt 1$ , $\phi _\pi \gt 1$ and $\phi _y \gt 0$ . Since $R_t^{re} = R_t^{D}$ , setting the rate on reserves would be equivalent to the central bank setting the deposit rate.

The asset market segmentation (or presence of a leverage constraint) implies that a central bank can also conduct monetary policy on the long end of the debt-maturity structure. That is, it can influence long rates in the economy without necessarily changing the short-term rates. The sale or purchases of government long-term debt, which we term unconventional or QE policies are financed using its reserves. Formally, the balance-sheet equation for the central bank is given by:

\begin{equation*}Q_{B,t} B_{cb,t} = RE_t \end{equation*}

where $Q_{B,t} B_{cb,t}$ denotes the total value of all government bonds acquired by the central bank till period $t$ , and $RE_t$ denotes the nominal value of period $t$ reserves issued by the central bank. We consider both exogenous and endogenous unconventional policies. For exogenous policy, we assume that the central bank’s government bond holdings follow an exogenous AR(1) process:

(24) \begin{equation} b_{cb,t} = (1-\rho _b) b_{cb} + \rho _b b_{cb,t-1} + s_b \epsilon _{b,t} \end{equation}

where $b_{cb}$ denotes the steady state of government bond holdings and $\epsilon _{b,t}$ is an i.i.d shock. Endogenous unconventional policy considered in Section 6 involves the central bank pegging the government bond term premium to its steady state level, thus making the level of debt endogenous.

Finally, the monetary authority remits any net revenue it makes to the government in the form of lump-sum transfers $T_{cb,t}$ . This can be expressed in real terms as

\begin{equation*}T_{cb,t} = \big(R_t^B - R_{t-1}^{re}\big) \Pi _t^{-1} Q_{B,t-1} b_{cb,t-1} \end{equation*}

where the right-hand side reflects the net revenue earned by the central bank on its asset holdings.

3.6 Aggregate and market clearing conditions

Since the labor market is competitive, labor supplied by households should equal the labor demanded by the wholesale firm, i.e.

\begin{equation*} L_t = L_{d,t} \end{equation*}

The market for long-term bonds of both the wholesale firm and the government should clear as follows:

\begin{equation*} f_{w,t} = f_t \end{equation*}

\begin{equation*} b_{G,t} = b_t + b_{cb,t} \end{equation*}

where $f_t$ and $b_t$ are the real aggregate values of all long bonds acquired by intermediaries from the wholesale firms and government, respectively.

Following (6), the aggregate real net worth of intermediaries (both surviving and new ones) at the start of date $t$ is given by:

(25) \begin{align} n_t & = \sigma \Pi _t^{-1} \left[\left(R_t^F - R_{t-1}^d\right)Q_{t-1} f_{t-1} + \left(R_t^B - R_{t-1}^d\right)Q_{B,t-1} b_{t-1}\right.\nonumber\\&\quad \left. + \left(R_{t-1}^{re} - R_{t-1}^d\right) re_{t-1} + R_{t-1}^d n_{t-1}\right] + X \end{align}

Since households own both financial and non-financial firms, we plug back the aggregate real dividends from these firms and $X$ from Equation (25) into the budget constraint of households to get the usual aggregate resource constraint given by:

\begin{equation*} Y_t = C_t + I_t + G_t \end{equation*}

4.1 Unconventional monetary policy shock

We begin by considering impulse responses under an exogenous positive QE shock. The central bank bond-buying program is described by Equation (24). Figure 4 shows that such a shock has opposing effects on leverage and regulatory constraints. Here, the vertical axes indicate deviations of the variables from the steady state, and the horizontal axes indicate time in quarters. The bond-buying program by the central bank decreases the banks’ government debt holdings, which tightens the regulatory constraint. The shortage of government debt increases its demand, which raises its price and reduces its expected return or yield.

Figure 4. Impulse responses to a positive quantitative easing shock in the presence of credit regulations.Note: All variables are in percentage points and all rates are annualized.

Under a binding regulatory constraint, the financial intermediary must hold government and private bonds in a fixed proportion. The intermediary, therefore, responds to this shock by rebalancing its portfolio and reducing its holding of private bonds. In the spirit of Chari et al. (Reference Chari, Dovis and Kehoe2020) and Kriwolutzy et al. (Reference Kriwolutzy, Muller and Scheer2018), this portfolio effect causes the yields of private bonds to increase and investment to fall. On the other hand, QE increases the reserves that intermediaries hold. Higher reserves, in turn, increase their net worth through the balance-sheet equation Footnote ⁹ and relax the leverage constraint. The positive net worth effect increases deposits and stimulates investment. For our baseline parameters, the portfolio effect seems to dominate, resulting in reduced investment and output.

Next, we perform the counterfactual, where we examine the impact of a QE shock on an economy without regulatory constraints. As we can see in Figure 5, in such an environment, a positive shock to QE, by increasing banks’ net worth, results in an expansion of production and investment in the economy. In other words, in the absence of the regulatory constraint, a QE shock is expansionary.

Figure 5. Impulse responses to a positive quantitative easing shock under no credit regulations.Note: All variables are in percentage points and all rates are annualized.

To summarize, the regulatory constraint can completely invert the monetary transmission mechanism under QE: an increase in bond buying by the central bank raises the yield on private bonds and causes a contraction instead of an expansion in the economy.

4.2 Conventional monetary policy shock

Figure 6 shows the impulse responses to a negative policy rate shock. Again, this conventional expansionary policy shock has opposing effects on leverage and regulatory constraints. The reduction in the deposit rate raises excess returns earned by the intermediary on their assets, causing their net worth to increase through the net worth accumulation Equation (6). The positive net worth effect relaxes the leverage constraint, increasing deposits, and raising the demand for private bonds. The rise in demand for private bonds reduces their yield and stimulates investment.

Figure 6. Impulse responses to a negative policy rate shock in the presence of credit regulations.Note: All variables are in percentage points and all rates are annualized.

However, it also tightens the regulatory constraint, as the intermediaries cannot increase their holdings of private bonds without proportionately raising their holdings of government bonds. Ultimately, it is the net worth effect that triumphs, causing investment and output to rise. In the counterfactual exercise where the regulatory constraint is absent, as shown in Figure 7, the results are qualitatively similar.

Figure 7. Impulse responses to a negative policy rate shock under no credit regulations.Note: All variables are in percentage points and all rates are annualized.

Figure 8. Impulse responses to a positive credit shock under different monetary policy regimes in the presence of credit regulations.Note: All variables are in percentage points and all rates are annualized.

Figure 9. Impulse responses to a positive productivity shock under different monetary policy regimes in the presence of credit regulations.Note: All variables are in percentage points and all rates are annualized.

Our result here is in contrast to Lahiri and Patel (Reference Lahiri and Patel2016), who, in a simple model without leverage constraints, show that the presence of regulatory constraints can invert monetary policy transmission under conventional policy. On the contrary, we show that their result is overturned with a leverage constraint.

6.1 Credit shock

An adverse credit shock (Figure 8), that is, a positive shock to $\theta _t$ , magnifies the agency cost problem by increasing the assets that financial intermediaries can divert. Consequently, the leverage constraint tightens, increasing the yields of government and private bonds. Under a term premium peg, the central bank responds by purchasing government bonds to lower the yields. It follows from quations (11) and (12) that, in the absence of regulatory constraints $(\zeta _{it}=0)$ , such a policy completely neutralizes credit shocks.

Intuitively, the purchase of government bonds under this regime relaxes the leverage constraint by changing the composition of the intermediary’s assets from government bonds to reserves (see Equation (8)). As shown in Carlstrom et al. (Reference Carlstrom, Fuerst and Paustian2017), the relaxation of the leverage constraint completely sterilizes the real economy from credit shocks. However, with credit regulations $(\zeta _{it} \neq 0)$ , the purchase of bonds by the central bank tightens the regulatory constraint. The financial intermediary responds by rebalancing its portfolio and reducing its holding of private bonds. As a consequence of this portfolio effect, the term premiums on private bonds increase, which causes investment and output to fall.

By contrast, under Taylor rule, the absence of a portfolio rebalancing effect relaxes the leverage constraint. Relative to the peg, the lower term premium on private bonds mitigates to some extent the impact of adverse credit shocks. As a result, the fall in investment and output is less in this regime. The comparison of welfare performances of the term-premium peg relative to the simple Taylor rule vindicates the above discussion. Table 2 shows that the simple Taylor rule welfare dominates the term premium peg.

Table 2. Comparison of welfare costs under alternative monetary policy regimes (measured in percentage points of steady state consumption stream)

Our results suggest that the presence of regulations reduces the efficacy of a term premium targeting policy in countering credit shocks. As emphasized earlier, these results are in contrast to Carlstrom et al. (Reference Carlstrom, Fuerst and Paustian2017), who find the term premium peg to be welfare enhancing.

Table 3. Comparison of welfare costs under alternative monetary policy regimes (measured in percentage points of steady state consumption stream) for different values of $\Gamma$

Note: (1) To compute welfare cost, we use second-order approximation on Dynare. It is calculated empirically by simulating a long series for 1,00,000 periods after the shock. (2) Similar to figure 3, the lower and upper bounds for regulatory parameter $\Gamma$ correspond to the limiting cases of $\Delta$ and $\theta$ approaching unity in the steady state, respectively.

6.2 Productivity shock

Figure 9 shows that a temporary rise in productivity by lowering the marginal cost of production causes inflation to fall. The central bank responds by reducing rates to stabilize inflation. As mentioned earlier, lower interest rates raise the financial intermediary’s net worth by increasing the excess returns earned on their assets. The positive impact on net worth relaxes the intermediary’s leverage constraint and raises their demand for private bonds. The increase in demand for private bonds raises their price, lowers the yield, and tightens the regulatory constraint. To maintain a fixed proportion under credit regulation, the demand for government bonds also increases, lowering their yield or term premium.

Under a term premium peg, the central bank responds to the falling term premium by selling government bonds. Such a policy relaxes the regulatory constraint, enabling the intermediary to increase its purchase of private bonds and stimulate investment. Under a Taylor rule, the absence of the above central bank intervention means that the rise in investment and output is lower under this regime. Table 2, which welfare ranks the rules, supports the intuition obtained from the impulse responses. The term premium peg outperforms a simple Taylor rule in the case of productivity shocks. This result is in contrast to those obtained in the case of a credit shock.

In the spirit of Poole (Reference Poole1970), our analysis suggests that the nature of the shock determines the choice of the monetary policy regime. In the presence of credit regulations, the model prescribes adopting a term premium peg when productivity shocks are predominant and a simple Taylor rule when credit shocks are predominant.

Next, we examine the impact on welfare when we change the regulatory constraint parameter $\Gamma$ . Table 3 indicates that our results are robust to this change—Taylor rule does better under credit shocks, while the term premium peg performs better under productivity shocks. Notice also that the reduction (rise) of the portfolio effect results in a relative improvement (reduction) in the performance of the term premium peg under credit shocks and the Taylor rule under productivity shocks.