QE in the Future: The Central Bank’s Balance Sheet in a Fiscal Crisis

Abstract

Analyses of quantitative easing (QE) typically focus on the recent past studying the policy’s effectiveness during a financial crisis when nominal interest rates are zero. This paper examines instead the usefulness of QE in a future fiscal crisis, modeled as a situation where the fiscal outlook is inconsistent with both stable inflation and no sovereign default. The crisis can lower welfare through two channels, the first via aggregate demand and nominal rigidities, and the second via contractions in credit and disruption in financial markets. Managing the size and composition of the central bank’s balance sheet can interfere with each of these channels, stabilizing inflation and economic activity. The power of QE comes from interest-paying reserves being a special public liability, neither substitutable by currency nor by government debt.

Appendix

1.1 Formal Statement of the Model and Equilibrium

This section collects all of the pieces of the model presented in the paper, starting with the decision problems of the agents, moving to the market clearing conditions, and ending with the definition of equilibrium.

Households The representative household chooses $\{c_t,l_t,b_t^h,B_t^h,z_t\}$ to maximize the utility function in Eq. (5) subject to the budget constraint:

$$\begin{aligned} p_t c_t + q_t b_t^h + Q_t B_t^h + p_t z_t= w_t l_t + \delta _t \left( b_{t-1}^h + q_t B_{t-1}^h\right) + R^z_t p_t z_t + p_t (1 - \kappa ) - p_t {\tilde{f}}_t, \end{aligned}$$

(24)

at all dates t, and a standard no Ponzi scheme condition. The bond holdings by households are denoted by $(b_t^h,B_t^h)$. While in principle the household can issue bonds similar to government bonds (but which could not be used as collateral by banks), the market clearing condition that these private bonds are in zero net supply is already incorporated into the notation. Deposits earn a real return $R^z_t$ and are subject to a no short-selling constraint as well as the fact that only working capital is deposited $0 \le z_t \le 1-\kappa$. Moreover, the household will take into account the incentive constraint of the banks in Eq. (4). Finally, $\tilde{f}_t$ are the net taxes paid by the household.

The sufficient and necessary optimality conditions for this dynamic problem are the Euler equations with respect to the two bonds in Eq. (6); the labor supply condition:

$$\begin{aligned} l_t^\alpha = w_t/p_t, \end{aligned}$$

(25)

where $w_t$ is the nominal wage rate; the budget constraint above holding with equality together with a transversality condition defining its intertemporal counterpart; and finally the condition that if $R^z_t \ge 1$ then $z_t$ will be the minimum of $1-\kappa$ or the value at which the bank’s incentive compatibility constraint in Eq. (4) binds.

Competitive final goods firms They maximize Eq. (7) every period subject to the budget constraint that total spending $E_t$ is equal to $E_t =\int _0^{k_t} p_t(j) y_t(j) \hbox {d}j$. Standard derivations reveal that the solution to this problem is:

$$\begin{aligned} y_t(j)&= \left( \frac{p_t(j)}{p_t} \right) ^{-\sigma } k_t^{\theta \sigma } y_t, \end{aligned}$$

(26)

$$\begin{aligned} p_t&= \left[ k_t^{\theta \sigma } \int _0^{k_t} p_t(j)^{1-\sigma } \hbox {d}j \right] ^{1/(1-\sigma )}, \end{aligned}$$

(27)

where $p_t$ is the static cost-of-living price index with the property that $E_t = p_t y_t$.

Intermediate goods firms A firm j that can change its price with full information this period will choose $\{y_t(j),l_t(j),p_t(j),k_t(j)\}$ to maximize profits:

$$\begin{aligned} X_t = \frac{1}{p_t} \left[ \left( \frac{\sigma }{\sigma -1} \right) p_t(j) y_t(j) - w_t l_t(j) - p_t (1+r_t) k_t(j) \right] \end{aligned}$$

(28)

subject to: the demand for the good in Eq. (26), the production technology $y_t(j)=a_t l_t(j)$, and the setup cost of working capital $k_t \in \{0,1\}$. $\sigma /(\sigma -1)$ is a standard sales subsidy to offset the monopoly distortion.

Simple algebra shows that the firm will choose the optimal nominal price:

$$\begin{aligned} p^*_t = \frac{w_t}{a_t}. \end{aligned}$$

(29)

In turn the zero profit condition from free entry, $X_t=0$ subject to $k_t \le 1$, implies that if a firm is in operation ($k_t(j)=1$) then:

$$\begin{aligned} 1+r_t = \left( \frac{1}{\sigma - 1 }\right) \left( \frac{w_t l_t(j)}{p_t} \right) . \end{aligned}$$

(30)

Turning to the uninformed firms, their problem is similar but now they maximize expected profits discounting uncertainty using the household’s stochastic discount factor so:

$$\begin{aligned} p^{*e}_t = \text {arg} \max _{p_t(j)} {{\mathrm{{\mathbb {E}}}}}_{t-1} \left( m_{t-1,t} X_t \right) = \frac{{{\mathrm{{\mathbb {E}}}}}_{t-1} \left( \frac{w_t k_t^{\theta \sigma } y_t}{a_t p_t^{1-\sigma } }\right) }{{{\mathrm{{\mathbb {E}}}}}_{t-1} \left( \frac{k_t^{\theta \sigma } y_t}{p_t^{1-\sigma }} \right) } . \end{aligned}$$

(31)

A log-linearization around the certainty case would give the familiar result: $p^{*e}_t= {{\mathrm{{\mathbb {E}}}}}_{t-1} (p^*_t)$.

Finally, to derive Eq. (9), first note that integrating the production function over all firms gives: $\int y_t(j)\hbox {d}j = a_t l_t$. Next, recalling that there are only two types of firms and that the demand of their good is given in Eq. (26), gives:

$$\begin{aligned} \int y_t(j)\hbox {d}j = k_t \left[ \lambda \left( \frac{p^*_t}{p_t} \right) ^{-\sigma } k_t^{\theta \sigma } y_t + (1-\lambda ) \left( \frac{p^{*e}_t}{p_t} \right) ^{-\sigma } k_t^{\theta \sigma } y_t \right] . \end{aligned}$$

(32)

Rearranging delivers the expression for $\Delta _t$.

Unproductive banks These banks are born in period $t-1$ with an ownership claim on period t working capital $(1-\omega ) \kappa$, and die at the end of period t distributing all of their gains as dividends in the form of consumption to the household.

At date t, their problem is simple. They enter the period with net worth $\tilde{n}_t$. They cannot find a firm to lend, and will be dead by the end of the period, so they do not buy or sell any financial assets. Therefore, all they do is to distribute their net worth entirely to the household to consume. This net worth is given by:

$$\begin{aligned} p_t \tilde{n}_t = R^x_{t-1} x_t + (1+i_{t-1} ) \tilde{v}_{t-1} + \delta _t \left( \tilde{b}^p_{t-1} + q_{t} \tilde{B}^p_{t-1}\right) + p_t \tilde{x}_t. \end{aligned}$$

(33)

This captures the returns earned: (i) on the interbank market from lending to good banks $x_t$ with promised repayment $R^x_{t-1}$, (ii) from the central bank from the deposits of reserves $\tilde{v}_{t-1}$, (iii) from selling bonds issued by the fiscal authority $(\tilde{b}^p_{t-1},\tilde{B}^p_{t-1})$, and (iv) on the claims to capital that it could potentially trade with other unproductive banks (but will not in equilibrium) $\tilde{x}_t$ at a shadow price $q^x_{t-1}$.

At date $t-1$ the bank chooses $(x_t,\tilde{v}_{t-1},\tilde{b}^p_{t-1},\tilde{B}^p_{t-1},\tilde{x}_t)$, to solve:

$$\begin{aligned}&\max {{\mathrm{{\mathbb {E}}}}}_{t-1} \left( m_{t-1,t} \tilde{n}_t \right) \quad \text { subject to:} \end{aligned}$$

(34)

$$\begin{aligned}&p_{t-1} x_t + \tilde{v}_{t-1} + q_{t-1} \tilde{b}^p_{t-1} + Q_{t-1} \tilde{B}^p_{t-1} \le p_{t-1} q^x_{t-1} [ (1-\omega ) \kappa - \tilde{x}_t]. \end{aligned}$$

(35)

and subject also to the upper bound on $x_t$ in Eq. (3) that maintains incentives for repayment.

Optimal behavior by the unproductive banks implies that for there to be a solution with non-infinite holdings of each asset the arbitrage conditions in Eq. (6) hold as well as the further two similar conditions:

$$\begin{aligned}&(1+i_t) {{\mathrm{{\mathbb {E}}}}}_t \left( \frac{m_{t,t+1} p_t}{ p_{t+1}} \right) = 1, \end{aligned}$$

(36)

$$\begin{aligned}&1+ i_t = R^x_{t}. \end{aligned}$$

(37)

Because the bad banks are indifferent between the use of all these funds, the amount invested in each is indeterminate. I break this indeterminacy by assuming that the bank lends in the interbank market all of its capital, or that $x_t$ is driven to its incentive upper bound.

Productive banks The problem facing these banks is more interesting, because they are engaged, sequentially, in the interbank and financial markets at date $t-1$, the deposit market at the beginning of period t, the loan market during period t, before returning their net worth to households at the end of period t. As with unproductive banks, I discuss these problems one at a time, moving backwards in time.

Starting with the loan market, the productive banks have available resources equal to their initial capital which they have kept, $\omega \kappa$, plus the amount they collected from the unproductive banks in the interbank market, $x_t$, plus the amount they collected in the deposit market from households $z_t$. The choice is whether to distribute these to household as consumption, or to lend them to firms for production at an interest rate $r_t$. Since there is no risk in the loan, as long as $r_t$ is larger than zero, the bank will lend all its capital out. Assumption 2 ensures this is the case when prices are flexible, and in all other cases in the paper I always verify that the condition holds. Therefore, total capital employed in the economy is equal to the resources available to banks:

$$\begin{aligned} k_t = \omega \kappa + x_t + z_t \le 1. \end{aligned}$$

(38)

This expression times $(1+r_t)$ is returned to the households at the end of the life of the bank.

Moving one step backwards in time, the banks have net worth of $n_t$ when they arrive at the deposit market. Since they will earn a return $1+r_t>1$ on capital and only pay $R^z_t = 1$ to deposit holders, they would like to collect as much in deposits as possible. However, the incentive constraint in Eq. (4) puts an upper bound on how much the banks can collect in this market. Optimal behavior implies that this constraint holds with equality.

Moving one further step in time, at the start of period t, the bank arrives with bonds and reserve holdings, as well as debts in the interbank market. Their available capital at this stage is $\omega \kappa + x_t$, after they sell their collateral, which no longer serves any useful role. The productive banks owe $x_t$ to the unproductive banks, and the incentive constraint and the seniority of interbank debt ensure that this debt is repaid in full. Therefore, it is the good bank that suffers the loss when this collateral does not pay in full. Therefore, the bank’s net worth after settling claims in the interbank market and selling its collateral is:

$$\begin{aligned} n_t = \omega \kappa - b^p_{t-1} (1-\delta _t)/p_t. \end{aligned}$$

(39)

Combining this with the incentive constraint for deposits gives:

$$\begin{aligned} z_t \le \left( \frac{\gamma (1+r)}{1 - \gamma (1+r)} \right) [ \omega \kappa - b^p_{t-1} (1-\delta _t)/p_t ]. \end{aligned}$$

(40)

Finally, consider the choices at period $t-1$. Similar to the unproductive banks, the productive banks go to the capital markets with their capital $\omega \kappa$ to maximize net worth, which is given by Eq. (39) plus the excess returns earned on holding bonds minus what it pays on the loans in the interbank market. This excess return though is zero in equilibrium, since what the good banks earn on bonds and on reserves is given to the unproductive banks in exchange for their capital.

The productive banks’ resource constraint is:

$$\begin{aligned} \hat{v}_{t-1} + q_{t-1} \hat{b}^p_{t-1} + Q_{t-1} \hat{B}^p_{t-1} + p_{t-1} \hat{x}_t \le p_{t-1} x_t + p_{t-1} \omega \kappa . \end{aligned}$$

(41)

Productive banks can use the capital that they have from their endowment and the interbank market to: (i) make deposits at the central bank, (ii) buy government bonds, or (iii) to hold the claims until next period.

Given the arbitrage conditions already derived for the unproductive banks, these banks want to maximize the amount of capital that they have next period that can earn return $r_t$. Therefore, they wish to maximize borrowing in the interbank market, and buy no government bonds or reserves. However, they face the incentive constraint in the interbank market in Eq. (4). This will bind so the banks’ demand for bonds, after rearranging Eq. (3) is:

$$\begin{aligned} b^p_{t-1} = \frac{(1- \xi ) p_t x_t - v_{t-1}}{q_{t-1}} \le b_{t-1} \end{aligned}$$

(42)

where the last inequality reflects the market constraint that banks cannot hold more government bonds than the ones outstanding.

Finally, for the amount of interbank lending, $x_t$, I have already determined that by paying an interest rate an epsilon above the market interest rate, then the unproductive banks would like to lend out all of their $(1-\omega )\kappa$ claims on capital. Then, there are two cases: either the bond holdings of productive banks are strictly below the amount of bonds outstanding, so

$$\begin{aligned} x_t&= (1-\omega )\kappa , \end{aligned}$$

(43)

$$\begin{aligned} b^p_{t-1}&= \frac{p_t (1- \xi ) (1-\omega )\kappa - v_{t-1}}{q_{t-1}} \le b_{t-1}, \end{aligned}$$

(44)

or instead the constraint on the total amount of bonds available binds and:

$$\begin{aligned} p_t x_t&= \frac{q_{t-1} b_{t-1} + v_{t-1}}{1- \xi }, \end{aligned}$$

(45)

$$\begin{aligned} b^p_{t-1}&= b_{t-1}. \end{aligned}$$

(46)

Fiscal policy The fiscal authorities choose $\{f_t,\delta _t,b_t,B_t\}$ subject to the flow of funds in equation (1). The no Ponzi scheme on government debt is $\lim _{T \rightarrow \infty } {{\mathrm{{\mathbb {E}}}}}_t [ \delta _{t+T} (b_{t+T-1} + q_{t+T} B_{t+T-1} )/p_{t+T} ] = 0$, which, using the arbitrage conditions on different assets to iterate forward the flow of funds, can be written as the intertemporal budget balance condition:

$$\begin{aligned} \left( \frac{\delta _t}{p_t} \right) ( b_{t-1} + q_t B_{t-1}) = {{\mathrm{{\mathbb {E}}}}}_t \left[ \sum _{\tau =0}^{\infty } m_{t,t+\tau } ( d_{t+\tau } + f_{t+\tau } - g_{t+\tau }) \right] . \end{aligned}$$

(47)

I take the choices of $\{b_t,B_t\}$ as given and consider different policy regimes for $\{\delta _t\}$. Therefore, the only choice for the government is to set $f_t \le \bar{f}_t$. Since fiscal crises come with welfare costs, and these strictly decline with $f_t$, in a crisis the fiscal authority will always want to set $f_t$ as high as possible. Note that:

$$\begin{aligned} f_t = \tilde{f}_t - \left( \frac{ 1}{\sigma -1} \right) \int p_t(j) y_t(j) \hbox {d}j . \end{aligned}$$

(48)

Central bank The central bank chooses $\{i_t,v_t,b^c_t,B^c_t\}$ subject to the flow of funds in Eq. (2) and the intertemporal constraint that follows from the no-Ponzi scheme on reserves:

$$\begin{aligned} \frac{(1+i_{t-1})v_{t-1} - \delta _t \left( b^c_{t-1} + q_t B^c_{t-1}\right) }{p_t} = {{\mathrm{{\mathbb {E}}}}}_t \left[ \sum _{\tau =0}^{\infty } m_{t,t+\tau } ( s_{t+\tau } - d_{t+\tau }) \right] . \end{aligned}$$

(49)

I restrict the set of policies to study quantitative easing, understood as changes in the balance sheet such that $\Delta v_t =q_t \Delta b^c_t + Q_t \Delta B^c_t$. Interest rates are either chosen following a rule, like the Taylor rule, that enforces the price-level target $p_t=1$, or instead accommodate the real interest rates and inflation determined elsewhere in the model, given the Fisher equation (37).

Market clearing There is no storage technology or any way to transfer resources over time, so the market clearing condition for goods is:

$$\begin{aligned} c_t + g_t + k_t = y_t + 1. \end{aligned}$$

(50)

In the labor and capital markets, clearing requires, respectively:

$$\begin{aligned} l_t&= \int l_t(j) \hbox {d}j, \end{aligned}$$

(51)

$$\begin{aligned} k_t&= \int k_t(j) \hbox {d}j. \end{aligned}$$

(52)

Market clearing in deposit, interbank and reserves is already implicit in the notation, while market clearing in the government bonds market requires:

$$\begin{aligned} b_t = b^c_t + b^p_t + \tilde{b}^p_t + b_t^h, \end{aligned}$$

(53)

$$\begin{aligned} B_t = B^c_t + \tilde{B}^p_t + B_t^h. \end{aligned}$$

(54)

Equilibrium Repeated from the text, an equilibrium is a collection of outcomes in goods markets $\{c_t,y_t,y_t(j),p_t,p_t(j)\}$, in labor markets $\{l_t,l_t(j),w_t\}$, in the credit, deposit and interbank markets $\{r_t,k_t,x_t,z_t,b^p_t\}$, and in bond markets $\{q_t,Q_t\}$, such that all agents behave optimally and all markets clear, and given exogenous processes for $\{\bar{f}_t,s_t,a_t,g_t\}$ together with choices for fiscal policy $\{f_t,\delta _t,b_t,B_t\}$ and monetary policy $\{i_t,v_t,b^c_t,B^c_t\}$.

1.2 Proof of Lemma 1: The Welfare Function

The first best of the economy is easy to describe since it solves:

$$\begin{aligned} \max&{{\mathrm{{\mathbb {E}}}}}_t \left[ \sum _{\tau =0}^{\infty } \beta ^\tau \left( c_{t+\tau } + g_{t+\tau } - \frac{l_{t+\tau }^{1+\alpha }}{1+\alpha } \right) \right] , \\ y_t&= \left( k_t^\theta \int _0^{k_t} y_t(j)^{\frac{\sigma -1}{\sigma }} \hbox {d}j \right) ^{\frac{\sigma }{\sigma -1}}, \\ y_t(j)&=\left\{ \begin{array}{c l} a_t l_t(j) &{} \text { if }\quad k_t(j)=1,\\ 0 &{} \text { if } \quad k_t(j)=0, \end{array}\right. \\ l_t&= \int _0^{k_t} l_t(j) \hbox {d}j, \\ k_t&= \int _0^1 k_t(j) \hbox {d}j, \\ c_t + g_t + k_t&= y_t + 1. \end{aligned}$$

It is easy to see that the optimal solution requires symmetry so $l_t(j),y_t(j)$ are the same for all j and so $l_t = k_t l_t(j)$ and $y_t = k_t^{(1+\theta )\sigma /(\sigma -1)} y_t(j)$. Replacing these into the objective function, together with the resource constraint and the production function, gives a static optimization problem at every date t to choose $k_t$ and $l_t$.

For that problem, as long as the two conditions in assumption 2 hold, then welfare is strictly increasing in capital in the [0, 1] interval. Therefore, $k_t(j)^*=k_t^* =1$. Optimality with respect to $l_t$ then implies that $l_t(j)^*=l_t^* = a_t^{1/\alpha }$ for all j, that $y_t(j)^* = y_t^* = a_t^{(1+\alpha )/\alpha }$ for all j, and that $c_t^*=y_t-g_t$.

Manipulating the utility function gives:

$$\begin{aligned} c_{t} + g_{t} - \frac{l_{t}^{1+\alpha }}{1+\alpha }&= y_t + 1 - k_t - \frac{\left( \frac{y_t k_t^{\frac{1+\sigma \theta }{1-\sigma }} \Delta _t}{a_t } \right) ^{1+\alpha }}{1+\alpha } \end{aligned}$$

(55)

$$\begin{aligned}&= y_t + k_t^* - k_t - \frac{\left( y_t k_t^{\frac{1+\sigma \theta }{1-\sigma }} \Delta _t \right) ^{1+\alpha }}{ y_t^{* \alpha } (1+\alpha )} \end{aligned}$$

(56)

$$\begin{aligned}&= y^*_t \left[ \frac{y_t}{y^*_t} + \frac{k_t^* - k_t }{y^*_t} - k_t^{-\alpha } \left( \frac{y_t}{y^*_t} \right) ^{1+\alpha } \frac{\Delta _t^{1+\alpha }}{1+\alpha } \right] . \end{aligned}$$

(57)

The first equality comes from using the market clearing condition for goods and the aggregate production function; the second equality from using the definition of $(y^*_t,k^*_t)$, and the third from simple division and from assumption 2.2. This proves the lemma.

1.3 Proof of Proposition 2

From date 2 onwards, there are no more shocks. Because there are no relevant dynamics, I drop the t subscript from all variables. I conjecture that $p_1=\delta _1=1$ and verify that this is consistent with equilibrium.

All firms have full information, so they all choose the same output level y(j) and the same prices, so there is no price dispersion $\Delta =1$. The equilibrium conditions in the real side of the economy are:

$$\begin{aligned} y&= k^{\frac{(1+\theta ) \sigma }{\sigma -1}} y(j), \end{aligned}$$

(58)

$$\begin{aligned} a l&= k y(j), \end{aligned}$$

(59)

$$\begin{aligned} l^{\alpha }&= w, \end{aligned}$$

(60)

$$\begin{aligned} k^{\frac{1+ \theta \sigma }{\sigma -1}}&= \frac{w}{a}, \end{aligned}$$

(61)

$$\begin{aligned} 1+r&= \left( \frac{1}{\sigma -1} \right) \frac{w y(j)}{a}. \end{aligned}$$

(62)

These 5 equations can be easily solved for the 5 unknown variables (y(j), y, l, w, r) as a function of k.

From the solution for the real interest rate:

$$\begin{aligned} 1+r = \left( \frac{a^{1+1/\alpha }}{\sigma -1} \right) k^{(1+1/\alpha )(1+ \theta \sigma )/(\sigma -1) - 1}. \end{aligned}$$

(63)

Assumption 2 implies that this expression is constant and that $r>0$. Therefore, banks want to maximize k subject to their incentive constraints, which must therefore bind.

Turning to the capital market, combining all the equilibrium conditions gives:

$$\begin{aligned} k&= \omega \kappa + x + z, \ \end{aligned}$$

(64)

$$\begin{aligned} z&= \min \left\{ \left( \frac{\gamma (1+r)}{1 - \gamma (1+r)} \right) ( \omega \kappa ) , 1-\kappa \right\} , \end{aligned}$$

(65)

$$\begin{aligned} x&= \min \left\{ (q b + v)/(1- \xi ) , (1-\omega )\kappa \right\} , \end{aligned}$$

(66)

$$\begin{aligned} b^p&= \min \left\{ b, \frac{(1- \xi ) x - v}{q} \right\} . \end{aligned}$$

(67)

The arbitrage conditions for bonds imply that:

$$\begin{aligned} q&= (1+i)^{-1} = \beta , \end{aligned}$$

(68)

$$\begin{aligned} Q&=\beta ^2. \end{aligned}$$

(69)

This set of conditions becomes simply: $k=1,z=1-\kappa ,x=(1-\omega )\kappa$, as long as:

$$\begin{aligned} b \ge \frac{(1- \xi ) (1-\omega )\kappa - v}{\beta }. \end{aligned}$$

(70)

Next turn to the intertemporal budget balance of the government evaluated at the bond prices above. From date 2 onwards, it must be that:

$$\begin{aligned} \frac{v_{t-1}}{\beta } + b_{t-1}-b^c_{t-1} + \beta \left( B_{t-1}-B^c_{t-1}\right) = \sum _{\tau =0}^{\infty } \beta ^\tau {f}_{t+\tau } + \frac{s-g}{1-\beta }. \end{aligned}$$

(71)

Given a default or inflation at date 1, the debt brought forward will satisfy this condition. This concludes the proof: $p_t=\delta _t=1$ satisfies all the equilibrium conditions from date 2 onwards.

1.4 Proof of Proposition 1

Proposition 1 is a corollary of proposition 2. With no fiscal crisis, all the steps in the proof can be restated for dates 0 and 1. Moreover, with the choice of $f_t$ in the proposition, and the condition on $b_t$, the two conditions stated in the proof of proposition 2 hold. Among all the equilibrium conditions, clearly the QE policy choices only appear in Eq. (71), so that QE is neutral.

1.5 Proof of Proposition 3

Given $\delta _1$, at date 0, the intertemporal budget balance of the government at dates 0 and 1, after replacing arbitrage conditions, is:

$$\begin{aligned}&\frac{v_{-1}/\beta + b_{-1} - b^c_{-1}}{p_0} + \beta (B_{-1}-B^c_{-1}) {{\mathrm{{\mathbb {E}}}}}_0\left( \frac{\delta _1}{p_1} \right) = \frac{\bar{f} -g + s}{1-\beta } - \beta \phi (1-\pi ), \end{aligned}$$

(72)

$$\begin{aligned}&\frac{v_0}{\beta p_0 p_1 {{\mathrm{{\mathbb {E}}}}}_0(1/p_1)} + \delta _1 \left[ \frac{(b_0-b^c_0)}{p_1} + \beta (B_0-B^c_0) \right] = \frac{\beta \bar{f}-g+s}{1-\beta } + f_1. \end{aligned}$$

(73)

Recall that there are two possible values of $f_1$ and so two versions of the last equation above, determining two values of $p_1$, one for each state. Therefore, these are three equations pinning down three values $p_0,p_1',p_1''$.

Turning to the capital markets at dates 0 and 1, as studied in the previous section, as long as $r_t>0$, which I will verify in every case, then

$$\begin{aligned} k_t&= \omega \kappa + x_t + z_t, \end{aligned}$$

(74)

$$\begin{aligned} z_t&= \min \left\{ \left( \frac{\gamma (1+r_t)}{1 - \gamma (1+r_t)} \right) [ \omega \kappa - b^p_{t-1} (1-\delta _t) ] , 1-\kappa \right\} , \end{aligned}$$

(75)

$$\begin{aligned} x_t&= \min \left\{ (q_{t-1} b_{t-1} + v_{t-1})/(1- \xi ) , (1-\omega )\kappa \right\} , \end{aligned}$$

(76)

$$\begin{aligned} b^p_{t-1}&= \min \left\{ b_{t-1}, \frac{(1- \xi ) x_t - v_{t-1}}{q_{t-1}} \right\} . \end{aligned}$$

(77)

At date 0, I assume that $k_0 =1, z_0 = 1-\kappa , x_0 = (1-\omega )\kappa$. At date 1, given the two states of the world, there are two possible values for $k_1,z_1$ and one for $x_1,b^p_{0}$, taking $\delta _1$ as given from the previous block. Focusing on the case where the incentive constraints bind and all channels are operative, this system becomes:

$$\begin{aligned} k_1&= \omega \kappa + x_1 + z_1, \end{aligned}$$

(78)

$$\begin{aligned} z_1&= \left( \frac{\gamma (1+r_1)}{1 - \gamma (1+r_1)} \right) \left[ \omega \kappa - b^p_{0} (1-\delta _1) \right] , \end{aligned}$$

(79)

$$\begin{aligned} x_1&= (q_{0} b_{0} + v_{0})/(1- \xi ), \end{aligned}$$

(80)

$$\begin{aligned} b^p_{0}&= b_{0}. \end{aligned}$$

(81)

Note that assuming that the first best is achieved at date 0, but at date 1 all constraints bind, puts restrictions on parameters $\gamma ,\omega ,\kappa ,b^p_0$.

Given the $k_1,p_0,p_1$ pinned down this way, one can finally turn to the real side of the economy. Combining all equilibrium conditions, and dropping the time index notation for convenience of the notation, the equilibrium in the real side of the economy at $t=0,1$ is:

$$\begin{aligned} y&= k^\frac{(1+\theta ) \sigma }{\sigma -1} \left( \lambda y^{*\frac{\sigma -1}{\sigma }} +(1-\lambda ) y^{*e\frac{\sigma -1}{\sigma }} \right) ^\frac{\sigma }{\sigma -1}, \end{aligned}$$

(82)

$$\begin{aligned} y^*&= \left( \frac{p^*}{p} \right) ^{-\sigma } k^{\theta \sigma } y, \end{aligned}$$

(83)

$$\begin{aligned} y^{*e}&= \left( \frac{p^{*e}}{p} \right) ^{-\sigma } k^{\theta \sigma } y, \end{aligned}$$

(84)

$$\begin{aligned} a l&= k \left( \lambda y^* + (1-\lambda ) y^{*e} \right) , \end{aligned}$$

(85)

$$\begin{aligned} \frac{w}{p}&= l^{\alpha }, \end{aligned}$$

(86)

$$\begin{aligned} p^*&= w/a, \end{aligned}$$

(87)

$$\begin{aligned} p^{*e}&= \frac{{{\mathrm{{\mathbb {E}}}}}\left( \frac{w k^{\theta \sigma } y}{a p^{1-\sigma }} \right) }{{{\mathrm{{\mathbb {E}}}}}\left( \frac{k^{\theta \sigma } y}{p^{1-\sigma }} \right) }. \end{aligned}$$

(88)

For all but the last equation, these hold for each of the two states of the world in date 1. So, in total, there are 13 of these equilibrium equations to solve for 2 values each of $y,y^*,y^{*e}, p^*,l,w$ and one value of $p^{*e}$, which are 13 unknowns, for given values of k, p. At date 0, the same system holds and likewise one can solve for it, noting that expectations at date $-1$ are that the economy would be in its steady state.

Finally, one must check that the value of r that comes out of this system is indeed larger than 0.

$$\begin{aligned} 1+r = \left( \frac{w}{(\sigma -1)ap} \right) \left[ \lambda y^* + (1-\lambda ) \left( \frac{\sigma {{\mathrm{{\mathbb {E}}}}}(w k^{\theta \sigma } y / p^{1-\sigma })}{w {{\mathrm{{\mathbb {E}}}}}(k^{\theta \sigma } y / p^{1-\sigma })} - \sigma +1 \right) y^{*e} \right] . \end{aligned}$$

(89)

1.6 Proof of Proposition 4

All that is needed is one counter-example. This proof provides several.

First, consider an equilibrium where $p_t=1$ always, and where in capital markets the deposit market constraint binds, but the interbank market constraint does not bind. Then:

$$\begin{aligned} q_t b^p_{t-1}&= (1- \xi ) (1-\omega ) \kappa - v_{t-1}, \end{aligned}$$

(90)

$$\begin{aligned} z_{t-1}&= \left( \frac{\gamma (1+r)}{1 - \gamma (1+r)} \right) \left[ \omega \kappa - b^p_{t-1} (1-\delta _1) \right] . \end{aligned}$$

(91)

A change in $(b_t,B_t)$ does not affect these equations. Therefore, debt management cannot prevent losses in net worth and contractions in deposits. All it does, from the market clearing condition in the deposit market, is to change the bond holdings of households since:

$$\begin{aligned} b_t^h&= b_t - b^c_t - b^p_t , \end{aligned}$$

(92)

$$\begin{aligned} B_t^h&= B_t - B^c_t . \end{aligned}$$

(93)

Second, note that $\delta _1$, solved for in the text depends on $v_t$ but not on $(b_t,B_t)$. Therefore, debt management cannot affect the extent of default on bonds.

Third, take the case where instead $\delta _t=1$. Then, allow the central bank to set an arbitrary $i_1$, possibly in disregard of its price-level target. In that case, equilibrium prices have to satisfy the following two equations:

$$\begin{aligned}&\frac{(1+i_0)v_0 + b_0 - b^c_0 + (1+i'_1)(B_0 - B_0^c)}{p'_1} = \frac{\bar{f} -g + s}{1-\beta } - \phi , \end{aligned}$$

(94)

$$\begin{aligned}&\frac{(1+i_0)v_0 + b_0 - b^c_0 + (1+i''_1)(B_0 - B_0^c)}{p''_1} = \frac{\bar{f} -g + s}{1-\beta }. \end{aligned}$$

(95)

Therefore, choices of $(b_0,B_0)$ cannot achieve desired values for the price level independently of $i'_1$ and $i''_1$.

1.7 Proof of Proposition 5

The model changes in only three ways. First, preferences now are:

$$\begin{aligned} {{\mathrm{{\mathbb {E}}}}}_t \left[ \sum _{\tau =0}^{\infty } \beta ^\tau \left[ \left( c_{t+\tau } + g_{t+\tau } - \frac{l_{t+\tau }^{1+\alpha }}{1+\alpha } \right) \right] + \varkappa U \left( \frac{h^h_{t+\tau }}{p_{t+\tau }} \right) \right] . \end{aligned}$$

(96)

This leads to a standard demand for money function:

$$\begin{aligned} U' \left( \frac{h^h_t}{p_t} \right) = \frac{1}{ (1-q_t)}. \end{aligned}$$

(97)

Second, the incentive constraint in the interbank market changes to

$$\begin{aligned} (1- \xi ) x_t \le q_{t-1} b^p_{t-1} + v_{t-1} + h^p_{t-1}. \end{aligned}$$

(98)

Third, the market clearing condition for money balances then is $h_t = h^h_t + h^p_t$, and seignorage now is:

$$\begin{aligned} s_t = \frac{h_t-h_{t-1}}{p_t}. \end{aligned}$$

(99)

Note that I already incorporated in the notation the result that unproductive banks will not want to hold currency, since it is strictly dominated by bonds for positive nominal interest rates.

The proof of the proposition is obvious. First, take the case where there is default. If $q_{t-1} b^0_{t-1} + v_{t-1} \ge (1-\xi )(1-\omega )\kappa$, then $h^p_{t-1}=0$. Then, from money demand, any change in $h_t$ will affect inflation. Therefore, any printing of money that depends on there being a fiscal crisis will generate unexpected inflation at date 1. However, unexpected inflation affects output, via the Phillips curve, and price dispersion. Thus, it affects outcomes and welfare in a way that QE did not, since QE led to no inflation in this case.

Second, consider the case when $\delta _t=1$. Then, during a fiscal crisis, increasing the money supply will both affect the value of the debt, but also generate extra seignorage revenues. Whereas more QE lowered the inflation surprise, more monetary financing will increase the inflation surprise.