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Moral Hazard Misconceptions: The Case of the Greenspan Put

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Abstract

Policy discussions on financial regulation tend to assume that whenever a corrective policy is used ex post to ameliorate the effects of a crisis, there are negative side effects in terms of moral hazard ex ante. This paper shows that this is not a general theoretical prediction, focusing on the case of monetary policy interventions ex post. In particular, we show that if the central bank does not intervene by monetary easing following a crisis, an aggregate demand externality makes borrowing ex ante inefficient. If instead the central bank follows the optimal discretionary policy and intervenes to stabilize asset prices and real activity, we show examples in which the aggregate demand externality disappears, reducing the need for ex ante intervention.

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Notes

  1. See also Benigno et al. (2016) in an open economy context.

  2. The aggregate price in the economy in period 1 is equal to \(\left( \int _0^1 p_{j1}^{1-\epsilon }\mathrm{d}j\right) ^{\frac{1}{1-\epsilon }}\).

  3. The formal proof that \(\partial Y/\partial D<0\) is in “Appendix”.

  4. Here we are thinking of the welfare benefits of producing more goods and assigning them to A agents. Given that there are two groups of agents and imperfect insurance, the notion would be different if we looked at the welfare benefits of assigning the extra goods to B agents. This would change slightly the decomposition and interpretation of our two main forces, but it would change nothing in the substance of the analysis.

  5. The presence of heterogeneous discount factors introduces a source of time inconsistency in optimal policy. By focusing on the welfare function (3), we leave aside that source of time inconsistency. Other interesting sources are present, as we shall see.

  6. An alternative source of real interest rate rigidity that would lead to related results would be to consider an open economy with a fixed exchange rate regime or a monetary union.

  7. Since prices are flexible in period 0, the central bank can choose the nominal interest rate \(i_{0}\) but has no power to affect the real interest rate between dates 0 and 1, \((1+i_{0})p_{0}/p_{1}\), which is determined at the level that clears the bond market.

  8. Using the optimality condition for labor supply ,we obtain the equilibrium real wage

    $$\begin{aligned} \frac{w_{1}}{p_{1}} u'\left( c_{1}^{A}\right) =v'\left( Y\right) . \end{aligned}$$

    Substituting in the optimality condition of price setters, we obtain

    $$\begin{aligned} E\left[ Y\left( \left( 1+\sigma \right) u'\left( c_{1}^{A}\right) -\frac{\epsilon }{\epsilon -1}v'\left( Y\right) \right) \right] =0. \end{aligned}$$

    The value of \(\sigma\) is chosen to satisfy this condition.

  9. Complementarity/substitutability is defined in this way in this policy space, because lowering the rate more in the bad state is a more aggressive intervention.

  10. This is different from a policy that subsidizes directly the asset price. If such policy could be financed by lump sum taxes on A agents, it would effectively transfer resources between the agents with no efficiency cost, thus allowing the planner to reach the first best.

  11. The presence of the collateral constraints does not alter the fact that the effect of a change in r on the utility of the B agents is captured by \(u'\left( c_{1}^{B}\right) d_{1}/\left( 1+r\right)\).

  12. For ease of notation, whenever possible, we omit the state s for the different endogenous variables. Note that these variables do vary with the state of the economy.

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Authors and Affiliations

Authors

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Correspondence to Guido Lorenzoni.

Additional information

For very useful comments, we thank Eduardo Davila, Pierre-Olivier Gourinchas, Olivier Jeanne, Maury Obstfeld, the editor, and two referees.

Appendix

Appendix

1.1 Characterization of Continuation Equilibrium

1.1.1 General Utility Function

Recall the value functions for A and B agents are given byFootnote 12

$$\begin{aligned} V^{A}(b,D,r,s)= & {} \max \nolimits _{c_{1}^{A},c_{2}^{A}} u(c_{1}^{A})-v\left( Y(D,r,s)\right) +u\left( c_{2}^{A}\right) \\&\text {s.t.} \quad c_{1}^{A}+\frac{1}{1+r}c_{2}^{A}=Y(D,r,s)+b,\\ V^{B}(d,D,r,s)= & {} \max \nolimits _{c_{1}^{B},c_{2}^{B}} u\left( c_{1}^{B}\right) +\beta u\left( c_{2}^{B}\right) \\&\text {s.t.}\quad c_{1}^{B}+\frac{1}{1+r}c_{2}^{B}=\delta _{1}\left( s\right) +\frac{1}{1+r}\delta _{2}(s)-d. \end{aligned}$$

The optimality conditions of the two types of agents are

$$u^{\prime } \left( {c_{1}^{A} } \right) = (1 + r)u^{\prime } \left( {c_{2}^{A} } \right),\quad u^{\prime } \left( {c_{1}^{B} } \right) = \beta (1 + r)u^{\prime } \left( {c_{2}^{B} } \right),$$

which together with their resource constraint characterize the continuation equilibrium. That is, given D and r, a continuation equilibrium is given by the quantities \(c_{1}^{A}\), \(c_{2}^{A}\), \(c_{1}^{B}\), \(c_{2}^{B}\), Y that satisfy:

$$\begin{aligned} & u^{\prime } \left( {c_{1}^{A} } \right) = (1 + r)u^{\prime } \left( {c_{2}^{A} } \right),\quad u^{\prime } \left( {c_{1}^{B} } \right) = \beta (1 + r)u^{\prime } \left( {c_{2}^{B} } \right), \\ & c_{1}^{A} + \frac{1}{{1 + r}}c_{2}^{A} = Y + D,\quad c_{1}^{B} + \frac{1}{{1 + r}}c_{2}^{B} = \delta _{1} + \frac{1}{{1 + r}}\delta _{2} - D, \\ \end{aligned}$$

and the goods market clearing conditions,

$$c_{1}^{A} + c_{1}^{B} = Y + \delta _{1} ,\quad c_{2}^{A} + c_{2}^{B} = \delta _{2} .$$

We use Y to denote produced output in period 1. Then, the definition of continuation equilibrium above defines a mapping \(Y\left( D,r,s\right)\), which corresponds to the equilibrium produced output in period 1 as a function of the debt level and the level of interest rate. In the next section we derive an explicit formula for \(Y\left( D,r,s\right)\) by considering a specific functional form for the utility of consumption, the CRRA form.

1.1.2 CRRA Utility

Here we provide a characterization of a continuation equilibrium that will be used for the proofs to follow and for the numerical examples. The utility function u has the CRRA form

$$\begin{aligned} u\left( c\right) =\frac{1}{1-\gamma }c^{1-\gamma }. \end{aligned}$$

From consumer optimization, we obtain

$$\begin{aligned} c_{1}^{A}=\frac{1}{1+\left( 1+r\right) ^{\frac{1}{\gamma }-1}}\left( Y+D\right) ,\quad c_{1}^{B}=\frac{1}{1+\beta ^{\frac{1}{\gamma }}\left( 1+r\right) ^{\frac{1}{\gamma }-1}}\left( \delta _{1}+\frac{\delta _{2}}{1+r}-D\right) . \end{aligned}$$

Aggregating and using goods market clearing, we have

$$\begin{aligned} Y=\frac{1}{1+\left( 1+r\right) ^{\frac{1}{\gamma }-1}}\left( Y+D\right) +\frac{1}{1+\beta ^{\frac{1}{\gamma }}\left( 1+r\right) ^{\frac{1}{\gamma }-1}}\left( \delta _{1}+\frac{\delta _{2}}{1+r}-D\right) - \delta _1. \end{aligned}$$

After rearranging and defining \({\tilde{R}}=(1+r)^{\frac{1-\gamma }{\gamma }}\) and \({\tilde{\beta }}=\beta ^{\frac{1}{\gamma }}\), the continuation equilibrium quantities in period 1 can be written as follows:

$$\begin{aligned} c_{1}^{A}&=\frac{1}{1+{\tilde{R}}}(Y+D), \end{aligned}$$
(19)
$$\begin{aligned} c_{1}^{B}&=\frac{1}{1+{\tilde{\beta }}{\tilde{R}}}\left( \delta _{1}+\frac{\delta _{2}}{1+r}-D\right) , \end{aligned}$$
(20)
$$\begin{aligned} Y&=-\frac{1-{\tilde{\beta }}}{1+{\tilde{\beta }}{\tilde{R}}}D -\left( 1-\frac{1-{\tilde{\beta }}}{1+{\tilde{\beta }}{\tilde{R}}}\right) \delta _1 +\frac{1+{\tilde{R}}}{{\tilde{R}}\left( 1+{\tilde{\beta }} {\tilde{R}}\right) }\frac{\delta _{2}}{1+r} . \end{aligned}$$
(21)

Taking derivatives with respect to D and r, we obtain:

$$\begin{aligned} \frac{\partial Y}{\partial D}= & {} -\frac{1-{\tilde{\beta }}}{1+{\tilde{\beta }}{\tilde{R}}}, \end{aligned}$$
(22)
$$\begin{aligned} \frac{\partial Y}{\partial r}= & {} \theta _{1}(r) (\delta _1-D) +\theta _{2}(r)\delta _2, \end{aligned}$$
(23)

where

$$\begin{aligned} \theta _{1}(r)= & {} -\frac{1}{1+r}\frac{1-\gamma }{\gamma }\frac{\left( 1-{\tilde{\beta }}\right) {\tilde{\beta }}{\tilde{R}}}{\left( 1+{\tilde{\beta }}{\tilde{R}}\right) ^{2}},\\ \theta _{2}(r)= & {} -\frac{1}{(1+r)^{2}}\frac{1}{{\tilde{R}}\left( 1+{\tilde{\beta }}{\tilde{R}}\right) ^{2}}\frac{1}{\gamma }\left[ 1+{\tilde{\beta }}{\tilde{R}}+\left( \gamma (1-{\tilde{\beta }})+{\tilde{\beta }}(1+{\tilde{R}})\right) {\tilde{R}}\right] . \end{aligned}$$

Notice that \(\theta _{2}(r)\) is negative, while the sign of \(\theta _{1}(r)\) depends on whether \(\gamma\) is greater or lower than 1. The intuition is as follows. An increase in the interest rate decreases the value of endowment of B agents in period 2. So the relative discounted income of A agents goes up. As A agents are more patient and output is demand-determined, this channel decreases the level of output in period 1. However, a change in the interest rate also affects B agents through their net endowment, \(\delta _1-D\), in period 1. On the one hand, the substitution effect incentivizes households to increase their demand for savings and lower consumption in period 1. On the other hand, there is an income effect incentivizing households to increase consumption in period 1. With log preferences (\(\gamma =1\)), the substitution and income effect cancel each other and the expressions above become:

$$\begin{aligned} c_{1}^{A}= & {} \frac{1}{2}\left( Y+D\right) ,\quad c_{1}^{B}=\frac{1}{1+\beta }\left( \delta _{1}+\frac{\delta _{2}}{1+r}-D\right) ,\\ Y= & {} -\frac{1-\beta }{1+\beta }D - \frac{2\beta }{1+\beta }\delta _1+\frac{2}{1+\beta }\frac{\delta _{2}}{1+r}. \end{aligned}$$

1.2 Proofs for Section 4

1.2.1 Proof of Proposition 1

Proof

Denote the asset price by Q, where

$$\begin{aligned} Q=\frac{\delta }{1+r}. \end{aligned}$$

We guess and verify that an equilibrium with constant asset prices, \({\bar{Q}}\), exists. Given the derivations in Sect. 8.1.2, it is immediate that \(c_{1}^{B}\) is constant across states when \(\delta /1+r\) is constant

$$\begin{aligned} c_{1}^{B}=\frac{1}{1+\beta }\left( \frac{\delta }{1+r}-D\right) . \end{aligned}$$

Equilibrium in the bonds market at \(t=0\) and equilibrium in the goods market at \(t=1\) then give the two equations

$$\begin{aligned}&u'\left( c_{1}^{A}\right) =\beta u'\left( c_{1}^{B}\right) ,\\&Y=c_{1}^{A}+c_{1}^{B}. \end{aligned}$$

With log preferences, these yield

$$\begin{aligned} c_{1}^{A}=\frac{1}{1+\beta }Y,\quad c_{1}^{B}=\frac{\beta }{1+\beta }Y. \end{aligned}$$

The zero-output-gap condition then becomes

$$\begin{aligned} u'\left( \frac{1}{1+\beta }Y\right) =v'\left( Y\right) , \end{aligned}$$

which has a unique solution Y. The value of D can be found using the condition

$$\begin{aligned} c_{1}^{A}=\frac{1}{1+\beta }Y=\frac{1}{2}\left( Y+D\right) , \end{aligned}$$

and the value for \({\bar{Q}}\) using the condition

$$\begin{aligned} c_{1}^{B}=\frac{\beta }{1+\beta }Y=\frac{1}{1+\beta }\left( \frac{\delta }{1+r}-D\right) . \end{aligned}$$

The interest rates are

$$\begin{aligned} 1+r\left( s\right) =\left( 1+\delta (s)\right) /{\bar{Q}}. \end{aligned}$$

It is immediate to check that all conditions for a zero-output-gap equilibrium are satisfied. It is also easy to check that the optimal monetary policy condition (10) is satisfied, since both of its terms are equal to zero in all states. \(\square\)

Note that when there is no uncertainty, i.e., \(\sigma =0\), we have that the inertial regime is equivalent to the proactive regime.

1.2.2 Proof of Proposition 2

Proof

Without loss of generality we let

$$\begin{aligned} \delta _2 = {\bar{\delta }} + \sigma z , \end{aligned}$$

where \({\bar{\delta }}\) is the mean of \(\delta _2\), z is a random variable with mean 0 and variance of 1, and \(\sigma\) is a positive scalar. In this proof ,we show that there exists a \(\underline{\sigma }\), such that if \(\sigma \in (0,\underline{\sigma })\) then the level of borrowing in the inertial regime is lower than in the proactive regime.

In this proof, we use a change of variables: \(m = \tfrac{1}{1+\beta }\tfrac{1}{1+r}\), and \(b = \tfrac{1}{1+\beta }D\). Since the proof uses only quantities in period 1, we use the notation \(c_a \equiv c_1^A\) and \(c_b \equiv c_1^B\). Using the change of variables we have that

$$\begin{aligned}&c_A = m({\bar{\delta }}+\sigma z) + \beta b, \\&c_B = m({\bar{\delta }}+\sigma z) - b, \\&Y = 2m({\bar{\delta }}+\sigma z)- (1-\beta ) b . \end{aligned}$$

The two equilibrium conditions are

$$\begin{aligned}&\mathbb {E}\left[ \frac{1}{m({\bar{\delta }}+\sigma z) + \beta b} - \beta \frac{1}{m({\bar{\delta }}+\sigma z) - b } \right] = 0 ,\nonumber \\&\mathbb {E}\left[ \left( \frac{1}{m({\bar{\delta }}+\sigma z) + \beta b} - \beta \frac{1}{m({\bar{\delta }}+\sigma z) - b }\right) \left( m({\bar{\delta }}+\sigma z) + \beta b\right) \right] \nonumber \\&\qquad =\mathbb {E}\left[ \left( \frac{1}{m({\bar{\delta }}+\sigma z) + \beta b}-\psi \left( 2m({\bar{\delta }}+\sigma z)-(1-\beta )b\right) ^{\phi }\right) 2m({\bar{\delta }}+\sigma z) \right] , \end{aligned}$$
(24)

where the first one is the equation which pins down the level of borrowing, and the second one is the optimality condition for the social planner’s choice of interest rate. We can rearrange the second equation as follows,

$$\begin{aligned}&\mathbb {E}\left[ \frac{m({\bar{\delta }}+\sigma z) + \beta b}{m({\bar{\delta }}+\sigma z) + \beta b} - \beta \frac{m({\bar{\delta }}+\sigma z) + \beta b}{m({\bar{\delta }}+\sigma z) - b } \right] \\&\qquad = 2\mathbb {E}\left[ \frac{m({\bar{\delta }}+\sigma z)}{m({\bar{\delta }}+\sigma z) + \beta b}-\psi \left( 2m({\bar{\delta }}+\sigma z)-(1-\beta )b\right) ^{\phi } m({\bar{\delta }}+\sigma z)\right] \end{aligned}$$

So that

$$\begin{aligned}&1 - \beta - (1+ \beta ) b\mathbb {E}\left[ \frac{\beta }{m({\bar{\delta }}+\sigma z) - b } \right] \\&\qquad = 2 - 2\beta b\mathbb {E}\left[ \frac{1}{m({\bar{\delta }}+\sigma z) + \beta b} \right] -\mathbb {E}\left[ 2\psi \left( 2m({\bar{\delta }}+\sigma z)-(1-\beta )b\right) ^{\phi } m({\bar{\delta }}+\sigma z)\right] \end{aligned}$$

Using Eq. (24) to substitute for \(\mathbb {E}\left[ \tfrac{\beta }{m({\bar{\delta }}+\sigma z) - b } \right]\) and rearranging, we obtain

$$\begin{aligned} \mathbb {E}\left[ 1+\beta +(1-\beta )\frac{b}{m({\bar{\delta }}+\sigma z) +\beta b} - 2\psi \left( 2m({\bar{\delta }}+\sigma z)-(1-\beta )b\right) ^{\phi } m({\bar{\delta }}+\sigma z) \right] = 0 \end{aligned}$$

Let’s define the following two functions,

$$\begin{aligned}&G(m,b,\sigma ) = \mathbb {E}\left[ \frac{1}{m({\bar{\delta }}+\sigma z) + \beta b} - \beta \frac{1}{m({\bar{\delta }}+\sigma z) - b } \right], \end{aligned}$$
(25)
$$\begin{aligned}&H(m,b,\sigma ) = \mathbb {E}\left[ 1+\beta +(1-\beta )\frac{b}{m({\bar{\delta }}+\sigma z) +\beta b} - 2\psi \left( 2m({\bar{\delta }}+\sigma z)-(1-\beta )b\right) ^{\phi } m({\bar{\delta }}+\sigma z) \right]. \end{aligned}$$
(26)

In equilibrium both \(G(m,b,\sigma )\) and \(H(m,b,\sigma )\) are equal to 0. So these two equations define an implicit equilibrium levels of m and b given \(\sigma\), \(m(\sigma )\) and \(b(\sigma )\):

$$\begin{aligned}&G\left( m(\sigma ),b(\sigma ),\sigma \right) =0, \end{aligned}$$
(27)
$$\begin{aligned}&H\left( m(\sigma ),b(\sigma ),\sigma \right) =0. \end{aligned}$$
(28)

A corollary of Proposition 1 is that the levels of m and b in the proactive regime are equal to m(0) and b(0), i.e., the inertial regime equilibrium when there is no uncertainty about \(\delta _2\). We want to show that \(b(\sigma )<b(0)\) for \(\sigma\) in the neighborhood of 0. As a by-product, we also show that \(m(\sigma )<m(0)\) for small enough \(\sigma\). To do so, we prove that \(m'(0)=b'(0)=0\) and that both \(m''(0)\) and \(b''(0)\) are negative. This, in turn, implies that around a neighborhood of \(\sigma =0\) both \(m(\sigma )\) and \(b(\sigma )\) are lower than m(0) and b(0).

Step I – showing \(m'(0)=b'(0)=0\). We take total derivative of equations (27) and (28):

$$\begin{aligned}&G_m(\sigma ) m'(\sigma ) + G_b(\sigma ) b'(\sigma ) + G_\sigma (\sigma ) = 0,\\&H_m(\sigma ) m'(\sigma ) + H_b(\sigma ) b'(\sigma ) + H_\sigma (\sigma ) = 0, \end{aligned}$$

where \(G_m(\sigma )\) is a short notation for \(G_m(m(\sigma ),b(\sigma ),\sigma )\), and similarly for the other partial derivatives. We can rearrange the following equations at \(\sigma =0\) to obtain

$$\begin{aligned} \begin{bmatrix} m'(0) \\ b'(0) \end{bmatrix} = - \begin{bmatrix} G_m(0)&G_b(0) \\ H_m(0)&H_b(0) \end{bmatrix}^{-1} \begin{bmatrix} G_\sigma (0) \\ H_\sigma (0) \end{bmatrix}. \end{aligned}$$

The partial derivatives of \(G(\cdot )\) and \(H(\cdot )\) are presented below. To save on notation, we use \(c_a\) and \(c_b\) when appropriate.

$$\begin{aligned} G_m(m,b,\sigma )&= \mathbb {E}\left[ \left( -\frac{1}{c_a^2} + \beta \frac{1}{c_b^2}\right) \left( {\bar{\delta }}+\sigma z\right) \right],\\ G_b(m,b,\sigma )&= \mathbb {E}\left[ -\beta \left( \frac{1}{c_a^2}+\frac{1}{c_b^2}\right) \right], \\ G_\sigma (m,b,\sigma )&= \mathbb {E}\left[ \left( -\frac{1}{c_a^2} + \beta \frac{1}{c_b^2}\right) mz\right],\\ H_m(m,b,\sigma )&= \mathbb {E}\left[ -\left( (1-\beta ) \frac{b}{c_a^2}+2\psi Y^{\phi } + 2\psi \phi Y^{\phi -1}m ({\bar{\delta }} +\sigma z)\right) ({\bar{\delta }} +\sigma z)\right], \\ H_b(m,b,\sigma )&= \mathbb {E}\left[ (1-\beta )\left( \frac{1}{c_a^2} + 2\psi \phi Y^{\phi -1} \right) m({\bar{\delta }} +\sigma z) \right],\\ H_\sigma (m,b,\sigma )&= \mathbb {E}\left[ -\left( (1-\beta )\frac{b}{c_a^2}+2\psi Y^{\phi } +4\psi \phi Y^{\phi -1} m({\bar{\delta }} +\sigma z)\right) mz \right]. \end{aligned}$$

Evaluating at \(\sigma =0\) using \(\tfrac{c_b(0)}{c_a(0)}=\beta\), together with \(\mathbb {E}(z)=0\) and \(\mathbb {E}\left( z^2\right) =1\),

$$\begin{aligned} G_m(0)&= -\left( \frac{1}{c_b(0)}\right) ^2 \left( \beta ^2-\beta \right) {\bar{\delta }}>0,\\ G_b(0)&= - \left( \frac{1}{c_b(0)}\right) ^2 \beta \left( \beta ^2 + 1 \right)<0, \\ G_\sigma (0)&= 0,\\ H_m(0)&= -{\bar{\delta }}\left( \frac{(1-\beta )b}{\left( c_a(0)\right) ^2}+2\psi \left( Y(0)\right) ^{\phi } + 2\psi \phi \left( Y(0)\right) ^{\phi -1} m {\bar{\delta }} \right) <0,\\ H_b(0)&=(1-\beta )\left( \frac{1}{\left( c_a(0)\right) ^2} + 2\psi \phi \left( Y(0)\right) ^{\phi -1} \right) m {\bar{\delta }} >0,\\ H_\sigma (0)&= 0. \end{aligned}$$

Define the matrix A as follows,

$$\begin{aligned} A = \begin{bmatrix} G_m(0)&G_b(0) \\ H_m(0)&H_b(0) \end{bmatrix} \equiv \begin{bmatrix} a_{1}&a_{2} \\ a_{3}&a_{4} \end{bmatrix}. \end{aligned}$$

We showed above that \(a_1>0\), \(a_2<0\), \(a_3<0\), and \(a_4>0\). So the determinant of A is positive and A is non-singular. In particular, the inverse of A is given by

$$\begin{aligned} A^{-1} = \frac{1}{a_1 a_4 - a_2 a_3} \begin{bmatrix} a_4&-a_2 \\ -a_3&a_1 \end{bmatrix}. \end{aligned}$$

So all the elements of \(A^{-1}\) are positive. For this step of the proof it is enough to know that A is non-singular, but in the next step we will use the fact that \(A^{-1}\) is a positive matrix. Since \(G_\sigma (0)=H_\sigma (0)=0\), we have that

$$\begin{aligned} \begin{bmatrix} m'(0) \\ b'(0) \end{bmatrix} = - A^{-1} \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. \end{aligned}$$

Step II – showing \(m''(0)\) and \(b''(0)\) are negative. Taking a second total derivative of equations (27) and (28), together with \(G_\sigma (0)=H_\sigma (0)=m'(0)=b'(0)=0\), we obtain

$$\begin{aligned}&G_m(0)m''(0)+G_b(0)b''(0) + G_{\sigma \sigma } = 0, \\&H_m(0)m''(0)+H_b(0)b''(0) + H_{\sigma \sigma } = 0. \end{aligned}$$

So that

$$\begin{aligned} \begin{bmatrix} m'(0) \\ b'(0) \end{bmatrix} = -A^{-1} \begin{bmatrix} G_{\sigma \sigma }(0) \\ H_{\sigma \sigma }(0) \end{bmatrix}. \end{aligned}$$

\(G_{\sigma \sigma }\) and \(H_{\sigma \sigma }\) are equal to

$$\begin{aligned} G_{\sigma \sigma }(m,b,\sigma )&= \mathbb {E}\left[ \left( \frac{2}{c_a^3} - \beta \frac{2}{c_b^3}\right) m^2z^2\right] , \\ H_{\sigma \sigma }(m,b,\sigma )&= \mathbb {E}\left[ \left( (1-\beta )\frac{2b}{c_a^3} + 8 \psi \phi Y^{\phi -2} \left( Y + (\phi -1)m({\bar{\delta }} +\sigma z)\right) \right) m^2 z^2 \right] . \end{aligned}$$

At \(\sigma =0\) we have,

$$\begin{aligned} G_{\sigma \sigma }(0)&= -\left( \frac{1}{c_b(0)}\right) ^3 2 \left( \beta ^3 - \beta \right) m^2>0, \\ H_{\sigma \sigma }(0)&= \left( (1-\beta )\frac{2b}{\left( c_a(0)\right) ^3} + 8 \psi \phi \left( Y(0)\right) ^{\phi -2} \left( c_b(0) +\phi m {\bar{\delta }} + \beta b\right) \right) m^2 >0. \end{aligned}$$

Since \(A^{-1}\) is a positive matrix and both \(G_{\sigma \sigma }(0)\) and \(H_{\sigma \sigma }(0)\) are positive, we conclude that \(m''(0)\) and \(b''(0)\) are negative:

$$\begin{aligned} \begin{bmatrix} m''(0) \\ b''(0) \end{bmatrix} = - A^{-1} \begin{bmatrix} G_{\sigma \sigma }(0) \\ H_{\sigma \sigma }(0) \end{bmatrix}<\begin{bmatrix} 0 \\ 0 \end{bmatrix}. \end{aligned}$$

\(\square\)

1.2.3 Proof of Proposition 3

Proof

Proposition 1 and Eq. (12) imply that there is no benefit from changing the level of borrowing, D, in the proactive regime. In what follows we prove that in the inertial regime a decrease in the level of borrowing has welfare benefits.

Step I Show that

$$\begin{aligned} E\left[ \left[ u^{\prime }\left( c_{1}^{A}\right) -\beta u^{\prime }\left( c_{1}^{B}\right) \right] \left( c_{1}^{B}+D\right) \right] >0. \end{aligned}$$

Notice that given CRRA and a constant interest rate the ratio \(c_{1}^{A}/\left( Y+D\right)\) is constant. Since \(Y=c_{1}^{A}+c_{1}^{B}\), this implies that there is a constant \(\xi\) such that

$$\begin{aligned} c_{1}^{A}=\xi \left( c_{1}^{B}+D\right) . \end{aligned}$$

We then want to evaluate

$$\begin{aligned} E\left[ \left[ \left( \xi \left( c_{1}^{B}+D\right) \right) ^{-\gamma }-\beta \left( c_{1}^{B}\right) ^{-\gamma }\right] \left( c_{1}^{B}+D\right) \right] . \end{aligned}$$

Notice that there is a unique cutoff \(\hat{c}_{2}^{B}\) such that

$$\begin{aligned} \left( \xi \left( c_{1}^{B}+D\right) \right) ^{-\gamma } -\beta \left( c_{1}^{B}\right) ^{-\gamma }\gtrless 0, \end{aligned}$$

iff \(c_{1}^{B}\gtrless \hat{c}_{1}^{B}\). Therefore

$$\begin{aligned} E\left[ \left[ \left( \xi \left( c_{1}^{B}+D\right) \right) ^{-\gamma } -\beta \left( c_{1}^{B}\right) ^{-\gamma }\right] \left( c_{1}^{B} -\hat{c}_{1}^{B}\right) \right] >0. \end{aligned}$$

Moreover, from consumers optimality at \(t=0\) we have

$$\begin{aligned} E\left[ \left[ \left( \xi \left( c_{1}^{B}+D\right) \right) ^{-\gamma } -\beta \left( c_{1}^{B}\right) ^{-\gamma }\right] \right] =0. \end{aligned}$$

Combining the last two equations we have

$$\begin{aligned} E\left[ \left[ \left( \xi \left( c_{1}^{B}+D\right) \right) ^{-\gamma } -\beta \left( c_{1}^{B}\right) ^{-\gamma }\right] \left( c_{1}^{B}+D\right) \right] >0. \end{aligned}$$

Step II From Step I and optimality of monetary policy we deduce that

$$\begin{aligned} E\left[ \left( u^{\prime }\left( c_{1}^{A}\right) -v'\left( Y\right) \right) \frac{\partial Y}{\partial r}\right] <0. \end{aligned}$$

With log preferences and a constant interest rate we deduce that

$$\begin{aligned} \frac{\partial Y}{\partial r}=-\Xi _{1}\frac{\delta }{\left( 1+{\bar{r}}\right) ^{2}} \text{ and } \frac{\partial Y}{\partial D}=-\Xi _{2}, \end{aligned}$$
(29)

for some positive constant terms \(\Xi _{1},\Xi _{2}\). Therefore, we have the inequality

$$\begin{aligned} E\left[ \left( u^{\prime }\left( c_{1}^{A}\right) -v'\left( Y\right) \right) \delta \right] >0. \end{aligned}$$

The expression \(u^{\prime }(c_{1}^{A})-v'\left( Y\right)\) is a monotone decreasing function of \(\delta\), so we have the chain of inequalities

$$\begin{aligned} E\left[ \left( u^{\prime }\left( c_{1}^{A}\right) -v'\left( Y\right) \right) \right] E\left[ \delta \right]>E\left[ \left( u^{\prime }(c_{1}^{A})-v'\left( Y\right) \right) \delta \right] >0. \end{aligned}$$

Using (29) we then conclude that

$$\begin{aligned} E\left[ \frac{\partial V^{A}}{\partial D}+\beta \frac{\partial V^{B}}{\partial D}\right] =E\left[ \left[ u^{\prime }\left( c_{1}^{A}\right) -v'\left( Y\right) \right] \frac{\partial Y}{\partial D}\right] <0. \end{aligned}$$

\(\square\)

1.2.4 Proof of Proposition 4

Proof

Throughout the proof all prices and quantities are in the inertial regime equilibrium. We want to prove

$$\begin{aligned} E\left[ \left( u'\left( c_{1}^{A}\right) -v'(Y)\right) \frac{\partial Y}{\partial D}\right] <0, \end{aligned}$$

and Eq. (22) shows that \(\partial Y/\partial D\) is negative and constant across states in the inertial regime. So we need to prove

$$\begin{aligned} E\left[ u'(c_{1}^{A})-v'\left( Y\right) \right] >0. \end{aligned}$$
(30)

Since interest rate \({\bar{r}}\) is chosen optimally, the following condition holds

$$\begin{aligned} E\left[ \frac{1}{1+r}\left( u'\left( c_{1}^{A}\right) -\beta u'(c_{1}^{B})\right) \left( c_{1}^{B}+D\right) +\left( u'\left( c_{1}^{A}\right) -v'\left( Y\right) \right) \frac{\partial Y}{\partial r}\right] =0. \end{aligned}$$

Using the optimal debt choice condition and the fact that D and r are independent of the state, we can rearrange this into

$$\begin{aligned} \frac{1}{1+r}E\left[ \left( u'\left( c_{1}^{A}\right) -\beta u'\left( c_{1}^{B}\right) \right) c_{1}^{B}\right] +E\left[ \left( u'\left( c_{1}^{A}\right) -v'\left( Y\right) \right) \frac{\partial Y}{\partial r}\right] =0. \end{aligned}$$
(31)

Rewrite (19) as

$$\begin{aligned} c_{1}^{A}=\frac{1}{1+{\tilde{R}}}\left( c_{1}^{A}+c_{1}^{B}+D\right) , \end{aligned}$$

to get

$$\begin{aligned} c_{1}^{A}=\frac{c_{1}^{B}+D}{{\tilde{R}}}. \end{aligned}$$

This implies

$$\begin{aligned} u'\left( c_{1}^{A}\right) -\beta u'\left( c_{1}^{B}\right) ={\tilde{R}}^{\gamma }\left( c_{1}^{B}+D\right) ^{-\gamma }-\beta \left( c_{1}^{B}\right) ^{-\gamma }. \end{aligned}$$

This expression satisfies single crossing in \(c_{1}^{B}\), that is, there exists a level \(\hat{c}_{1}^{B}\) such that the expression is zero at \(c_{1}^{B}=\hat{c}_{1}^{B}\) and is positive if and only if \(c_{1}^{B}>\hat{c}_{1}^{B}\). This, together with \(E\left( u'(c_{1}^{A})-\beta u'\left( c_{1}^{B}\right) \right) =0\), implies that

$$\begin{aligned} E\left[ \left( u'\left( c_{1}^{A}\right) -\beta u'\left( c_{1}^{B}\right) \right) c_{1}^{B}\right] =E\left[ \left( u'\left( c_{1}^{A}\right) -\beta u'\left( c_{1}^{B}\right) \right) \left( c_{1}^{B}-\hat{c}_{1}^{B}\right) \right] >0. \end{aligned}$$

Equation (31) then implies

$$\begin{aligned} E\left[ \left( u'\left( c_{1}^{A}\right) -v'\left( Y\right) \right) \frac{\partial Y}{\partial r}\right] <0. \end{aligned}$$

Substituting in the derivative of output with respect to the interest rate (23), we obtain

$$\begin{aligned} E\left[ \left( u'\left( c_{1}^{A}\right) -v'\left( Y\right) \right) \left( \theta _{1}(r)D+\theta _{2}(r)\delta \right) \right] <0. \end{aligned}$$

Rearranging the expression above, using \(\theta _{2}(r)<0\), we have

$$\begin{aligned} \frac{\theta _{1}(r)D+\theta _{2}(r)E \left( \delta \right) }{\theta _{2}(r)}E \left[ u'\left( c_{1}^{A}\right) -v' \left( Y\right) \right] >-cov\left[ u'\left( c_{1}^{A}\right) -v'\left( Y\right) ,\delta \right] . \end{aligned}$$
(32)

Since Y and \(c_{1}^{A}\) are increasing in \(\delta\), together with concavity of u and convexity of v, we have that \(cov\left[ u'(c_{1}^{A})-v'\left( Y\right) ,\delta \right] <0.\) Therefore, a necessary and sufficient condition for Eq. (30) to hold is

$$\begin{aligned} \theta _{1}(r)D+\theta _{2}(r)E\left( \delta \right) <0. \end{aligned}$$

Substituting in the expressions for \(\theta _{1}(r)\) and \(\theta _{2}(r)\) and rearranging we have

$$\begin{aligned} \left( \frac{1+{\tilde{\beta }}{\tilde{R}}}{{\tilde{R}}}+\gamma (1-{\tilde{\beta }})+{\tilde{\beta }} (1+{\tilde{R}})\right) E\left( \frac{\delta }{1+r}\right) > \left( 1-\gamma \right) \left( 1-\tilde{{{\beta }}}\right) {\tilde{\beta }}{\tilde{R}}D. \end{aligned}$$
(33)

This holds immediately if \(\gamma \ge 1.\) Let us show it also holds for \(\gamma <1.\) Recall that D is implicitly defined by the following condition

$$\begin{aligned} {\tilde{R}}E\left[ \left( c_{1}^{B}+D\right) ^{-\gamma }\right] ^{\frac{1}{\gamma }}={\tilde{\beta }}E\left[ \left( c_{1}^{B}\right) ^{-\gamma }\right] ^{\frac{1}{\gamma }}. \end{aligned}$$

Substituting in the expression for \(c_{1}^{B}\) from Eq. (20) and rearranging we get

$$\begin{aligned} {\tilde{R}}E\left[ \left( \frac{\delta }{1+r}+{\tilde{\beta }}{\tilde{R}}D\right) ^{-\gamma }\right] ^{\frac{1}{\gamma }}={\tilde{\beta }}E\left[ \left( \frac{\delta }{1+r}-D\right) ^{-\gamma }\right] ^{\frac{1}{\gamma }}. \end{aligned}$$

Notice this equation (looking at the right-hand side) implies that \(D<\frac{\delta }{1+r}\) for any realization of \(\delta\), so that

$$\begin{aligned} D<E\left( \frac{\delta }{1+r}\right) . \end{aligned}$$

Therefore, a sufficient condition for Eq. (33) to hold when \(\gamma <1\) is that

$$\begin{aligned} \frac{1+{\tilde{\beta }}{\tilde{R}}}{{\tilde{R}}}+\gamma (1-{\tilde{\beta }})+{\tilde{\beta }}(1+{\tilde{R}})>\left( 1-\gamma \right) \left( 1-\tilde{{{\beta }}}\right) {\tilde{\beta }}{\tilde{R}}. \end{aligned}$$

Rearranging this equation, we have

$$\begin{aligned} \left( 1+{\tilde{\beta }}{\tilde{R}}\right) \left[ \frac{1}{{\tilde{R}}}+\gamma \left( 1-{\tilde{\beta }}\right) +{\tilde{\beta }}\right] >0, \end{aligned}$$

which holds as all terms on the left-hand side are positive. This completes the argument. \(\square\)

1.3 Sensitivity Analysis for Numerical Results

Our baseline numerical example features the following parameterization:

$$\begin{aligned} \beta =0.5,\quad \gamma =0.5,\quad \psi =1,\quad \phi =1,\quad \delta _{1}=0. \end{aligned}$$

Figure 4 displays the results of the model for different parameter values. The plots in the top row present the level of overborrowing: the difference between socially optimal D and D under no tax, both for the inertial and for the proactive regime. The plots in the bottom row present the optimal macroprudential tax \(\tau\) under both regimes. In each column, we vary one parameter at a time, holding constant all the other parameters.

The fact that the red lines are consistently above the blue lines shows that in all the examples considered the inertial regime displays more severe overborrowing.

Fig. 4
figure 4

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Sensitivity analysis.

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Bornstein, G., Lorenzoni, G. Moral Hazard Misconceptions: The Case of the Greenspan Put. IMF Econ Rev 66, 251–286 (2018). https://doi.org/10.1057/s41308-018-0052-x

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