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
See also Benigno et al. (2016) in an open economy context.
The aggregate price in the economy in period 1 is equal to \(\left( \int _0^1 p_{j1}^{1-\epsilon }\mathrm{d}j\right) ^{\frac{1}{1-\epsilon }}\).
The formal proof that \(\partial Y/\partial D<0\) is in “Appendix”.
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.
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.
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.
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.
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.
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.
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.
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 \(u'\left( c_{1}^{B}\right) d_{1}/\left( 1+r\right)\).
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.
References
Benigno, G., H. Chen, C. Otrok, A. Rebucci, and E.R. Young. 2013. Financial crises and macro-prudential policies. Journal of International Economics 89: 453–470.
Benigno, G., H. Chen, C. Otrok, A. Rebucci, and E.R. Young. 2016. Optimal capital controls and real exchange rate policies: A pecuniary externality perspective. Journal of Monetary Economics 84: 147–165.
Bernanke, B., and M. Gertler. 1999. Monetary policy and asset price volatility. Economic Review, Federal Reserve Bank of Kansas City, 17–51.
Chari, V., and P.J. Kehoe. 2016. Bailouts, time inconsistency, and optimal regulation: A macroeconomic view. The American Economic Review 106: 2458–2493.
Davila, E., and A. Korinek. 2017. Pecuniary Externalities in Economies with Financial Frictions. Review of Economic Studies 85: 352–395.
Diamond, D.W., and R.G. Rajan. 2012. Illiquid banks, financial stability, and interest rate policy. Journal of Political Economy 120: 552–591.
Farhi, E., and J. Tirole. 2012. Collective moral hazard, maturity mismatch and systemic bailouts. American Economic Review 102: 60–93.
Farhi, E., and I. Werning. 2016. A theory of macroprudential policies in the presence of nominal rigidities. Econometrica 84: 1645–1704.
Korinek, A. and O. Jeanne. 2016. Macroprudential regulation versus mopping up after the crash. Technical report, NBER Working Paper 18675.
Korinek, A., and A. Simsek. 2016. Liquidity trap and excessive leverage. American Economic Review 106: 699–738.
Lorenzoni, G. 2001. Essays on Liquidity in Macro. Ph.D. thesis, MIT.
Schmitt-Grohé, S., and M. Uribe. 2016. Downward nominal wage rigidity, currency pegs, and involuntary unemployment. Journal of Political Economy 124: 1466–1514.
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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
The optimality conditions of the two types of agents are
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:
and the goods market clearing conditions,
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
From consumer optimization, we obtain
Aggregating and using goods market clearing, we have
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:
Taking derivatives with respect to D and r, we obtain:
where
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:
1.2 Proofs for Section 4
1.2.1 Proof of Proposition 1
Proof
Denote the asset price by Q, where
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
Equilibrium in the bonds market at \(t=0\) and equilibrium in the goods market at \(t=1\) then give the two equations
With log preferences, these yield
The zero-output-gap condition then becomes
which has a unique solution Y. The value of D can be found using the condition
and the value for \({\bar{Q}}\) using the condition
The interest rates are
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
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
The two equilibrium conditions are
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,
So that
Using Eq. (24) to substitute for \(\mathbb {E}\left[ \tfrac{\beta }{m({\bar{\delta }}+\sigma z) - b } \right]\) and rearranging, we obtain
Let’s define the following two functions,
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 \(b(\sigma )\):
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):
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
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.
Evaluating at \(\sigma =0\) using \(\tfrac{c_b(0)}{c_a(0)}=\beta\), together with \(\mathbb {E}(z)=0\) and \(\mathbb {E}\left( z^2\right) =1\),
Define the matrix A as follows,
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
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
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
So that
\(G_{\sigma \sigma }\) and \(H_{\sigma \sigma }\) are equal to
At \(\sigma =0\) we have,
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:
\(\square\)
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
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
We then want to evaluate
Notice that there is a unique cutoff \(\hat{c}_{2}^{B}\) such that
iff \(c_{1}^{B}\gtrless \hat{c}_{1}^{B}\). Therefore
Moreover, from consumers optimality at \(t=0\) we have
Combining the last two equations we have
Step II From Step I and optimality of monetary policy we deduce that
With log preferences and a constant interest rate we deduce that
for some positive constant terms \(\Xi _{1},\Xi _{2}\). Therefore, we have the inequality
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
Using (29) we then conclude that
\(\square\)
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
and Eq. (22) shows that \(\partial Y/\partial D\) is negative and constant across states in the inertial regime. So we need to prove
Since interest rate \({\bar{r}}\) is chosen optimally, the following condition holds
Using the optimal debt choice condition and the fact that D and r are independent of the state, we can rearrange this into
Rewrite (19) as
to get
This implies
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
Equation (31) then implies
Substituting in the derivative of output with respect to the interest rate (23), we obtain
Rearranging the expression above, using \(\theta _{2}(r)<0\), we have
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
Substituting in the expressions for \(\theta _{1}(r)\) and \(\theta _{2}(r)\) and rearranging we have
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
Substituting in the expression for \(c_{1}^{B}\) from Eq. (20) and rearranging we get
Notice this equation (looking at the right-hand side) implies that \(D<\frac{\delta }{1+r}\) for any realization of \(\delta\), so that
Therefore, a sufficient condition for Eq. (33) to hold when \(\gamma <1\) is that
Rearranging this equation, we have
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:
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.