Collateral and the efficiency of monetary policy

Abstract

This paper argues that in a monetary Real Business Cycle economy where a complete set of nominal contingent claims exist, the requirement to collateralize loans, alone, does not affect the equilibrium allocation when monetary policy is chosen optimally: The Pareto optimal allocation can be supported. A Friedman rule (r = 0), which would be optimal in the absence of collateral constraints, here is not. At the resulting prices, collateral constraints bind and the allocation is inefficient. However, positive interest rates (through an inflation tax on money balances) support the Pareto optimal allocation when the collateral constraint binds.

Appendix

1.1 Proof of proposition 1

Proof

For the first part, take all the interest rates in Eq. 23 to zero. This gives

$$\begin{aligned} 1/\beta \le \frac{k(1)}{y(0)} \end{aligned}$$

(29)

Note that

$$\begin{aligned} \frac{k(1)}{y(0)} = \frac{\left\{ \alpha \beta \sum _s \pi (s)A_1(s) \right\} ^{1/(1-\alpha )}}{y(0)}. \end{aligned}$$

Now let $\beta \rightarrow 0$. The left-hand side of 29 approaches infinity, while the right-hand side approaches zero; hence, they must cross for some level of $\beta$. In general, it is easy to see that there are enough degrees of freedom for the parameters in the problem such that the collateral constraint will not be satisfied. $\square$

1.2 Proof of proposition 2

Proof

Let the monetary authority choose portfolio weights $f(s)=f$ such that $\forall s \in S$, $p_1(s) = p_1$. Take the first-order condition for the asset trade of this risk-less portfolio: $\frac{p_1}{p(0)} = \beta (1+r(0))$. Consider for some state $s' \in S$, the collateral constraint:

$$\begin{aligned} b_1(s')&\le p_1(s')k(1)\nonumber \\ p(0)y(0) - p_1(s') y_1(s') \frac{r_1(s')}{1+r_1(s')}&\le p_1(s')k(1)\nonumber \\ \frac{y(0)}{k(1) + y_1(s') \frac{r_1(s')}{1+r_1(s')}}&\le \frac{p_1(s')}{p(0)} \nonumber \\ \frac{y(0)}{k(1) + y_1(s') \frac{r_1(s')}{1+r_1(s')}}&\le \beta (1+r(0)). \end{aligned}$$

(30)

The left-hand side is independent of the date 0 interest rate. Using assumption 2 it must be greater than

$$\begin{aligned} \frac{1}{1+A_1(s')y(0)^{\alpha -1}\frac{r_1(s')}{1+r_1(s')}}. \end{aligned}$$

Assumption 1 gives us that this is highest when the productivity shock is lowest. The right-hand side is linearly increasing on $r(0)$ and has a minimum at $\beta$. Hence, if the constraint is to bind, then

$$\begin{aligned}&\frac{1}{1+A_1(s')y(0)^{\alpha -1}\frac{r_1(s')}{1+r_1(s')}} > \beta \\&\quad A_1(s')\frac{r_1(s')}{1+r_1(s')} < y(0)^{1-\alpha }\left\{ \frac{1}{\beta }-1\right\} \end{aligned}$$

The left-hand side has a minimum at $A_1(L)$, and using assumption 4 the above inequality is satisfied. Hence, under a policy of Inflation Targeting, the collateral constraint binds after a negative productivity shock. Furthermore, the collateral constraint binds at a positive date 0 interest rate. $\square$

1.3 Proof of proposition 3

Proof

In this case, $\forall s \in S$, $M_1(s) = M_1(s)$. The no-arbitrage condition gives us $\frac{M(0)}{M_1}\sum _s\beta \frac{y_1(s)}{y(0)} = \frac{1}{1+r(0)}$ or $(1+r(0))\sum _s\beta \frac{\pi (s)y_1(s)}{y(0)} = \frac{M_1}{M(0)}$. Consider the collateral constraint:

$$\begin{aligned} b_1(s')&\le p_1(s')k(1)\\ M(0) - \frac{r_1(s')}{1+r_1(s')}M_1(s')&\le M_1(s') \frac{k(1)}{y_1(s')}\\ \frac{1}{\frac{k(1)}{y_1(s')}+\frac{r_1(s')}{1+r_1(s')}}&\le \frac{M_1(s)}{M(0)}\\ \frac{1}{\frac{k(1)}{y_1(s')}+\frac{r_1(s')}{1+r_1(s')}}&\le (1+r(0)) \beta \sum _s\frac{\pi (s)y_1(s)}{y(0)},\\ \frac{1}{\sum _s\frac{\pi (s)y_1(s)}{y(0)}\left\{ \frac{k(1)}{y_1(s')} +\frac{r_1(s')}{1+r_1(s')}\right\} }&\le (1+r(0)) \beta . \end{aligned}$$

The right-hand side is minimized at $r(0)=0$. Without loss of generality, we can set $r_1(s') \rightarrow 0$ as this minimizes the cost of seigniorage on the optimal investment decision. The inequality if the constraint is to bind then is $\sum _s\frac{\pi (s)y_1(s)}{y(0)}\frac{k(1)}{y_1(s')} \le \frac{1}{\beta }$.

$$\begin{aligned} \frac{k(1)}{y_1(s')}\sum _s\frac{\pi (s)y_1(s)}{y(0)}&= \frac{k(1)^{1-\alpha }}{A_1(s')}\frac{k(1)^\alpha }{{y(0)}}\sum _s\pi (s)A_1(s)\\&= \frac{k(1)}{A_1(s'){y(0)}}\sum _s\pi (s)A_1(s)\\&< \frac{\sum _s\pi (s)A_1(s)}{A_1(s')} \end{aligned}$$

where the last step comes from assumption 2. Finally, note that from assumption 1, this is the smallest following a high productivity shock. The condition which guarantees that the constraint binds here is assumption 5: $\frac{A_1(H)}{\sum _s\pi (s)A_1(s)} > \beta$. Hence, if the collateral constraint is to bind, it will be when there is a positive productivity shock. Additionally, there exists a positive date 0 interest rate at which the constraint first binds. $\square$

1.4 Proof of proposition 4

Proof

From 27 $1 -\alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}>0$. If the constraint did not bind, it would be zero. As $\alpha <1$, it follows that investment is higher when the collateral constraint binds. $\square$

1.5 Proof of proposition 5

Proof

Let $\forall s \in S$, $p_1(s) = p_1$ and combine 27 and 28

$$\begin{aligned} \sum _s q(s)&= \frac{p(0)}{p_1(s^*)}\left[ 1 + \beta \pi (s^*) - \alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}\right] \nonumber \\&\quad + \sum _{s^{**} \in S/s^*} \beta \pi (s^{**})\frac{p(0)}{p_1(s^{**})}\nonumber \\ \frac{1}{1+r(0)}&= \frac{p(0)}{p_1}\left[ \beta + 1 - \alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}\right] \nonumber \\ \frac{p_1}{p(0)}&= (1+r(0))\beta \left[ 1 + \frac{1}{\beta } \left\{ 1- \alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}\right\} \right] . \end{aligned}$$

(31)

As the rate of inflation when the constraint does not bind is $(1+r(0))\beta$, it follows that inflation is higher when the constraint binds than when it does not. $\square$

1.6 Proof of proposition 6

Proof

Let $\forall s \in S$, $M_1(s) = M_1$ and combine 27 and 28

$$\begin{aligned} \sum _s q(s)&= \frac{p(0)}{p_1(s^*)}\left[ 1 + \beta \pi (s^*) - \alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}\right] \nonumber \\&\quad + \sum _{s^{**} \in S/s^*}\beta \pi (s^{**})\frac{p(0)}{p_1(s^{**})}\nonumber \\ \frac{1}{1+r(0)}&= \frac{M(0)}{M_1}\left\{ \frac{y_1(s^*)}{y(0)}\left[ 1 + \beta \pi (s^*) - \alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}\right] + \beta \pi (s^*)\frac{y_1(s^{**})}{y(0)}\right\} \nonumber \\ \frac{M_1}{M(0)}&= (1+r(0))\beta \left[ \sum _s \frac{y_1(s)}{y(0)} \pi (s) + \frac{y_1(s^*)}{y(0)}\frac{1}{\beta } \left\{ 1- \alpha k(1)^{\alpha -1} \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}\right\} \right] . \end{aligned}$$

(32)

As the rate of money growth when the constraint does not bind is

$$\begin{aligned} (1+r(0))\beta \sum _s \frac{y_1(s)}{y(0)} \pi (s), \end{aligned}$$

(33)

it follows that money growth is higher when the constraint binds than when it does not. $\square$

1.7 Proof of proposition 7

Proof

Inflation Targeting Consider equation 27

$$\begin{aligned} 1 -\alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)} = \left\{ \frac{q(s^*)p_1(s^*)}{p(0)} - \beta \pi (s^*)\right\} \end{aligned}$$

Using $q(s^*)=\frac{1}{1+r(0)}-\sum _{s^{**} \in S/s^*}\beta \pi (s^{**})\frac{p(0)}{p_1(s^{**})}$, the right-hand side can be written as:

$$\begin{aligned} \frac{1}{1+r(0)}\frac{p_1(s^*)}{p(0)}-\sum _{s^{**} \in S/s^*}\beta \pi (s^{**})\frac{p_1(s^*)}{p_1(s^{**})}- \beta \pi (s^*). \end{aligned}$$

Under Inflation Targeting, the monetary authority will choose its portfolio such that $\forall s \in S$, $p_1(s)=p_1$, thus the equation becomes: $\frac{1}{1+r(0)}\frac{p_1(s^*)}{p(0)}-\beta$.

Moreover, $p_1(s^*)=\frac{M_1(s^*)}{y_1(s^*)}$, $p(0)=\frac{M(0)}{y(0)}$ and $y_1(s^*)=A_1(s^*)k(1)^a$, thus the right-hand side is $\frac{1}{1+r(0)}\frac{M_1(s^*)}{M(0)}\frac{y(0)}{y_1(s^*)}-\beta$.

Consider the binding collateral constraint for state $s^*$:

$$\begin{aligned}&b_1(s^*)=p_1(s^*)k(1)\\&M(0)-\frac{r_1}{1+r_1}M_1(s^*)=\frac{M_1(s^*)}{y_1(s^*)}k(1)\\&M(0)=M_1(s^*)\left( \frac{k(1)}{y_1(s^*)}+\frac{r_1}{1+r_1}\right) \\&\frac{M_1(s^*)}{M(0)}=\frac{1}{\frac{k(1)}{y_1(s^*)}+\frac{r_1}{1+r_1}} \end{aligned}$$

Substituting the above equation, 27 becomes:

$$\begin{aligned}&1 -\alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s)A_1(s)}{1+r_1(s)}=\frac{1}{1+r(0)}\frac{1}{\frac{k(1)}{y_1(s^*)}+\frac{r_1}{1+r_1}}\frac{y(0)}{y_1(s^*)}-\beta \nonumber \\&1 -\alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}=\frac{1}{1+r(0)}y(0)\frac{1}{k(1)+A_1(s^*)k(1)^a\frac{r_1}{1+r_1}}-\beta \end{aligned}$$

(34)

Total differentiation with respect to $r(0)$ yields:

$$\begin{aligned}&\frac{\partial k(1)}{\partial r(0)}\left[ \alpha (1-\alpha ) k(1)^{\alpha -2} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}\right] \\&\quad =-\frac{1}{(1+r(0))^2}\frac{y(0)}{k(1)+A_1(s^*)k(1)^{1-a}}\\&\quad \quad -\frac{1}{1+r(0)}y(0)\frac{\partial k(1)}{\partial r(0)}\frac{1+A_1(s^*)k(1)^{\alpha -1}\frac{r_1}{1+r_1}}{\left( k(1)+A_1(s^*)k(1)^a\frac{r_1}{1+r_1}\right) ^2} \end{aligned}$$

Collecting the terms for $\frac{\partial k(1)}{\partial r(0)}$ we find that $\frac{\partial k(1)}{\partial r(0)}<0$. That is, raising the date zero interest rate lowers the rate of capital accumulation. The monetary authority can choose a high enough period 0 interest rate to fully relax the collateral constraint. Conversely, it can choose lower levels of $r(0)$ such that the constraint is more binding and capital investment higher.

Nominal GDP Targeting Under a policy of Nominal GDP Targeting the monetary authority and equal interest states in the intermediate period, $\forall s \in S$, money supplies $M_1(s)=M_1$. Thus, the right-hand side of equation 27 becomes:

$$\begin{aligned}&\frac{1}{1+r(0)}\frac{p_1(s^*)}{p(0)}-\sum _{s^{**} \in S/s^*}\beta \pi (s^{**})\frac{p_1(s^*)}{p_1(s^{**})}- \beta \pi (s^*)\\&\quad =\frac{1}{1+r(0)}\frac{M_1(s^*)}{M(0)}\frac{y(0)}{y_1(s^*)}-\sum _{s^{**}\in S/s^*}\beta \pi (s^{**})\frac{M_1(s^*)}{M_1(s^{**})}\frac{A_1(s^{**})}{A_1(s^*)}-\beta \pi (s^*)\\&\quad =\frac{1}{1+r(0)}\frac{M_1(s^*)}{M(0)}\frac{y(0)}{y_1(s^*)}-\sum _{s^{**} \in S/s^*}\beta \pi (s^{**})\frac{A_1(s^{**})}{A_1(s^*)}- \beta \pi (s^*).\\ \end{aligned}$$

Using the collateral constraint, we get

$$\begin{aligned} \frac{1}{1+r(0)}\frac{y(0)}{k(1)+\frac{r_1}{1+r_1}y_1(s^*)}-\sum _{s^{**} \in S/s^*}\beta \pi (s^{**})\frac{A_1(s^{**})}{A_1(s^*)}- \beta \pi (s^*). \end{aligned}$$

(35)

In addition, the second and third terms are constants, and thus, they will drop in the total differentiation with respect to $r(0)$. As a result, the analysis is as before and $\frac{\partial k(1)}{\partial r(0)}<0$. The monetary authority can relax (tighten) the collateral constraint by choosing high (low) period 0 interest rates, and thus, it can control the level of investment. $\square$

1.8 Proof of proposition 8

Proof

Taking the total utility in the constrained economy

$$\begin{aligned} U&= y(0) - k(1) + \beta k(1)^{\alpha }\sum _s \pi (s) A_1(s)\end{aligned}$$

(36)

$$\begin{aligned} \frac{\partial U}{r(0)}&=-\frac{\partial k(1)}{r(0)}\left\{ 1-\alpha \beta k(1)^{\alpha -1}\sum _s \pi (s) A_1(s)\right\} . \end{aligned}$$

(37)

Under Inflation Targeting

For simplicity and without loss of generality, assume that $\forall s \in S$, $r_1(s)=r_1$. The optimal date zero interest rate, $\hat{r}(0)$, occurs when $\alpha \beta k(1)^{\alpha -1}\sum _s \pi (s) A_1(s)=1$. Using Eq. 34:

$$\begin{aligned} \hat{r}(0) = \frac{\frac{y(0)}{k(1)+A_1(s^*)k(1)^a\frac{r_1}{1+r_1}}}{\beta + \frac{r_1}{1+r_1}}-1. \end{aligned}$$

(38)

When the constraint just binds, from Eq. 30,

$$\begin{aligned} \overline{r}(0)&= \frac{\frac{y(0)}{k(1)+A_1(s^*)k(1)^a\frac{r_1}{1+r_1}}}{\beta }-1> \hat{r}(0). \end{aligned}$$

The last step follows from the result that in the constrained economy, the level of capital accumulation is decreasing on date zero interest rates.

Under Nominal GDP Targeting Using 35, Eq. 27 becomes

$$\begin{aligned} 1 -\alpha k(1)^{\alpha -1} \beta \sum _s \frac{\pi (s) A_1(s)}{1+r_1(s)}&= \frac{1}{1+r(0)}\frac{y(0)}{k(1)+\frac{r_1}{1+r_1}y_1(s^*)}\\&\quad -\sum _{s^{**} \in S/s^*}\beta \pi (s^{**})\frac{A_1(s^{**})}{A_1(s^*)}- \beta \pi (s^*). \end{aligned}$$

As before, at the optimal interest rate $\alpha \beta k(1)^{\alpha -1}\sum _s \pi (s) A_1(s)=1$ so the above becomes

$$\begin{aligned}&\frac{1}{1+\hat{r}(0)}\frac{y(0)}{k(1)+\frac{r_1}{1+r_1}y_1(s^*)}= \frac{r_1(s)}{1+r_1(s)} + \frac{\beta \sum _s\pi (s)y_1(s)}{A_1(s^*)k(1)^\alpha } \nonumber \\&\hat{r}(0) = \frac{\frac{y(0)}{k(1)+\frac{r_1}{1+r_1}y_1(s^*)}}{ \frac{r_1}{1+r_1}+\frac{\beta \sum _s\pi (s)y_1(s)}{y_1(s^*)}}-1 \end{aligned}$$

(39)

The interest rate at which the constraint just binds occurs when the rate of money growth is given by Eq. 33. Using the collateral constraint, it turns out that the interest rate which supports this is

$$\begin{aligned} \overline{r}(0) = \frac{\frac{y(0)}{k(1)+\frac{r_1}{1+r_1}y_1(s^*)}}{\frac{\beta \sum _s\pi (s)y_1(s)}{y_1(s^*)}} - 1. \end{aligned}$$

(40)

The result is thus the same: Given date 1 interest rates, the optimal interest rate allows for the collateral constraint to bind. $\square$