# RDP 2019-10: Emergency Liquidity Injections 5. Liquidity Support through Securities Purchases

Under a securities purchase policy, the authority ensures that the liquidity value of banks' portfolios remains sufficiently high to avoid failure. It does this by purchasing securities to lift their market-clearing price. The policy parameter is a purchase schedule mS(b) such that if without intervention market illiquidity m* would be above mS(b), the authority purchases enough securities to ensure the market price is 1–mS(b).

A particular purchase schedule can be represented by the level of ${\overline{s}}_{S}$ such that any bank with ${s}_{i}\le {\overline{s}}_{S}$ is saved by the policy. Specifically, provided that no bank fails,

(22) $m S ( b )= l−b s ¯ S$

and the authority intervenes at all b > bS such that

(23) $b S =l− s ¯ S m*( b S , s −i )$

However, the authority's priority of preventing bank failures implies that if ${s}_{i}>{\overline{s}}_{S}$ for any i, it will respond to banks' choices by raising ${\overline{s}}_{S}$ to maxi{si}. The date 1 realisations of bS and ${\overline{s}}_{S}$ can be written ${b}_{S}^{1}\left(s\right)$ and ${\overline{s}}_{S}^{1}\left(s\right)$ to incorporate this requirement. Bank i's pay-off is

(24) $Π i S ( s i s −i )=( 1−λ )( l+ s i r s ) +λ( ∫ 0 b S 1 ( s ) l− s i m*( b, s −i )−b 1−m*( b, s −i ) f( b )db+ ∫ b S 1 ( s ) l l−b− s i l−b s ¯ S 1 ( s ) 1− l−b s ¯ S 1 ( s ) f( b )db )$

Proposition 6. Say an unsecured lending policy with ru > rpen induces an equilibrium ${S}_{u}^{*} . Under a securities purchase policy, the authority cannot credibly induce an equilibrium at or below ${S}_{u}^{*}$.

Proposition 6 is a result of banks understanding that the authority will not let them fail, which permits each bank to lower ${b}_{S}^{1}$ by choosing ${s}_{i}>{\overline{s}}_{S}$ . The preceding analysis has been intentionally agnostic about the functional form of the cost of failure to banks, but the result would be the same for any positive cost, because banks know the authority will not let them face these costs. Moreover, banks benefit from raising ${\overline{s}}_{S}^{1}$ , because it lifts the price that the authority offers in the case that $b>{b}_{S}^{1}$ . Therefore, banks have incentives to raise their liquidity risk-taking to levels of ${\overline{s}}_{S}$ that are credible.

Proposition 6 does not require that the authority is unwilling to let any banks fail. Consider an alternative objective function W in which $\left(\mathbb{𝟙}\left(fail\right)\right)$ = 1 if and only if some particular positive measure of banks fails, and assume the cost of failure to banks is sufficiently high that banks always choose to avoid it. This would produce multiple equilibria, with one at ${\overline{s}}_{S}$ and one at higher S.[13] Only the highest equilibrium is time consistent for the authority, because the authority will concede to liquidity risk levels above ${\overline{s}}_{S}$ provided that the sufficient measure of banks coordinates above it. In other words, individual banks' liquidity risk choices are strategic complements, as in the model of Farhi and Tirole (2012).

This need to be more lenient to banks when they take more liquidity risk, and the consequential lack of credible deterrence, occurs because the policy does not require banks to commit future assets. The authority only receives compensation (i.e. securities) for its liquidity provision at the height of the crisis, when banks have little to compensate without failing. The lending policies, on the other hand, do not require payment from banks until conditions improve. The model differs from that of Farhi and Tirole (2012) by having a disconnect between the market's date 1 provision of liquidity and the date 2 value of banks' assets. This disconnect presents an opportunity for a liquidity-rich authority. It can require date 2 repayments that are large enough to deter ex ante liquidity risk-taking, without undermining its primary objective of preventing bank failures during the peak of the crisis.

## Footnote

Continuity of $d{\Pi }_{i}^{S}/d{s}_{i}$ dsi (Equations (B17) and (B18)) in bs and S ensures existence of the higher equilibrium, potentially at S = l. [13]