b_i(n) = \frac{ D^{'-1} ( 1-q + q \frac{R}{P}) }{n} -1. These two results are intimately related, and they are both driven by how banks respond to a capital requirement, as shown in Proposition 4. The second inequality, |$R<1/(1-q)$|⁠, implies that scrapping investment in the bad state yields negative expected profits, and, thus, it is never optimal. Thus, we isolate the strategic motive—the opportunity to purchase the assets of failing banks at a deep discount—as the sole motive for holding liquidity for the ugly state. Therefore, similar to Farhi, Golosov, and Tsyvinski (2009), we argue that efficient regulations should have a wide scope and apply to all relevant financial institutions. Such fire sales lead to excessive liquidity holding in Acharya, Shin, and Yorulmazer (2011), too little debt and underinvestment in risky assets in Gale and Gottardi (2015), and underused deposits (versus equity) and overinvestment in risky assets in Gale and Yorulmazer (2019). \Pi _{i}(n_{i},b_i) &=& \Gamma (n_i, b_i) - q(R-P)Q_{i}^{s}(P,n_{i},b_{i}). There are two types of goods, a consumption and an investment good (the liquid and illiquid assets). The equilibrium price, |$P^{c}$|⁠, is increasing in the probability of the liquidity shock, |$q$|⁠, and the size of the shock, |$c$|⁠, but decreasing in the return on the risky assets, |$R$|⁠. One main factor was the argument that capital and liquidity requirements are substitutes. 23 Note that a planner endowed with only capital regulation tools would be willing to tolerate some reduction in liquidity when all banks decrease their risky investment level because there is a substitution between capital and liquidity ratios from the planner’s perspective. If we instead model liquidity shock as a proportion of deposits, we would then need capital regulation to limit the size of deposits and liquidity requirement to increase the high-quality liquid assets. In this section we investigate whether capital regulation alone can restore the second-best allocations. Therefore, the liquidity requirement can be written as |$b_i n_i / c n_i \geq b^{s} n^{s} / c n^{s}$|⁠. |$F^{\prime }(y)>0 \ \text{and} \ F^{\prime \prime }(y)<0 \ \text{for all} \ y\geq0, \ \text{with} \ R \geq F^{\prime}(0) > \nu \equiv qR(1+c)/(R-1+q).$|. When banks expect to incur a smaller additional cost for the investment, or when they face this cost with a lower probability, they increase risky investment and decrease liquidity buffers, as we show in the next proposition. Briefly, decreasing fire sales using capital and liquidity regulations hurts firms in the traditional sector, the fire buyers.

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