RDP 2019-10: Emergency Liquidity Injections Appendix B: Proofs

Definition. Throughout these proofs, $\widehat{d}f\left(x\right)/\widehat{d}xand\widehat{\partial }f\left(x\right)/\widehat{\partial }x$ refer to generalised derivatives, defined, as in Clarke (1975), as the convex hull of the set of limits of the form df (x + h_i)/dx and $\partial f\left(x+h_i\right)/\partial x$ where $h_i\to 0asi\to \infty$ . In any neighbourhood such that f is continuously differentiable, the generalised derivative collapses to the standard derivative.

Remark. In most cases throughout these proofs, ${\Pi }_i$ is a function of the almost everywhere continuously differentiable functions m* and $\overline{b}$ . In such cases the generalised derivatives of ${\Pi }_i$ with respect to s_i or S are equal to the interval in $ℝ$ between the left-hand and right-hand derivatives

Lemma 9. Under the unsecured lending policy, market illiquidity m* is Lipschitz continuous, almost everywhere differentiable, and non-decreasing in b, and, given s_−i = S, in S.

Proof. Market illiquidity m* is defined implicitly by g_m = 0 where

If $b\ge \underset{\_}{b}\left(s\right)$ defined in Equation (13), then L_D is whatever value that makes m* constant at its upper bound r_u /(1 + r_u), by the arbitrage condition discussed in Section 3. The rest of this proof fixes r_u and considers $b<\underset{\_}{b}\left(s\right)$ . In these cases L_D is defined by

where

If $b\le l-S$ then, in aggregate, banks have sufficient cash to meet the liquidity shock, so no securities are sold to securities buyers $\left(\text{i}\text{.e}\text{.}{L}_D\le 0\right)$ and m* = 0. Market illiquidity m* continuously increases in b from zero as banks' aggregate cash shortage L_D = b + S – l continuously increases in b through zero, by the properties of L_S. At positive m*, bank i can be liquidity deficient such that L_i = s_i (1 – m) as in Equation (B3). The next paragraph considers positive m*.

First assume (almost-) symmetric s, i.e. s_−i = S. If S is high enough then there is some $b^{\prime }<\underset{\_}{b}\left(s\right)$ such that the unit measure of banks is just liquidity deficient, and M (b' + S – l) = (l – b') / S. (If S is not high enough, then L_D = b + S – l for all $b<\underset{\_}{b}\left(s\right)$ and the following arguments hold more trivially.) At this b' liquidity demand is L_D = b + S – l = S(1 – m), so L_D defined by Equations (B2) and (B3) is continuous in a neighbourhood of b. The left-hand derivative is d^–L_D/db = 1, and the right-hand derivative is d⁺L_D/db = 0, which are both bounded, so L_D is Lipschitz continuous. Further, given that $l-S<b<\underset{\_}{b}\left(s\right)$ , the function g_m is strictly increasing in m whether L_D = l + b – S or L_D = S(1 – m). Therefore g_m satisfies the Lipschitz implicit function theorem and m* is Lipschitz continuous.

Total differentiation of g_m at g_m = 0 shows that $\widehat{d}m*/\widehat{d}b\text{and}\widehat{d}m*/\widehat{d}S$ have the same signs as $\widehat{d}{L}_D/\widehat{d}b\text{and}\widehat{d}{L}_D/\widehat{d}S$ , so, from Equations (B2) and (B3), m* is non-decreasing in b and S.

Observe that L_D is continuously differentiable at all b such that $b\ne l-Sm*\left(b,S\right)$ , from Equations (B2) and (B3). Therefore, by the implicit function theorem, when s_–i = S, market illiquidity m* is almost everywhere continuously differentiable.

Now consider asymmetric s. It is possible that for some $l-S<b<\underset{\_}{b}\left(s\right)$ , only a proportion of banks are liquidity deficient. Banks' aggregate liquidation can therefore be expressed as $L_D=\alpha \left(m,b,s\right)\left(b+S_1-l\right)+\left(1-\alpha \left(m,b,s\right)\right)S_2\left(1-m\right)$ , where $0\le \alpha \left(m,s,b\right)\le 1$ , and S₁ and S₂ are the mean securities holdings for banks with $s_i<\left(l-b\right)/m\text{and}{s}_i\ge \left(l-b\right)/m$ respectively. For any (m, b) such that a positive measure of banks holds s_i = (l – b)/m, the proportion of liquidity-deficient banks $1-\alpha$ is discontinuous. Nevertheless, s_i = (l – b)/m implies b + s_i – l = s_i(1 – m), so L_D is still continuous in all variables. To see Lipschitz continuity and the derivative signs for L_D, the same reasoning as for the symmetric case can be applied to the points of non-differentiability. It follows that if $l-S<\underset{\_}{b}\left(s\right)$ then, in general, g_m is Lipschitz continuous, strictly increasing in m, and non-decreasing in b, so m* is Lipschitz continuous and non-decreasing in b. Because there is a finite measure of banks, the set of b such that $\alpha$ is discontinuous has measure zero, so m* is almost everywhere continuously differentiable.

Lemma 10. Under the unsecured lending policy, for all $b<\underset{\_}{b}\left(s\right)$ , the liquidity deficiency threshold ${\overline{b}}_i$ is Lipschitz continuous, almost everywhere differentiable, and strictly decreasing in s_i, and, if s_–i = S, also in S.

Proof. The liquidity deficiency threshold ${\overline{b}}_i$ is defined implicitly by g_b = 0 where

Lemma 9 shows that m* is non-decreasing, Lipschitz continuous, and almost everywhere continuously differentiable in b and (when s_–i = S) in S. It follows that g_b is strictly decreasing in b. Therefore, by Equation (B4) and implicit function theorems, ${\overline{b}}_i$ is Lipschitz continuous and almost everywhere continuously differentiable in b and S.

From Equation (B4) g_b is clearly continuously differentiable in s_i. Total differentiation of g_b shows that $\widehat{d}{\overline{b}}_i/\widehat{d}{s}_i<0\text{and}\widehat{d}{\overline{b}}_i/\widehat{d}{s}_i\le 0.$

Lemma 11. Assume $b\le l-\epsilon$, where $\epsilon$ is an arbitrarily small constant. References to S assume s_–i = S. Under the secured lending policy, market illiquidity m* is Lipschitz continuous and almost everywhere differentiable in b and in S, and non-decreasing in S.

Proof. Under the secured lending policy each bank's securities liquidation satisfies

Most of the reasoning in the proof of Lemma 9 also applies here (excluding proof that m* is non-decreasing in b, which is not part of this Lemma). The only two points of departure are that: (i) while $d{L}_D^{+}/db$ is bounded for the unsecured lending policy, it remains to be shown for the secured lending policy, due to the difference between Equations (B3) and (B5); and (ii) given that m* is not necessarily non-decreasing in b, it remains to be shown that the set of b such that m* is not differentiable has measure zero.

1. If b > l – s_im then dL_i/db = 1 – 1/m. As b approaches l – s_im from above, this derivative approaches 1 – s_i/(l – b). It follows from $l-b\ge \epsilon$ that $d^{+}{L}_i/db\ge 1-s_i/\epsilon \ge l/\epsilon$ , and therefore $d{L}_D^{+}/db$ is bounded.

2. Lipschitz continuity of L_D is given by (i). Since L_i and therefore L_D is non-increasing in m, L_S – L_D is strictly increasing in m, so the Lipschitz implicit function theorem applies to define m*. Then, by the (standard) implicit function theorem, m* is continuously differentiable in any (b, S) such that L_D is continuously differentiable. As in the proof of Lemma 9, non-differentiable points of L_D can only occur at (b, S) such that a positive measure of banks is on the threshold of liquidity deficiency. These points must therefore satisfy b = l – s_im*(b, S) where s_i equals some s₀ such that a positive measure of banks holds s_i = s₀. The following shows that the measure of such (b, S) is zero.

   Say there is a non-degenerate interval of b or S such that b = l – s₀m*(b, S). Inside the bounds of this interval, the proportion of liquidity-deficient banks $\left(1-\alpha \right)$ is constant, so $L_D=\alpha \left(b+S_1-l\right)+\left(1-\alpha \right)\left(1-b\right)\left(1/m-1\right)$ , which is continuously differentiable. Therefore non-differentiable points in L_D are limited to the boundary points of such intervals. Further, there is a finite measure of banks, so the set of s_i satisfying the definition of s₀ has measure zero, and the number of such intervals is countable. Therefore the set of non-differentiable points of L_D has measure zero.

Remark. Under the secured lending policy, as b → l, banks' ability to handle market illiquidity diminishes to zero. Therefore their securities selling also diminishes to zero, as they instead use all their securities as collateral to borrow from the authority. For simplicity the model has assumed all market illiquidity is generated by banks, so b → l then implies m* → 0, and at the limit some of the reasoning in the proof of Lemma 11 breaks down (i.e. $d{L}_D^{+}/db$ is not bounded). This is clearly an unrealistic outcome – for the highest possible liquidity shocks, securities markets approach full liquidity – so it is assumed away for Lemma 11, because it has an arbitrarily small probability. This could be justified by an assumption that $b\in \left[0,l-\epsilon \right]$ while banks believe that $b\in \left[0,l\right)$ .

Alternatively, the situation can be ruled out by assuming that whenever there is a liquidity shock, there is also $\gamma b$ exogenous liquidation in the securities market, where $\gamma$ is arbitrarily small, and r_s is above some lower bound (which compensates banks for the increased liquidity risk from $\gamma b$ ). All the results in this paper would be maintained under this assumption.

Lemma 12. Under the secured lending policy, for all $b<\underset{\_}{b}\left(s\right)$ , the liquidity constraint ${\overline{b}}_i$ is Lipschitz continuous, almost everywhere differentiable, and strictly decreasing in s_i, and, if s_–i = S, weakly decreasing in S.

Proof. The liquidity constraint ${\overline{b}}_i$ is defined by g_b = 0, where g_b has the same form as in Equation (B4). The proof of Lemma 10 also applies here, once it is shown that

which is required for applying the Lipschitz function theorem to show that $\overline{b}$ is unique given (s_i, s_–i). Re-expressing the L_D function allows us to evaluate this derivative. Fix s, and denote the total securities held by banks with i ≤ x as F_s(x), remembering that s_i is non-decreasing in i (see Section 2.1). Define $\overline{i}\left(b\right)\in \left[0,1\right]$ such that $i<\overline{i}\Rightarrow s_i<\left(l-b\right)/m*\left(b\right)$ and $i\ge \overline{i}\Rightarrow s_i\ge \left(l-b\right)/m*\left(b\right)$ . This permits the expressions for L_D and L_i to be combined into

The proof of Lemma 11 shows that L_D is almost everywhere continuously differentiable, with any non-differentiable points aligning with non-differentiable points of m*. Therefore the left-hand and right-hand derivatives of L_D are defined for all b. To solve them, first observe that, where dm*/db is defined,

Then, using Equation (B4), and acknowledging that g_b = 0 implies m* > 0,

Positivity of this derivative means that the Lipschitz implicit function theorem applies, and ${\overline{b}}_i$ is unique and Lipschitz continuous. Further, discontinuities in the derivatives of g_b occur only at discontinuities in the derivative of m*, so ${\overline{b}}_i$ is almost everywhere continuously differentiable by the implicit function theorem. The signs of the derivatives of ${\overline{b}}_i$ then follow from total differentiation of g_b.

Proof of Proposition 1. This proof will show that: (i) $d{\Pi }_i/d{s}_i$ is continuous; (ii) ${\widehat{d}}^2{\Pi }_i/\widehat{d}{s}_i^2$ is non-positive; (iii) ${\widehat{d}}^2{\Pi }_i/\widehat{d}s,\widehat{d}S$ is negative; and (iv) if S > 0 then ${\widehat{d}}^2{\Pi }_i/\widehat{d}{s}_i\widehat{d}{r}_u$ is negative. From (i), (ii) and (iii) it follows that if $s_i^{*}\left(S=l,r_u\right)<l\text{and}{s}_i^{*}\left(S=0,r_u\right)>0$ , there is a unique fixed point such that $0<s_i^{*}\left(S,r_u\right)=S<1$ . From (iv), this fixed point is continuously and strictly decreasing in r_u. Finally, the proof will demonstrate that (v) S* is bound below only at zero.

1. The integrands in Equation (16) are equal at $b=\min \left\{\underset{\_}{b},{\overline{b}}_i\right\}$ , so

   (B9) $$\begin{array}{l}\frac{d{\Pi }_i^u\left(s_i,s_{-i}\right)}{d{s}_i}=\left(1-\lambda \right)r_s+\\ +\lambda \left({\displaystyle {\int }_0^{\min \left\{\underset{\_}{b}\left(s_{-i},r_u\right),\overline{b}\left(s_i,s_{-i}\right)\right\}}\frac{-m*\left(b,s_{-i}\right)}{1-m*\left(b,s_{-i}\right)}f\left(b\right)db+{\displaystyle {\int }_{\min \left\{\underset{\_}{b}\left(s_{-i},r_u\right),\overline{b}\left(s_i,s_{-i}\right)\right\}}^l-m*\left(b,s_{-i}\right)\left(1+r_u\right)f\left(b\right)db}}\right)\end{array}$$

   where $\min \left\{\underset{\_}{b},{\overline{b}}_i\right\}$ and m* are Lipschitz continuous functions by Lemmas 9 and 10. Therefore Equation (B9) is continuous.

2. For high enough s_i it is the case that ${\overline{b}}_i<\underset{\_}{b}$ and

   (B10) $$\frac{{\widehat{d}}^2{\Pi }_i^u\left(s_i,s_{-i}\right)}{\widehat{d}{s}_i^2}=\lambda \frac{\widehat{d}{\overline{b}}_i\left(s_i,s_{-i}\right)}{\widehat{d}{s}_i}m*\left({\overline{b}}_i,s_{-i}\right)\left(1+r_u-\frac{1}{1-m*\left({\overline{b}}_i,s_{-i}\right)}\right)f\left({\overline{b}}_i\right)$$

   Lemmas 9 and 10 show that $\widehat{d}{\overline{b}}_i/\widehat{d}{s}_i<0\text{and}m*\left({\overline{b}}_i,s_{-i}\right)>0$ . Additionally, ${\overline{b}}_i<\underset{\_}{b}$ implies $1+r_u>1/\left(1-m*\left({\overline{b}}_i\right)\right)$ . Therefore ${\overline{b}}_i<\underset{\_}{b}$ implies Equation (B10) is negative. Alternatively, if ${\overline{b}}_i\ge \underset{\_}{b}$ then, from Equation (B9) and the fact that $d\underset{\_}{b}/d{s}_i=0$ , it follows that ${\widehat{d}}^2{\Pi }_i/\widehat{d}{s}_i^2=0$ .

3. If ${\overline{b}}_i<\underset{\_}{b}$ then

   (B11) $$\begin{array}{l}\frac{{\widehat{d}}^2{\Pi }_i^u\left(s_i,S\right)}{\widehat{d}{s}_i\widehat{d}S}=\lambda [{\displaystyle {\int }_0^{\overline{b}\left(s_i,S\right)}\frac{-\frac{\widehat{d}m*\left(b,S\right)}{\widehat{d}S}}{{\left(1-m*\left(b,S\right)\right)}^2}f\left(b\right)db+{\displaystyle {\int }_{\overline{b}\left(s_i,S\right)}^l-\frac{\widehat{d}m*\left(b,S\right)}{\widehat{d}S}\left(1+r_u\right)f\left(b\right)db}}\\ +\frac{\widehat{d}\overline{b}\left(s_i,S\right)}{\widehat{d}S}m*\left({\overline{b}}_i,S\right)\left(1+r_u-\frac{1}{1-m*\left({\overline{b}}_i,S\right)}\right)f\left({\overline{b}}_i\right)]\end{array}$$

   Alternatively, if ${\overline{b}}_i\ge \underset{\_}{b}$ then each ${\overline{b}}_i$ in Equation (B11) is replaced by $\underset{\_}{b}$ , and the third term is zero because $b=\underset{\_}{b}$ implies 1 + r_u = 1/(1–m*). Lemma 9 shows that $\widehat{d}m*/\widehat{d}S\ge 0$ , and if b is just above l – S, then $\widehat{d}m*/\widehat{d}S>0$ . Therefore the sum of the first two terms in Equation (B11) is negative. Further, if ${\overline{b}}_i<\underset{\_}{b}$ then 1 + r_u > 1/(1–m*), and Lemma 10 shows that $\widehat{d}{\overline{b}}_i/\widehat{d}S>0$ , so the third term in Equation (B11) is also negative. Therefore ${\widehat{d}}^2{\Pi }_i/\widehat{d}s,\widehat{d}S<0$ .

4. Whether $\underset{\_}{b}<{\overline{b}}_i\text{or}\underset{\_}{b}>{\overline{b}}_i,$

   (B12) $$\frac{{\widehat{d}}^2{\Pi }_i^u\left(s_i,s_{-i}\right)}{\widehat{d}{s}_i\widehat{d}{r}_u}={\displaystyle {\int }_{\min \left\{\underset{\_}{b}\left(s_{-i},r_u\right),\overline{b}\left(s_i,s_{-i}\right)\right\}}^l-m*\left(b,s_{-i}\right)f\left(b\right)db}$$

   This is true because if $\underset{\_}{b}<{\overline{b}}_i$ then $1+r_u=1/1\left(1-m*\left(\underset{\_}{b}\right)\right)$ and the two integrands in Equation (B9) are equal at $\underset{\_}{b}$ , so the integral-limit terms in the derivative drop out. If S > 0 then at high b, m* > 0, and whenever s_i > 0 it is the case that $\min \left\{\overline{b},\underset{\_}{b}\right\}<l$ , so S > 0 implies that Equation (B12) is negative for all s_i > 0.

5. If S = 0 then m* = 0 for all b so ${\Pi }_i^u=l+\left(1+\lambda \right)s_i{r}_s-\lambda E\left[b\right]$ and $s_i^{*}=l$ . Therefore there can be no equilibrium at S* = 0. However, for any given $s>0,{\overline{b}}_i$ is less than l and Equation (B9) is decreasing without bound in r_u. Therefore, high enough r_u can generate negativity of Equation (B9) whenever s > 0.

Proof of Lemma 2. Banks' pay-offs, expressed in Equation (16), are decreasing in market illiquidity m*, which, by Lemma 9, is increasing in S whenever m* > 0. The marginal return to securities in an aggregated pay-off function includes this negative effect, whereas Equation (B9) does not. Therefore, for any given S, the collective marginal return to securities is lower than the individual marginal return, so if the model equilibrium is interior, then the collective equilibrium is lower.

Proof of Lemma 3. This result is illustrated in Figure 4. Fix symmetric s > 0 across both policies. For low b, securities liquidation satisfies L_D = b + S – l and market illiquidity satisfies $m_R^{*}\left(b\right)=m_u^{*}\left(b\right)$ . As b increases, this holds true until m* hits an upper bound, either at r_P / (1 + r_P) or at M(S(1 – m*)).

At higher $b,m_u^{*}\left(b\right)$ is constant, whereas $m_R^{*}\left(b\right)$ is weakly decreasing and, for some b, strictly decreasing. Specifically, $m_R^{*}\left(b\right)$ is constant if bound at r_R / (1 + r_n) and decreasing otherwise. The result then follows from r_R ≤ r_u and from M(S(1 – m*)) being equal across policies. It only remains to be shown that $m_R^{*}\left(b\right)$ is strictly decreasing for some b.

Binding collateral constrains are the cause of decreasing $m_R^{*}\left(b\right)$ . To see that this necessarily occurs, first observe that if ${\overline{b}}_i<\underset{\_}{b}\left(r_R\right),\text{i}\text{.e}\text{.}\text{if}{r}_R>r_{pen}$ , then Equation (17) binds at all $b>{\overline{b}}_i$ (see the discussion in Section 4). If $\underset{\_}{b}\left(r_R\right)<{\overline{b}}_i$ then for $m_R^{*}$ to simultaneously satisfy Equation (3) and $m_R^{*}=r_R/\left(1+r_R\right)$ , the total quantity of securities sales S_m must satisfy $S_m={\overline{S}}_m$ such that

Given this, collateral available for borrowing is S – S_m, which is less than the required borrowing if

This condition can be rearranged to $b>\overline{B}$ where

The properties of L_S imply that $L_S\left(r_R/\left(1+r_R\right)\right)>0\text{so}\overline{B}<l$ . Therefore Equation (B13) is necessarily violated for some b.

Proof of Proposition 4. The first two sentences of Proposition 4 have a similar proof to Proposition 1. Only the following changes are required:

• References to u, Equation (16) and Lemmas 9 and 10 are replaced by references to R, Equation (21) and Lemmas 11 and 12, respectively.
• In Equations (B9), (B10) and (B11), the term 1 + r_u is replaced by r_R /m*(b,s_–i). In some cases the m* in this denominator cancels another m* in the expression. For instance, Equation (B9) is replaced by

• In Equation (B12), the –m* term becomes –1.

The following proves the third sentence of Proposition 4. It is shown that if $S_R^{*}=S_u^{*}<l$ , then, in equilibrium, ${\Pi }_i^R>{\Pi }_i^u$ . This implies that for any interior optimal unsecured policy equilibrium, there is a secured policy equilibrium at the same S* with higher W. Therefore W is maximised at a higher level under the secured policy.

Fix symmetric $S_u^{*}=S_R^{*}\equiv \widehat{S}<l$ , which sets ${\overline{b}}_i$ constant and equal in both policies, denoted $\overline{b}$. Note that for $b>\min \left\{\underset{\_}{b},\overline{b}\right\}$ , under the unsecured policy m*(b) is constant. Denote this ${\overline{m}}_u^{*}$ . Denote market illiquidity under the secured lending policy $m_R^{*}\left(b\right)$ . From Equations (16) and (21), the secured policy has a higher pay-off than the unsecured policy if and only if

Further, equal interior equilibrium S implies $d{\Pi }_i^u/d{s}_i=d{\Pi }_i^R/d{s}_i=0$ . Therefore, from Equations (B9) and (B14),

Substituting Equation (B16) into Equation (B15) and rearranging gives the condition

This condition holds if ${\overline{m}}_u^{*}\ge m_R^{*}\left(b\right)$ for all $b>\min \left\{\overline{b},\underset{\_}{b}\right\}$ , and with a strict inequality for some positive measure of b. This is true if r_u ≥ r_R by Lemma 3. In turn, r_u ≥ r_R is implied by Equation (B16) and the arbitrage condition $r_u\ge {\overline{m}}_u^{*}/\left(1-{\overline{m}}_u^{*}\right)$ from Equation (12). That is, combining these two conditions implies that either $r_R=r_u={\overline{m}}_u^{*}/\left(1-{\overline{m}}_u^{*}\right),\text{or}{r}_u>{\overline{m}}_u^{*}/\left(1-{\overline{m}}_u^{*}\right)$ and r_u > r_R.

Proof of Corollary 5. Given fixed S and the inequalities r_P > r_pen and $b>{\overline{b}}_i$ , the ex post (i.e. conditional on b) marginal return to securities $d{\Pi }_i/d{s}_i\text{is}-r_R$ under the secured lending policy and $-m_u^{*}\left(b\right)\left(1+r_u\right)$ under the unsecured lending policy. The condition r_u > r_pen implies that $r_u>m_u^{*}\left(b\right)/\left(1-m_u^{*}\left(b\right)\right)$ for all b, which can be rearranged to $r_u>m_u^{*}\left(b\right)\left(1+r_u\right)$ . Therefore r_R = r_u implies $r_R>m_u^{*}\left(b\right)\left(1+r_u\right)$ . So if $S_u^{*}<l$ , under the secured lending policy the marginal return to securities at $S=S_u^{*}$ is negative, and, by concavity of ${\Pi }_i^R$ (shown in Proposition 4), $S_R^{*}$ must be lower.

Proof of Proposition 6. Without loss of generality set ${\overline{s}}_S={\min }_i\left\{s_{-i}\right\}$ . Under a securities purchase policy the marginal return to securities can take two forms, depending on whether $s_i\le {\overline{s}}_S\text{or}{s}_i\ge {\overline{s}}_S$ . From Equation (24), if $s_i\le {\overline{s}}_S$ it is

and if $s_i\ge {\overline{s}}_S$ it is

Holding (s_i, s_–i) constant, when r_u > r_pen, which implies $\underset{\_}{b}=l$ , both Equations (B17) and (B18) are greater than Equation (B9). To see this, consider the ex post marginal returns, i.e. Equations (B9), (B17) and (B18) conditional on a specific realisation of b. They are equal across policies for $b\le \min \left\{b_S,{\overline{b}}_i\right\}$ . For higher b, the ex post marginal return necessarily decreases under the unsecured lending policy, by r_u > r_pen and the fact that m* is increasing in b when positive (Lemma 9). For higher b under the securities purchase policy, however, the ex post marginal return does not decrease, because (i) when $s_i<{\overline{s}}_S,-\left(l-b\right)/\left(s_S+b-l\right)$ is not less than –m*(b_S) and is increasing in b at higher b, and (ii) when $s_i\ge {\overline{s}}_S$ it is zero. Therefore, in expectation across b, both Equations (B17) and (B18) are greater than Equation (B9).

Now fix the $S_u^{*}$ induced by an unsecured lending policy with r_u > r_pen. Note that $d{\Pi }_i^u/d{s}_i$ is positive for any (s_i, S) satisfying $s_i<S_u^{*}\text{and}S<S_u^{*}$ , by negativity of Equations (B10) and (B11), and because $d{\Pi }_i^u/d{s}_i=0\text{at}{s}_i=S=S_u^{*}$ . Therefore, by the reasoning in the previous paragraph, $d{\Pi }_i^S/d{s}_i$ is positive for any (s_i, S) satisfying $s_i\le S_u^{*}$ , and so at $S\le S_u^{*}$ and there can be no equilibrium at $S<S_U^{*}$ .

Proof of Proposition 7. The ex post marginal return to securities for given $b>{\overline{b}}_i\text{is}-m^{*}{\varphi }^{\prime }{a}_i$ which, for given s, is strictly decreasing in a_i. Therefore, the ex ante marginal return to securities, and any interior optimum, are also strictly decreasing in a_i.

Proof of Corollary 8. The result follows directly from Equation (25) the assumption that $\varphi <1$ .