RDP 201910: Emergency Liquidity Injections Appendix B: Proofs
October 2019
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Definition. Throughout these proofs, $\widehat{d}f\left(x\right)/\widehat{d}x\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\widehat{\partial}f\left(x\right)/\widehat{\partial}x$ 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 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}as\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\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 $\mathbb{R}$ between the lefthand and righthand derivatives
Lemma 9. Under the unsecured lending policy, market illiquidity m* is Lipschitz continuous, almost everywhere differentiable, and nondecreasing 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 $b\le lS$ then, in aggregate, banks have sufficient cash to meet the liquidity shock, so no securities are sold to securities buyers $\left(\text{i}\text{.e}\text{.}\text{\hspace{0.17em}}{L}_{D}\le 0\right)$ 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 lefthand derivative is d^{–}L_{D}/db = 1, and the righthand derivative is d^{+}L_{D}/db = 0, which are both bounded, so L_{D} is Lipschitz continuous. Further, given that $lS<b<\underset{\_}{b}\left(s\right)$ , 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 $\widehat{d}m*/\widehat{d}b\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\widehat{d}m*/\widehat{d}S$ have the same signs as $\widehat{d}{L}_{D}/\widehat{d}b\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\widehat{d}{L}_{D}/\widehat{d}S$ , so, from Equations (B2) and (B3), m* is nondecreasing in b and S.
Observe that L_{D} is continuously differentiable at all b such that $b\ne lSm*\left(b,S\right)$ , 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 $lS<b<\underset{\_}{b}\left(s\right)$ , only a proportion of banks are liquidity deficient. Banks' aggregate liquidation can therefore be expressed as ${L}_{D}=\alpha \left(m,b,s\right)\left(b+{S}_{1}l\right)+\left(1\alpha \left(m,b,s\right)\right){S}_{2}\left(1m\right)$ , where $0\le \alpha \left(m,s,b\right)\le 1$ , and S_{1} and S_{2} are the mean securities holdings for banks with ${s}_{i}<\left(lb\right)/m\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{s}_{i}\ge \left(lb\right)/m$ respectively. For any (m, b) such that a positive measure of banks holds s_{i} = (l – b)/m, the proportion of liquiditydeficient 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 nondifferentiability. It follows that if $lS<\underset{\_}{b}\left(s\right)$ then, in general, g_{m} is Lipschitz continuous, strictly increasing in m, and nondecreasing in b, so m* is Lipschitz continuous and nondecreasing 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 nondecreasing, 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 $\widehat{d}{\overline{b}}_{i}/\widehat{d}{s}_{i}<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\widehat{d}{\overline{b}}_{i}/\widehat{d}{s}_{i}\le 0.$
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 nondecreasing 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 nondecreasing 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 nondecreasing in b, it remains to be shown that the set of b such that m* is not differentiable has measure zero.
 If b > l – s_{i}m then dL_{i}/db = 1 – 1/m. As b approaches l – s_{i}m from above, this derivative approaches 1 – s_{i}/(l – b). It follows from $lb\ge \epsilon $ that ${d}^{+}{L}_{i}/db\ge 1{s}_{i}/\epsilon \ge l/\epsilon $ , and therefore $d{L}_{D}^{+}/db$ is bounded.

Lipschitz continuity of L_{D} is given by (i). Since L_{i} and therefore L_{D} is nonincreasing 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, nondifferentiable 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_{i}m*(b, S) where s_{i} equals some s_{0} such that a positive measure of banks holds s_{i} = s_{0}. The following shows that the measure of such (b, S) is zero.
Say there is a nondegenerate interval of b or S such that b = l – s_{0}m*(b, S). Inside the bounds of this interval, the proportion of liquiditydeficient banks $\left(1\alpha \right)$ is constant, so ${L}_{D}=\alpha \left(b+{S}_{1}l\right)+\left(1\alpha \right)\left(1b\right)\left(1/m1\right)$ , which is continuously differentiable. Therefore nondifferentiable 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 s_{0} has measure zero, and the number of such intervals is countable. Therefore the set of nondifferentiable 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}). Reexpressing 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 nondecreasing in i (see Section 2.1). Define $\overline{i}\left(b\right)\in \left[0,1\right]$ such that $i<\overline{i}\Rightarrow {s}_{i}<\left(lb\right)/m*\left(b\right)$ and $i\ge \overline{i}\Rightarrow {s}_{i}\ge \left(lb\right)/m*\left(b\right)$ . 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 nondifferentiable points aligning with nondifferentiable points of m*. Therefore the lefthand and righthand 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 nonpositive; (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 ${s}_{i}^{*}\left(S=l,{r}_{u}\right)<l\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{s}_{i}^{*}\left(S=0,{r}_{u}\right)>0$ , 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.

The integrands in Equation (16) are equal at $b=\mathrm{min}\left\{\underset{\_}{b},{\overline{b}}_{i}\right\}$ , so
(B9) $$\begin{array}{l}\frac{d{\Pi}_{i}^{u}\left({s}_{i},{s}_{i}\right)}{d{s}_{i}}=\left(1\lambda \right){r}_{s}+\\ +\lambda \left({\displaystyle {\int}_{0}^{\mathrm{min}\left\{\underset{\_}{b}\left({s}_{i},{r}_{u}\right),\overline{b}\left({s}_{i},{s}_{i}\right)\right\}}\frac{m*\left(b,{s}_{i}\right)}{1m*\left(b,{s}_{i}\right)}f\left(b\right)db+{\displaystyle {\int}_{\mathrm{min}\left\{\underset{\_}{b}\left({s}_{i},{r}_{u}\right),\overline{b}\left({s}_{i},{s}_{i}\right)\right\}}^{l}m*\left(b,{s}_{i}\right)\left(1+{r}_{u}\right)f\left(b\right)db}}\right)\end{array}$$where $\mathrm{min}\left\{\underset{\_}{b},{\overline{b}}_{i}\right\}$ and m* are Lipschitz continuous functions by Lemmas 9 and 10. Therefore Equation (B9) is continuous.

For high enough s_{i} it is the case that ${\overline{b}}_{i}<\underset{\_}{b}$ and
(B10) $$\frac{{\widehat{d}}^{2}{\Pi}_{i}^{u}\left({s}_{i},{s}_{i}\right)}{\widehat{d}{s}_{i}^{2}}=\lambda \frac{\widehat{d}{\overline{b}}_{i}\left({s}_{i},{s}_{i}\right)}{\widehat{d}{s}_{i}}m*\left({\overline{b}}_{i},{s}_{i}\right)\left(1+{r}_{u}\frac{1}{1m*\left({\overline{b}}_{i},{s}_{i}\right)}\right)f\left({\overline{b}}_{i}\right)$$Lemmas 9 and 10 show that $\widehat{d}{\overline{b}}_{i}/\widehat{d}{s}_{i}<0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}m*\left({\overline{b}}_{i},{s}_{i}\right)>0$ . Additionally, ${\overline{b}}_{i}<\underset{\_}{b}$ implies $1+{r}_{u}>1/\left(1m*\left({\overline{b}}_{i}\right)\right)$ . 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 ${\widehat{d}}^{2}{\Pi}_{i}/\widehat{d}{s}_{i}^{2}=0$ .

If ${\overline{b}}_{i}<\underset{\_}{b}$ then
(B11) $$\begin{array}{l}\frac{{\widehat{d}}^{2}{\Pi}_{i}^{u}\left({s}_{i},S\right)}{\widehat{d}{s}_{i}\widehat{d}S}=\lambda [{\displaystyle {\int}_{0}^{\overline{b}\left({s}_{i},S\right)}\frac{\frac{\widehat{d}m*\left(b,S\right)}{\widehat{d}S}}{{\left(1m*\left(b,S\right)\right)}^{2}}f\left(b\right)db+{\displaystyle {\int}_{\overline{b}\left({s}_{i},S\right)}^{l}\frac{\widehat{d}m*\left(b,S\right)}{\widehat{d}S}\left(1+{r}_{u}\right)f\left(b\right)db}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\widehat{d}\overline{b}\left({s}_{i},S\right)}{\widehat{d}S}m*\left({\overline{b}}_{i},S\right)\left(1+{r}_{u}\frac{1}{1m*\left({\overline{b}}_{i},S\right)}\right)f\left({\overline{b}}_{i}\right)]\end{array}$$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 ${\widehat{d}}^{2}{\Pi}_{i}/\widehat{d}s,\widehat{d}S<0$ .

Whether $\underset{\_}{b}<{\overline{b}}_{i}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\underset{\_}{b}>{\overline{b}}_{i},$
(B12) $$\frac{{\widehat{d}}^{2}{\Pi}_{i}^{u}\left({s}_{i},{s}_{i}\right)}{\widehat{d}{s}_{i}\widehat{d}{r}_{u}}={\displaystyle {\int}_{\mathrm{min}\left\{\underset{\_}{b}\left({s}_{i},{r}_{u}\right),\overline{b}\left({s}_{i},{s}_{i}\right)\right\}}^{l}m*\left(b,{s}_{i}\right)f\left(b\right)db}$$This is true because if $\underset{\_}{b}<{\overline{b}}_{i}$ then $1+{r}_{u}=1/1\left(1m*\left(\underset{\_}{b}\right)\right)$ and the two integrands in Equation (B9) are equal at $\underset{\_}{b}$ , so the integrallimit 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 $\mathrm{min}\left\{\overline{b},\underset{\_}{b}\right\}<l$ , so S > 0 implies that Equation (B12) is negative for all s_{i} > 0.

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' payoffs, 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 payoff 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 $b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{u}^{*}\left(b\right)$ 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 ${\overline{b}}_{i}<\underset{\_}{b}\left({r}_{R}\right),\text{\hspace{0.17em}}\text{i}\text{.e}\text{.}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{r}_{R}>{r}_{pen}$ , 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 ${L}_{S}\left({r}_{R}/\left(1+{r}_{R}\right)\right)>0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{so}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{B}<l$ . 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 $b>\mathrm{min}\left\{\underset{\_}{b},\overline{b}\right\}$ , 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 payoff 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 $b>\mathrm{min}\left\{\overline{b},\underset{\_}{b}\right\}$ , 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 ${r}_{R}={r}_{u}={\overline{m}}_{u}^{*}/\left(1{\overline{m}}_{u}^{*}\right),\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{r}_{u}>{\overline{m}}_{u}^{*}/\left(1{\overline{m}}_{u}^{*}\right)$ 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 $d{\Pi}_{i}/d{s}_{i}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}{r}_{R}$ 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 ${\overline{s}}_{S}={\mathrm{min}}_{i}\left\{{s}_{i}\right\}$ . Under a securities purchase policy the marginal return to securities can take two forms, depending on whether ${s}_{i}\le {\overline{s}}_{S}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}{s}_{i}\ge {\overline{s}}_{S}$ . 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 $b\le \mathrm{min}\left\{{b}_{S},{\overline{b}}_{i}\right\}$ . 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 ${s}_{i}<{\overline{s}}_{S},\left(lb\right)/\left({s}_{S}+bl\right)$ 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 ${s}_{i}<{S}_{u}^{*}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}S<{S}_{u}^{*}$ , by negativity of Equations (B10) and (B11), and because $d{\Pi}_{i}^{u}/d{s}_{i}=0\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}{s}_{i}=S={S}_{u}^{*}$ . 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 $b>{\overline{b}}_{i}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}{m}^{*}{\varphi}^{\prime}{a}_{i}$ 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 $\varphi <1$ .