A Model of Secular Migration from Centralized to Decentralized Trade

Abstract

A large number of financial assets are traded in both exchanges and over-the-counter markets (i.e., centralized and decentralized markets, CM and DM hereafter, respectively). Moreover, as documented by Biais and Green (Rev Econ Dyn 33:250–271, 2019), the twentieth century has witnessed a secular migration of asset trade from CM to DM. To this end, this paper develops a tractable model on strategic selection of venue trading to study the causes and consequences of the endogenous coexistence of CM and DM. In the model, traders’ choice of venue is shaped by the trade-off between information frictions in the CM and matching frictions in the DM. Closed-form solutions are obtained and used to characterize the endogenous share of trade across the two venues. We then use the model to evaluate two potential explanations of the migration from CM to DM: improvements in matching technologies and increases in the number of institutional investors. Surprisingly, while both forces could lead to more trade in DM, there exist parameter regions where the increase in the number of institutional investors leads to less trade in DM. We also obtain empirically testable implications that differentiate the two explanations.

Appendices

Appendix

Proofs

Proof of Proposition 1 and 2:

Substituting $c_{1}$ and $c_{2}$ into the objective function yields

$$\begin{aligned} U^{S}\left( x,\delta \right)= & {} \underset{a\in \{0,1\}}{\max }\left\{ \max \,\left\{ \frac{m\left( b,s\right) }{s}\eta x,\,p(x)\right\} a+\delta \left[ a\mathbf {1}_{\left\{ \frac{m\left( b,s\right) }{s}\eta x>p(x)\right\} }\left( 1-\frac{m\left( b,s\right) }{s}\right) x+\left( 1-a\right) x\right] \right\} \\= & {} {\left\{ \begin{array}{ll} \underset{a\in \{0,1\}}{\max }\left\{ \max \,\left\{ \frac{m\left( b,s\right) }{s}\eta x,\,p(x)\right\} a\right\} , &{} \delta =0 \\ \underset{a\in \{0,1\}}{\max }\left\{ \max \,\left\{ \frac{m\left( b,s\right) }{s}\eta x,\,p(x)\right\} a+a\mathbf {1}_{\left\{ \frac{m\left( b,s\right) }{s}\cdot \eta x>p(x)\right\} }\left( 1-\frac{m\left( b,s\right) }{s}\right) x+\left( 1-a\right) x\right\} , &{} \delta =1 \end{array}\right. } . \end{aligned}$$

As a result, when $\delta =0$, $a^{*}=1$, i.e., investors with preference shock have to sell the claim of their projects. Investors with $\delta =1$, however, could either participate in centralized or decentralized market ($a=1$) or simply wait till $t=2$ ($a=0$). However, the above optimization implies that investors would never try centralized market due to search friction and bargaining.

First of all, competitive buyers set $p(x)=x$ in complete information. In this scenario, $p(x)>\frac{m(b,s)}{s}\eta x$ for all sellers-$(x,\delta =0)$ and thus they trade in centralized market. Moreover, sellers-$(x,\delta =1)$ would be indifferent between selling in centralized market at $t=1$ and waiting till $t=2$.

Secondly, in the presence of information asymmetry, $p(x)=p$ for all sellers pooling in centralized market. On one hand, for sellers with $\delta =0$, if decentralized market does not exist, their only choice is the centralized market. If the decentralized market exists, however, they would compare $\frac{m(b,s)}{s}\eta x$ with p. Furthermore, if $\frac{m(b,s)}{s} \eta x_{1}>p$, we would also have $\frac{m(b,s)}{s}\eta x_{2}>p$ provided $x_{2}>x_{1}$. Thus there may exist a cut-off point $\widetilde{x}$ on the choice of trading venues. If $\widetilde{x}\in (x_{L},x_{H})$, then $\frac{ m(b,s)}{s}\eta \widetilde{x}=p$ holds by definition. On the other hand, for sellers with $\delta =1$, as argued above, they would never consider trading in decentralized market even though it would be available. Instead, they simply compare p and x. As a result, those with $x<p$ would sell their asset claims in the centralized market at $t=1$ while those with $x\ge p$ would enter either markets and wait till $t=2$.

Finally, based the above two pieces of observation, we have

$$\begin{aligned} U^{s}\left( x,\delta \right)= & {} {\left\{ \begin{array}{ll} p &{} if\,\,\delta =0\,\,and\,\,x\le \widetilde{x} \\ \frac{x}{\widetilde{x}}\cdot p &{} if\,\,\delta =0\,\,and\,\,x>\widetilde{x} \\ p &{} if\,\,\delta =1\,\,and\,\,x\le p \\ x &{} if\,\,\delta =0\,\,and\,\,x\le \widetilde{x} \end{array}\right. } \\= & {} \max \left\{ \frac{x}{p+\left( \widetilde{x}-p\right) \cdot \mathbf {1} _{\{\delta =0\}}},\,\,1\right\} \cdot p \end{aligned}$$

Proof of Corollary 1:

It is immediately obtained by using Proposition 1 and 2. $\square$

Proof of Lemma 1:

The results in the general case in proved as below.

First of all, we show that $p\le \widetilde{x}$. Eq. (3) suggests that

$$\begin{aligned} p= & {} \frac{\pi F(\widetilde{x})\mathbb {E}\left( x|x\le \widetilde{x}\right) +(1-\pi )F(p)\mathbb {E}\left( x|x\le p\right) }{\pi F(\widetilde{x})+(1-\pi )F(p)} \\= & {} \frac{\pi \int _{x_{L}}^{\widetilde{x}}x\mathrm{d}F(x)+(1-\pi )\int _{x_{L}}^{p}x\mathrm{d}F(x)}{\pi F(\widetilde{x})+(1-\pi )F(p)} \\\le & {} \frac{\pi \int _{x_{L}}^{\widetilde{x}}\widetilde{x}\mathrm{d}F(x)+(1-\pi )\int _{x_{L}}^{p}p\mathrm{d}F(x)}{\pi F(\widetilde{x})+(1-\pi )F(p)} \\= & {} \frac{\pi \widetilde{x}F(\widetilde{x})+(1-\pi )pF(p)}{\pi F(\widetilde{x} )+(1-\pi )F(p)}, \end{aligned}$$

where the inequality strictly holds iff $x>x_{L}$. Thus we have

$$\begin{aligned} p\le \frac{\pi \widetilde{x}F(\widetilde{x})+(1-\pi )pF(p)}{\pi F( \widetilde{x})+(1-\pi )F(p)}. \end{aligned}$$

Multiplying both side of this inequality with $\pi F(\widetilde{x})+(1-\pi )F(p)$ and rearranging then yields $p\le \widetilde{x}$, where the equality holds iff $\widetilde{x}=x_{L}(=p)$.

Secondly, Eq. (3) can be rewritten as

$$\begin{aligned} G(p,\widetilde{x},\pi )\equiv \pi \int _{x_{L}}^{\widetilde{x}}x\mathrm{d}F(x)+(1-\pi )\int _{x_{L}}^{p}x\mathrm{d}F(x)-\pi pF(\widetilde{x})-(1-\pi )pF(p)=0. \end{aligned}$$

Thus we have

$$\begin{aligned} G_{p}\equiv \frac{\partial G}{\partial p}= & {} -[\pi F(\widetilde{x})+(1-\pi )F(p)]<0 \\ G_{\widetilde{x}}\equiv \frac{\partial G}{\partial \widetilde{x}}= & {} \pi ( \widetilde{x}-p)f(\widetilde{x})>0 \end{aligned}$$

According to Implicit Function Theorem, we have

$$\begin{aligned} \frac{dp}{d\widetilde{x}}=-\frac{G_{\widetilde{x}}}{G_{p}}=>0. \end{aligned}$$

Thus we can denote the above result as $p=P_{AS}(\widetilde{x},\pi )$, which is an increasing function of $\widetilde{x}$. Furthermore, since $\widetilde{x}\ge x_{L}$, we immediately have $p\ge x_{L}$. When $\widetilde{x}=x_{L}$, Eq. (3) is reduced as follows.

$$\begin{aligned} p=\frac{\int _{x_{L}}^{p}x\mathrm{d}F(x)}{F(p)}=\mathbb {E}(x|x\le p), \end{aligned}$$

which is a classic problem of adverse selection by Akerlof (1970) and the unique solution is $p=x_{L}$. As a result, $P_{AS}(\widetilde{x} =x_{L})=x_{L}$ and thus $p\ge x_{L}$. So far we finish the proof that $x_{L}\le p\le \widetilde{x}$, where both inequality strictly holds if $\widetilde{x}>x_{L}$.

Moreover, we have

$$\begin{aligned} \frac{\partial G}{\partial \pi }=\left[ \int _{x_{L}}^{\widetilde{x} }x\mathrm{d}F(x)-pF(\widetilde{x})\right] -\left[ \int _{x_{L}}^{p}x\mathrm{d}F(x)-pF(p)\right] \end{aligned}$$

Define $H\left( a;p\right) \equiv \int _{x_{L}}^{a}x\mathrm{d}F(x)-pF\left( a\right)$. Then we have $\frac{\partial H}{\partial a}=\left( a-p\right) f\left( a\right)$ and thus $H\left( a;p\right)$ increases with a when $a>p$. Since $\widetilde{x}>p$, we have

$$\begin{aligned} G_{\pi }\equiv \frac{\partial G}{\partial \pi }=H\left( \widetilde{x} ;p\right) -H\left( p;p\right) >0, \end{aligned}$$

which in turn, by using Implicit Function Theorem again, implies that

$$\begin{aligned} \frac{dp}{d\pi }=-\frac{G_{\pi }}{G_{p}}>0. \end{aligned}$$

Denote $p=P_{AS}(\widetilde{x},\pi )$. Then we know that $p\le P_{AS}( \widetilde{x},\pi =1)=\mathbb {E}(x|x\le \widetilde{x})\le \mathbb {E} (x|x\le x_{H})$.

Finally, when $\pi =0$, Eq. (3) is reduced to

$$\begin{aligned} p=\frac{\int _{x_{L}}^{p}x\mathrm{d}F(x)}{F(p)}=\mathbb {E}(x|x\le p), \end{aligned}$$

which has been discussed above in the case when $\widetilde{x}=x_{L}$. The only solution is $p=P_{AS}(\widetilde{x},\pi =0)=x_{L}$ and CM totally collapses.

Finally, when $x\overset{U}{\sim }X\left[ x_{L},x_{H}\right] =\left[ 0,1 \right]$, we have

$$\begin{aligned} F(x)= {\left\{ \begin{array}{ll} 0 &{} if\,\,x<x_{L} \\ \frac{x-x_{L}}{x_{H}-x_{L}}=x &{} if\,\,x_{L}\le x\le x_{H} \\ 1 &{} if\,\,x>x_{H} \end{array}\right. } . \end{aligned}$$

Substituting F(x) into Eq. (3) and making some algebraic manipulation yields Eq. (4). $\square$

Proof of Lemma 2:

We immediately obtain the results by combining the free entry condition in Sect. 2.5 with Cobb–Douglas matching function. $\square$

Proof of Proposition 3:

Combining Proposition 2, Lemma 1 , and Lemma 2 yields the closed-form solutions in Proposition 3. $\square$

Proof of Proposition 4, Proposition 5, Proposition 6 and Corollary 2:

With some algebraic manipulation, we can obtain the results as we did for proofs in the baseline model. $\square$

Proof of Proposition 7:

The proof is similar to that for Lemma 2, and thus omitted here. $\square$

Proof of Lemma 3 and Proposition 8:

The first part is proved as below. To ease illustration while preserving the key insights, we have assumed that information cost $\kappa$ is low enough such that $b>s$ is always true in equilibrium. In turn, we have $\frac{m(b,s) }{s}=\lambda \in (0,1)$, a constant. This would help us focus on characterizing optimal contract by buyers. We break down the proof into the following steps.

First of all, since sellers could always seller their assets at price p in CM and $\delta \in [0,1]$, buyers in DM would have no customers if $U(x,\delta )<p$. Meanwhile, to recover information cost, buyers in DM ex ante would never accept sellers with $x<p$.

Secondly, for seller-$(x,\delta )$ self-selecting into DM and is allowed to trade with buyers there, denote $\widetilde{\delta }(x)=\min \left\{ \frac{p}{x},1\right\} =\frac{p}{x}$. Since Then we can check that $\underline{\delta }(x)\le \widetilde{\delta }(x)\le \overline{\delta }(x)$ , where $\underline{\delta }(x)$ and $\overline{\delta }(x)$ are characterized in the proof of Lemma 1. For each x, buyers launch direct mechanism for two groups respectively. One is $\delta \in \varDelta _{1}=\left[ \underline{\delta }(x),\widetilde{\delta }(x)\right]$ while the other group is $\delta \in \varDelta _{2}=[\widetilde{\delta }(x),\overline{\delta }(x)]$. On one hand, for each group, buyers make sure IR and IC conditions are satisfied. On the other hand, buyers have to make sure sellers in group $\varDelta _{1}\cup \varDelta _{2}$ would have no incentive to deviate the other group. After all, even though x is verifiable after buyers incur information cost, $\delta$ is still unobservable. As a result, incentive compatibility of not deviating to another group has to be additionally taken into account. In the next two pieces of analysis, we first solve the within-group contract and then go to discussion of IC on across-group.

Buyer’s objective function for group $\varDelta _{1}$ is

$$\begin{aligned} \varPi _{B}(x)|_{\varDelta _{1}}\equiv \underset{\left\{ q(x,\delta )\in [0,1],\tau (x,\delta )\in [0,\infty )\right\} }{max}\left\{ \int _{ \underline{\delta }(x)}^{\widetilde{\delta }(x)}\left[ -\tau (x,\delta )+q(x,\delta )\cdot x\right] \right\} . \end{aligned}$$

Meanwhile, for group with $\delta \in \varDelta _{1}$, the outside option is simplified as $V\left( x,\delta ,p\right) \equiv \delta x+\frac{1}{\lambda } \max \left\{ p-\delta x,0\right\} =\frac{p}{\lambda }-\left( \frac{1}{ \lambda }-1\right) \delta x$. That is, for sellers in this group, the outside option decreases with $\delta$. Following Maggi and Rodriguez-Clare (1995), among others, we define $\varUpsilon (x,\delta )=U(x,\delta )-V(x,\delta ,p).$ Envelope Theorem suggests

$$\begin{aligned} \frac{\partial \varUpsilon }{\partial \delta }=\left[ \frac{1}{\lambda } -q\left( x,\delta \right) \right] \cdot x. \end{aligned}$$

Thus

$$\begin{aligned} \left[ 1-q\left( x,\delta \right) \right] \delta x+\tau \left( x,\delta \right) -\left[ \frac{p}{\lambda }-\left( \frac{1}{\lambda }-1\right) \delta x\right] =\int _{\underline{\delta }(x)}^{\delta }\left[ \frac{1}{\lambda } -q(x,\delta ^{\prime })\right] x\mathrm{d}\delta ^{\prime }. \end{aligned}$$

Expressing the above equation for $\tau (x,\delta )$ and substituting it into the buyer’s objective function for group $\varDelta _{1}$ mentioned above, we can easily prove that, for group $\varDelta _{1}$, $q^{*}(x,\delta )|_{\varDelta _{1}}=1$. Substituting it into the above equation suggests that $\tau ^{*}(x,\delta )|_{\varDelta _{1}}$ has nothing to with $\delta$ and is thus denoted as $\tau ^{*}(x)|_{\varDelta _{1}}$.

Similarly, we can show that $q^{*}(x,\delta )|_{\varDelta _{2}}=1$ and $\tau ^{*}(x,\delta )|_{\varDelta _{2}}$ also has nothing to do with $\delta$ and is thus denoted as $\tau ^{*}(x)|_{\varDelta _{2}}$. Finally, to make sure the IC condition of across-group is satisfied, we have to make sure $\tau ^{*}(x,\delta )|_{\varDelta _{1}}=\tau ^{*}(x,\delta )|_{\varDelta _{1}}\equiv \tau (x)|_{\varDelta _{1}\cup \varDelta _{2}}=\tau (x)$. In sum, given $x>p$ and buyers and sellers are matched in DM, the optimal contract would take the form as $\{q^{*}(x,\delta )=1,\tau ^{*}(x,\delta )=\tau (x)\}$. It is obvious that $\tau (x)\le x$ is always held.

In turn, we have $U(x,\delta )=\tau (x)$ and thus

$$\begin{aligned} \underline{\delta }(x)= & {} \max \left\{ 0,\frac{p-\lambda \cdot \tau (x)}{ (1-\lambda )\cdot x}\right\} \\ \overline{\delta }(x)= & {} \min \left\{ 1,\frac{\tau (x)}{x}\right\} =\frac{ \tau (x)}{x}. \end{aligned}$$

As a recap, buyer’s profit function focusing on sellers with x is

$$\begin{aligned} \varPi _{B}(x)\equiv \underset{\left\{ q(x,\delta )\in [0,1],\tau (x,\delta )\in [0,\infty )\right\} _{\mathbf {Z}_{DM}|x}}{\max } \left\{ \int _{\delta \in \mathbf {Z}_{DM}|x}\left[ -\tau (x,\delta )+q(x,\delta )\cdot x\right] \right\} . \end{aligned}$$

Using the optimal contract and cut-off values just obtained above, $\varPi _{B}(x)$ is refined as below.

$$\begin{aligned} \varPi _{B}(x)=\underset{\tau (x,\delta )\in [0,x]}{\max }\left[ x-\tau (x)\right] \left[ G(\overline{\delta }(x))-G(\underline{\delta }(x))\right] . \end{aligned}$$

subject to

$$\begin{aligned} \underline{\delta }(x)= & {} \max \left\{ 0,\frac{p-\lambda \cdot \tau (x)}{ (1-\lambda )\cdot x}\right\} , \\ \overline{\delta }(x)= & {} \min \left\{ 1,\frac{\tau (x)}{x}\right\} =\frac{ \tau (x)}{x}, \end{aligned}$$

where G denotes the CDF of $\delta$ with support [0, 1]. If we further assume $\delta \overset{U}{\sim }\varDelta =[0,1]$, then we obtain $\tau (x)$ as that in Proposition 5. In turn, we obtain $\underline{\delta }(x)$ and $\overline{\delta }(x)$ as in the second part of this proposition. $\square$

Proof of Lemma 4, Proposition 9 and Corollary 3:

The right panel features the payoff in (10) of the extended model. Since the matching probability of the informed CM tends to be higher than that of the DM, the payoff slope of the former case is steeper than that of the latter case. Then we immediately obtain Lemma 4.

Now we move on to prove Proposition 9. Given its existence, the equilibrium consists of the cutoffs $\left\{ \tilde{x}_{1}, \tilde{x}_{2}\right\}$, an equilibrium price p, and the market tightness $\left\{ b_{1}/s_{1},b_{2}/s_{2}\right\}$ such that:

$$\begin{aligned}&p=\frac{\pi F(\widetilde{x}_{1})\mathbb {E}(x|x\le \widetilde{x_{1}})+(1-\pi )F(p)\mathbb {E}(x|x\le p)}{\pi F(\widetilde{x_{1}})+(1-\pi )F(p)},\\&q_{b}^{DM}(1-\eta )\mathbb {E}(x|\widetilde{x}_{2}\ge x\ge \widetilde{x} _{1})=\kappa ,\\&q_{b}^{CM}(1-\eta )\left[ \mathbb {E}(x|x\ge \widetilde{x}_{2})\right] =\kappa ,\\&q_{s}^{DM}\eta \widetilde{x}_{1}=p,\\&q_{s}^{CM}\eta \widetilde{x}_{2}-\kappa _{I}=q_{s}^{DM}\eta \widetilde{x} _{2}, \end{aligned}$$

where the matching probability of buyers and sellers are respectively given by $q_{b}^{DM}=\lambda _{1}\left( \frac{b_{1}}{s_{1}}\right) ^{m-1}$, $q_{s}^{DM}=\lambda _{1}\left( \frac{b_{1}}{s_{1}}\right) ^{m},q_{b}^{CM}=\lambda _{2}\left( \frac{b_{2}}{s_{2}}\right) ^{m-1}$ and $q_{s}^{CM}=\lambda _{2}\left( \frac{b_{2}}{s_{2}}\right) ^{m}$. The first equation is the same as pricing equation as in the benchmark model, and can be solved as

$$\begin{aligned} \frac{p}{\widetilde{x}_{1}}=\frac{\sqrt{\pi }}{1+\sqrt{\pi }}=\varphi \left( \pi \right) , \end{aligned}$$

One can plug this relation into the second equation:

$$\begin{aligned} \lambda _{1}^{\frac{1}{m}}\left( \frac{\varphi \left( \pi \right) }{\eta } \right) ^{\frac{m-1}{m}}(1-\eta )\frac{\widetilde{x}_{1}+\widetilde{x}_{2}}{2 }=\kappa \end{aligned}$$

This delivers

$$\begin{aligned} \widetilde{x}_{1}=\frac{2\kappa }{\lambda _{1}^{\frac{1}{m}}\left( \frac{ \varphi \left( \pi \right) }{\eta }\right) ^{\frac{m-1}{m}}(1-\eta )}- \widetilde{x}_{2}, \end{aligned}$$

(A.1)

This expression is analogous to the benchmark model but now the upper bound is replaced by $\widetilde{x}_{2}$.

Then we move to characterizing $\widetilde{x}_{2}.$ The equation is given by the buyer’s zero profit condition, with the matching probability substituted in using seller’s indifference condition:

$$\begin{aligned} \varGamma \left( \widetilde{x}_{2}\right) =\kappa \end{aligned}$$

(A.2)

where

$$\begin{aligned} \varGamma \left( \widetilde{x}_{2}\right) \equiv \lambda _{2}^{\frac{1}{m} }\left( \frac{\eta }{\varphi \left( \pi \right) +\frac{\kappa _{I}}{ \widetilde{x}_{2}}}\right) ^{\frac{1-m}{m}}\left( 1-\eta \right) \frac{1+ \widetilde{x}_{2}}{2}. \end{aligned}$$

This is a nonlinear equation with respect to $\widetilde{x}_{2}$. Since the matching elasticity $m\in \left( 0,1\right)$, we can easily verify that $\varGamma \left( \widetilde{x}_{2}\right)$ strictly increases with $\widetilde{x}_{2}$, and thus $\widetilde{x}_{2}$ is unique if any solution to $\widetilde{x}_{2}$ exists.

In the remaining part, we need to characterize the parameter region that sustains the conjecture that $0<\widetilde{x}_{1}<\widetilde{x}_{2}<1$. To start with, substitute expression of $\widetilde{x}_{1}$ (equation (A.1 )) into the inequality, we obtain a series of inequalities regarding $\widetilde{x}_{2}$:

$$\begin{aligned} 0<\widetilde{x}_{1}=\frac{2\kappa }{\lambda _{1}^{\frac{1}{m}}\left( \frac{ \varphi \left( \pi \right) }{\eta }\right) ^{\frac{m-1}{m}}(1-\eta )}- \widetilde{x}_{2}<\widetilde{x}_{2}<1, \end{aligned}$$

The next lemma shows that the series of inequalities are equivalent to the following bound restrictions on $\widetilde{x}_{2}$: $\square$

Lemma 5

Given the existence of $\left( \widetilde{x}_{1},\widetilde{x}_{2}\right)$, the sufficient and necessary condition for $0<\widetilde{x}_{1}<\widetilde{x} _{2}<1$ is given by

$$\begin{aligned} x^{*}<\widetilde{x}_{2}<x^{**}. \end{aligned}$$

(A.3)

where

$$\begin{aligned} x^{*}\equiv & {} \frac{\kappa }{\lambda _{1}^{\frac{1}{m}}\left( \frac{ \eta }{\varphi \left( \pi \right) }\right) ^{\frac{1-m}{m}}(1-\eta )}, \\ x^{**}\equiv & {} \min \left\{ 2\frac{\kappa }{\lambda _{1}^{\frac{1}{m }}\left( \frac{\eta }{\varphi \left( \pi \right) }\right) ^{\frac{1-m}{m} }(1-\eta )},1\right\} . \end{aligned}$$

Note that $x^{*}$ is derived under the second inequality $\frac{2\kappa }{\lambda _{1}^{\frac{1}{m}}\left( \frac{\varphi \left( \pi \right) }{\eta } \right) ^{\frac{m-1}{m}}(1-\eta )}-\widetilde{x}_{2}<\widetilde{x}_{2}$ while $x^{**}$ is derived under the first inequality $0<\frac{ 2\kappa }{\lambda _{1}^{\frac{1}{m}}\left( \frac{\varphi \left( \pi \right) }{\eta }\right) ^{\frac{m-1}{m}}(1-\eta )}-\widetilde{x}_{2}$ and the last inequality $\widetilde{x}_{2}<1.$

Given the above, lemma, we can apply $\varGamma \left( .\right)$ operation to the inequality (A.3). This yields $\varGamma \left( x^{*}\right)<\varGamma \left( \widetilde{x}_{2}\right) <\varGamma \left( x^{**}\right)$, or equivalently, $\varGamma \left( x^{*}\right)<\kappa <\varGamma \left( x^{**}\right)$. Plugging in the expressions of $x^{*}$ and $x^{**},$ one can show, after algebraic manipulation, that this equation is equivalent to the following restriction on $\lambda _{2}$:

$$\begin{aligned} \lambda ^{*}<\lambda _{2}<\lambda ^{**}, \end{aligned}$$

where

$$\begin{aligned} \lambda ^{*}\equiv & {} \left[ \frac{2\kappa }{\left( \frac{\eta }{\varphi \left( \pi \right) +\kappa _{I}/x^{**}}\right) ^{\frac{1-m}{m} }\left( 1-\eta \right) \left( 1+x^{**}\right) }\right] ^{m}, \\ \lambda ^{**}\equiv & {} \left[ \frac{2\kappa }{\left( \frac{\eta }{ \varphi \left( \pi \right) +\kappa _{I}/x^{*}}\right) ^{\frac{1-m}{m} }\left( 1-\eta \right) \left( 1+x^{*}\right) }\right] ^{m}. \end{aligned}$$

The above condition for $\lambda$ summarizes the condition under which those three markets coexist with each other. Similarly, we can obtain the other two conditions as in Proposition 9.

Finally, we address the effect of $\lambda _{1}$ on the total volume of trade in DM. Lemma 4 reveals the trade volume in DM is given by $F\left( \widetilde{x}_{2}\right) -F\left( \widetilde{x}_{1}\right)$. Combining Eq. (A.1) and (A.2) immediately reaches Corollary 3.