Abstract
We study asset pricing within a general equilibrium model where unsecured credit is ruled out, and a real asset helps agents carry out mutually beneficial transactions by serving as collateral. A unique feature of our model is that the agent who provides the loan might have a low valuation for the collateral asset. Nevertheless, the lender rationally chooses to accept the collateral because she can access a secondary asset market where she can sell the asset. Following a recent strand of the finance literature, based on the influential work of Duffie et al. (Econometrica 73(6):1815–1847, 2005), we model this secondary asset market as an overthecounter market characterized by search and bargaining frictions. We study how the asset’s property to serve as collateral affects its equilibrium price, and how the asset price and the economy’s welfare are affected by the degree of liquidity in the secondary asset market.
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Notes
The argument that a lender might have a “low valuation” for an asset such as a car or a house, as in the earlier example, is selfexplanatory. When it comes to financial assets, such as Tbills, the notion of a low asset valuation is less straightforward, since these assets typically pay a predetermined cash flow. One way to motivate the concept of low valuation for a financial asset is to think that this asset has a long maturity horizon (i.e., it pays in the future), while the lender might have an immediate liquidity need.
The source of this quote is “A Review of Treasury’s Debt Management Policy,” June 3, 2002, available at http://www.treas.gov/press/releases/po3149.htm. It should be pointed out that Treasury bills are among the assets that are most widely used as collateral.
Interestingly, Lagos (2011) makes precisely this point. He suggests that as long as an asset can help an untrustworthy buyer obtain what he wants from a seller, either by serving as means of payment or as collateral, that asset will have liquidity properties (implying that it should also be priced accordingly). Despite making this interesting suggestion, the author also studies a model where assets serve as media of exchange. Hence, a contribution of this paper is to formalize Lagos’s (2011) suggestion.
This is a standard trick used in search theory in order to ensure that the measure of agents (here buyers) remains fixed. For example, see the marriage market model of Burdett and Coles (1997).
Since our goal is to study the pricing properties of assets that serve as collateral, we focus on an environment where the asset demand comes from the agents who actually use the asset as collateral, i.e., the buyers. If sellers were longlived (and did not die with probability l, like buyers do), then they might absorb all the asset supply simply because they are more patient than buyers [the effective discount rate of a buyer is \(\beta (1l)\), while that of a seller is \(\beta \)]. But this would kill off all the interesting asset pricing results that stem from the asset’s property to facilitate trade in the LW market. To avoid this, we assume that sellers (and investors) live only for one period. Alternatively, one could assume that sellers and investors are longlived, but they also die with probability l in every period. Then, these agents would choose to not purchase any assets in the CM because they cannot take advantage of the liquidity properties of the asset (for a formal proof, see Rocheteau and Wright 2005). In this case, all the main results of the paper would remain unaltered.
The meaning of these terms is explained in more detail in Sect. 3.2. In short, \(q^*\) is the amount of good that maximizes the current surplus of a match between a buyer and a seller, i.e., \(u(q)q\), while the \(q^{**}\) is the amount that maximizes surplus taking under consideration that the buyer effectively discounts future between the current subperiod (LW market) and the third subperiod (CM) at rate \(1l\), due to the probability of death.
Duffie et al. offer several interpretations of the agents with low valuations. More precisely, they claim that “[...] a lowtype investor may have (1) low liquidity (that is, a need for cash), (2) high financing costs, (3) hedging reasons to sell, (4) a relative tax disadvantage, or (5) a lower personal use of the asset.” In our story, the leading interpretation is (5), although some of the other interpretation [like (1),(2), or (3)] might also be relevant.
It should be pointed out that agents are free to use the asset as a medium of exchange (as they do with money), but they choose not to because the asset has a bad property as a means of payment: It is worth more to the buyer than it is to the seller. Hence, agents rationally choose to use the asset as collateral in the way described above.
The fact that all the assets go to the possession of the seller in case of the buyer’s death is a result rather than an assumption. The only assumption made here is that buyers do not obtain any utility from leaving bequests to the clones who replace them. Given this, it is easy to see that buyers are happy to leave their assets to the seller in the case of death, since this allows them to purchase more good in the LW market now (see Eq. 3.8), and it has no cost later (they do not care what happens to the assets when they are dead).
We assume that the agents that sellers meet in the LW (buyers) are not the same as the ones that they meet in the OTC market (investors). This is necessary because our assumption that there is no anonymity in the OTC market may create some troubles for the essentiality of money (in the LW) and open the door to folktype theorem arguments as the ones highlighted in Aliprantis et al. (2007) and Araujo et al. (2012).
This assumption seems to be realistic for most realworld asset markets. In terms of modeling, frictions such as anonymity in the LW market are necessary in order to generate a role for a medium of exchange or collateral in that market. Assuming that trade in the OTC market is also anonymous would only add unnecessary complications without delivering many interesting economic insights.
More formally, this seller’s problem is to choose X, H in order to maximize her net utility \(XH\), subject to the budget constraint \(X = H + \varphi m + b\). Replacing the term \(XH\) into \(W^\mathrm{SN}\) yields the expression reported in (3.3).
As a clarification, notice that the term c is a repayment promise from an investor (in the OTC market) while b is a repayment promise from a buyer (in the LW market). Recall that the former is an unbacked promise, while the latter is a secured (collateralized) promise, due to the assumption that, unlike the OTC, the LW market is characterized by anonymous trade.
With probability l the seller is matched with a buyer who will vanish after the LW market closes. This seller will seize the buyer’s asset holdings, and she will attempt to sell them in the OTC market. With probability \(1l\), the seller is matched with a buyer who honors her debt. This seller proceeds to the CM with money holdings d and a credit of b units of fruit.
The buyer is happy to sign this contract. If she dies, she does not care who gets the assets, while promising these assets to the seller induces the latter to work harder now and produce a higher q. See also footnote 10.
In papers such as Geromichalos et al. (2007) and Jacquet and Tan (2012), money and assets are (perfect) substitutes, and the amount of good that the buyer can purchase in the LW market is simply an increasing function of the value of money and assets summed together. Our bargaining problem has a much richer solution. It depends on the value of money that the seller receives on the spot, and the value of the asset that she rationally expects to receive by selling it in the OTC market, which, in turn, depends on the microstructure of that market.
To see this point more formally, assume that initially \((m,a)=(m_0,a_0)\), satisfying \(\pi (m_0,a_0)+(1l)a_0=q^*\), so that \(q(m_0,a_0)=q^*\), \(b(m_0,a_0)=a_0\), and \(d(m_0,a_0)=m_0\). Now consider an increase in assets to \(a'=a_0+\epsilon \), while money holdings remain unaltered and focus on the following two plans of action for the buyer. Under Plan A, the buyer still pledges all of her assets in an attempt to keep her consumption of q at the highest possible level. Under this scenario \(b_A(m_0,a')=a'\) and \(q_A(m_0,a')=\pi (m_0,a')+(1l)a'=q^*+\epsilon (1l+x)\). Under Plan B, the buyer keeps her consumption at \(q^*\) and uses the extra units of asset in order to decrease the value of her loan b. It is easy to check that in this case the value of the loan satisfies \((1l)b_B(m_0,a')=q^*\varphi m_0x a'\). Defining \(\tilde{\epsilon }\equiv \epsilon (1l+x)>0\), it is now straightforward to verify that the increase in surplus that the buyer can achieve by following Plan B is greater than the one that she can achieve under Plan A, if and only if \(u(q^*)+\tilde{\epsilon }>u(q^*+\tilde{\epsilon })\). Of course, this inequality is always satisfied since \(u'(q)<1\) for all \(q>q^*\).
Unlike the assets of deceased buyers, which will go to the possession of the seller with whom that buyer matched, we have not assumed that the same will happen with the money of deceased buyers. Hence, a question that arises is what happens to the money of deceased buyers. However, this issue will never arise in equilibrium: When it is costly to carry money (as we assume is the case here), buyers will never carry money holdings that bring them in the interior of Region 1.
More precisely, the net cost of holding one unit of money for the buyer is given by the term \(\varphi + \beta \hat{\varphi }(1l)\). This cost will be positive if and only if \(\varphi /(\beta (1l) \hat{\varphi })>1\), which, as we clarified in Sect. 2, is a maintained assumption of the model. The limiting case where \(\varphi /(\beta (1l) \hat{\varphi })\rightarrow 1\) is simply the modified Friedman rule (modified due to the fact that buyers in our model die with probability l within periods).
The terms \(m_{23},m_{34}\) are indicated in Fig. 3 as the levels of money holdings that bring the buyer right on the boundary of Regions 2,3 (\(m_{23}\)) and 3,4 (\(m_{34}\)), given \(\hat{a}=a_\mathrm{L}\). Then, from Lemma 3.2, we know that when \((\hat{m},\hat{a})=(m_{23},a_\mathrm{L})\) the buyer has managed to bring her loan down to zero (this is the very definition of entering Region 2). On the other hand, with asset holdings \((\hat{m},\hat{a})=(m_{34},a_\mathrm{L})\), the buyer is so constrained that she is still pledging all of her assets as collateral. Clearly, the buyer is not only indifferent between points A and B, but also between any other portfolio \((\hat{m},a_\mathrm{L})\), \(\hat{m}\in [m_{34},m_{23}]\), which will be combined with a loan equal to \(b=(1l)^{1}[q^*\hat{\varphi }\hat{m}l(1\delta +\alpha _\mathrm{_S}\lambda \delta )a_\mathrm{L}]\) (so that the buyer purchases \(q^*\) in all cases).
This is an immediate consequence of the indeterminacy of money demand, and it can be seen more clearly by inspection of Fig. 3: It follows from that graph that the buyer’s asset holdings will bring her in Region 4 if and only if \(\varphi /[ \beta \hat{\varphi }(1l)]>1/(1l)\), which is equivalent to \(\varphi >\beta \hat{\varphi }\), and reduces to \(\mu >\beta 1\) in steady state equilibrium. Similarly, the buyer’s asset holdings will bring her in Region 2 if and only if \(\varphi <\beta \hat{\varphi }\), which is equivalent to \(\mu <\beta 1\).
The net rate of return on money is \((\hat{\varphi }\varphi )/\varphi =1/(1+\mu )1\), and the net rate of return on the asset is \((1\psi )/\psi \). Equality of these rates would require \(\psi =1+\mu \), which is clearly not the case here. However, the two rates are positively related, in the sense that an increase in \(\mu \) decreases (directly) the rate of return on money, but it also decreases (indirectly) the rate of return on the asset by increasing its price.
The example of houses being used as collateral fits well with our story in a conceptual sense. Recall that our story entails agents who use assets as collateral, but these assets might not be valued per se by lenders. However, lenders still accept them as collateral, because they understand that the borrowers have an incentive to repay their debt and keep the asset, and because they know that even if the borrower has no choice but to default, they could sell off the collateral in a secondary market. These are all features that also characterize the types of transactions that take place in our realworld example.
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We are grateful to Jaerim Choi, Jae Wook Jung, Guillaume Rocheteau, Kevin Salyer, Akira Sasahara, Ina Simonovska, Inmo Sung, and Randall Wright for useful comments and suggestions.
Appendix
Appendix
Proof of Lemma 3.2
We define the Langrangian function for the LW bargaining problem as
where \(\lambda _1\), \(\lambda _2\), \(\lambda _3\), and \(\lambda _4\) are Lagrange multipliers. The FOCs are

Case 1: \(\lambda _1 = 0\), \(\lambda _2 = 0\), \(\lambda _3 = 0\) (\(d < m\), \(0 < b < a\))
Equations (6.1) and (6.2) imply \(u'(q) = 1l\). On the other hand, Eqs. (6.1) and (6.3) imply \(u'(q) = 1\). These contradict each other.

Case 2: \(\lambda _1 = 0\), \(\lambda _2 = 0\), \(\lambda _3 > 0\) (\(d < m\), \(0 < b = a\))
Equations (6.1) and (6.2) imply \(u'(q) = 1l\). On the other hand, Eqs. (6.1) and (6.3) imply \(\lambda _3 = (1l)[ u'(q)  1]\). Since \(\lambda _3 > 0\), this implies \(u'(q) > 1\). Again, these contradict each other.

Case 3: \(\lambda _1 = 0\), \(\lambda _2 > 0\), \(\lambda _3 = 0\) (\(d < m\), \(0 = b < a\))
Equations (6.1) and (6.2) imply \(u'(q) = 1l\). Hence, \(q=q^{**}\). Equations (6.1) and (6.3) imply \(\lambda _2 = (1l)[ 1  u'(q^{**})]\), which is consistent with \(\lambda _2>0\). Equation (6.5) implies \(b=0\). Then, from Eq. (6.7), we get \(d = [q^{**}  l(1\delta +\alpha _\mathrm{_S} \lambda \delta )a]/\varphi \), i.e., \(d=(q^{**}xa)/\varphi \). To sum up, in Case 3 the solution is \(q= q^{**}\), \(d = (q^{**}  xa)/\varphi \), and \(b= 0\). Notice that in this case we have \(\pi (m,a) > q^{**}\), which is consistent with the fact that \(d<m\).

Case 4: \(\lambda _1 = 0\), \(\lambda _2 > 0\), \(\lambda _3 > 0\) (\(d < m\), \(0 = b = a\))
This case is similar to Case 3, with the exception that here \(a=0\). It is easy to check that the solution is \(q = q^{**}\), \(d=q^{**}/\varphi \), and \(b = 0\). Since here \(d<m\) and \(a=0\), we have \(\pi (m,0)>q^{**}\).

Case 5: \(\lambda _1 > 0\), \(\lambda _2 = 0\), \(\lambda _3 = 0\) (\(d = m\), \(0 < b < a\))
Equations (6.1) and (6.3) imply \(u'(q) = 1\), and so \(q=q^*\). Notice that \(q^{**}>q^{*}\) because \(u'' <0\). Equations (6.1) and (6.2) imply \(\lambda _1 = \varphi [ u'(q^*)  (1l) ]\), which is consistent with \(\lambda _1>0\). Equation (6.4) implies \(d=m\). Then, from Eq. (6.7), we get \(b = \big [q^{*} \varphi m  l(1\delta +\alpha _\mathrm{_S} \lambda \delta )a\big ]/(1l)\), i.e., \(b=\big [q^*\pi (m,a)\big ]/(1l)\). To sum up, the solution is \(q = q^{*}\), \(d= m\), and \(b=[q^*\pi (m,a)]/(1l)\). Since here \(0 < b < a\), we have \(q^*(1l)a<\pi (m,a)< q^*\).

Case 6: \(\lambda _1 > 0\), \(\lambda _2 = 0\), \(\lambda _3 > 0\) (\(d = m\), \(0 < b = a\))
Equations (6.1) and (6.3) imply \(u'(q) > 1\). Equations (6.1) and (6.2) imply \(u'(q) > 1  l\). Therefore, the solution of q must satisfy \(q < q^{*}\). Equation (6.4) implies \(d=m\). Equation (6.6) implies \(b=a\). Then, from Eq. (6.7), we get \(q = \varphi m + [1l+l(1\delta +\alpha _\mathrm{_S} \lambda \delta )] a\), i.e., \(q=\pi (m,a)+(1l)a\). To sum up, in this case the solution is \(q = \pi (m,a) + (1l)a\), \(d= m\), and \(b = a\). Notice that since here \(q<q^*\), we have \(\pi (m,a) < q^*(1l)a\).

Case 7: \(\lambda _1 > 0\), \(\lambda _2 > 0\), \(\lambda _3 = 0\) (\(d = m\), \(0 = b < a\))
Equations (6.1) and (6.3) imply \(u'(q) < 1\). Equations (6.1) and (6.2) imply \(u'(q) > 1  l\). Thus, \(1l < u'(q) < 1\), which means that \(q^{*} < q < q^{**}\). Equation (6.4) implies \(d=m\). Equation (6.5) implies \(b=0\). Then, from Eq. (6.7), we get \(q = \varphi m + l(1\delta +\alpha _\mathrm{_S} \lambda \delta ) a\), i.e., \(q=\pi (m,a)\). To sum up, in this case the solution is \(q= \pi (m,a)\), \(d = m\), and \(b = 0\). Since here \(q^*< q < q^{**}\), we have \(q^* < \pi (m,a) < q^{**}\).

Case 8: \(\lambda _1 > 0\), \(\lambda _2 > 0\), \(\lambda _3 > 0\) (\(d = m\), \(0 = b = a\))
Equations (6.1) and (6.2) imply \(u'(q) > 1  l\). Thus, \(q < q^{**}\). Equation (6.4) implies \(d=m\). Equation (6.5) implies \(b=0\). Then, from Eq. (6.7), we get \(q = \varphi m\). Summing up, we have \(q = \varphi m\), \(d= m\), and \(b = 0\). Since here \(q < q^{**}\), we have \(q^{**} > \pi (m,0)\).
\(\square \)
Proof of Lemma 3.3
As a first step, we highlight some important properties of \(J: \mathbf {R}^2_+ \rightarrow \mathbf {R}\). First, this function is continuous throughout its domain. To see why this is the case, recall that the solution to the LW market bargaining problem is continuous, and so is u. Hence, J is also continuous.
Second, the function J is differentiable within each of the four regions defined in Lemma 3.2. To see this point, notice that the solution to the bargaining problem is differentiable within each of the four regions defined in Lemma 3.2, and u is also (twice continuously) differentiable. Therefore, J is differentiable within each of the four regions.
Third, J is weakly concave in both arguments everywhere. To see this point, recall that J is continuous everywhere and differentiable within each region, thus \(J_1\) and \(J_2\) are well defined within each region. Moreover, \(J_1\) and \(J_2\) are constant in Regions 1 and 3, and strictly decreasing in Regions 2 and 4 in \(\hat{m}\) and \(\hat{a}\) because \(u''<0\). Thus, J is weakly concave everywhere.
Given the properties of J, we have the following:

(a)
Since J is weakly concave and differentiable (with the exception of the boundary points), the optimal choice (\(\hat{m}\), \(\hat{a}\)) in each region satisfies \(\nabla J(\hat{m},\hat{a})=\mathbf {0}\).

(b)
If the asset price is equal to the fundamental value, i.e., \(\psi =\beta (1l)\), Regions 2, 3, and 4 are ruled out, because \(\psi = \beta (1l)\) implies that \(J_2^1=0\), while \(J_2^2\), \(J_2^3\) and \(J_2^4 > 0\) for any bundle (\(\hat{m}\), \(\hat{a}\)). There is no reason to choose less \(\hat{a}\) than \(q^{**}/x\) in this case. On the other hand, the fact that \(J_1^1 < 0\) in the case where \(\varphi > \beta \hat{\varphi } (1l)\), i.e., the cost of holding money is positive means that \(\hat{m}=0\) because a buyer can purchase the amount \(q^{**}\) without using any money. Hence, any bundle (\(\hat{m}\), \(\hat{a}\)) with \(\hat{m}=0\) and \(\hat{a}\ge q^{**}/x\) is optimal.

(c)
When \(\psi > \beta (1l)\), any choice within Region 1 is ruled out because \(J^1_2 < 0\). In Regions 2 and 4, \(J_1\) and \(J_2\) are strictly decreasing in \(\hat{m}\) and \(\hat{a}\), therefore the optimal choice (\(\hat{m}\), \(\hat{a}\)) satisfying \(J_1=0\) and \(J_2=0\) is unique (given that \(\varphi > \beta \hat{\varphi } (1l)\) and \(\psi > \beta (1l)\)). In Region 3, since any bundle (\(\hat{m}\), \(\hat{a}\)) satisfies \(J^3_1 = 0\) and \(J^3_2 = 0\), the optimal choice is indeterminate.
\(\square \)
Proof of Lemma 4.2
From Lemma 3.1, it is straightforward to show that \(\chi \) and c exist and are unique, given A, in the equilibrium. From Lemma 3.2 and 3.3, we have the following equations on each region in the steady state where \(\varphi /\hat{\varphi }=1+\mu \).

\(1+\mu =\beta (1l)\), \(\psi =\beta (1l)\), \(q=q^{**}\) and \(b=0\), in Region 1,

\(1+\mu =\beta u'(\tilde{q}_2(z))\), \(\psi =\beta \big [u'(\tilde{q}_2(z))x+1l]\) and \(b=0\), in Region 2,

\(1+\mu =\beta \), \(\psi =\beta (1l+x)\), \(q=q^{*}\), and \(b=\tilde{b}(z)\), in Region 3,

\(1+\mu =\beta u'(\tilde{q}_4(z))\), \(\psi =\beta u'(\tilde{q}_4(z)) (1l+x)\) and \(b=A\), in Region 4.
Clearly, \(\psi \), z, q, and b exist and are unique in Region 1. In Regions 2 and 4, \(\mu \) uniquely pins down z, q, and \(\psi \). In addition, b is unique in all regions. Hence, equilibrium exists and is unique in Regions 2 and 4. In Region 3, z can obtain any value in the interval of \([q^*(1l+x)A, q^*xA]\), even though \(\psi \) and q are unique. It follows that b is also indeterminate because it is a function of z. Hence, they all exist, but z and b are not unique in Region 3. \(\square \)
Proof
Derivation of expressions \(\bar{\mu }_i(A)\), \(i=2,4\), in Sect. 4.2.
Consider Region 2. For any given A, the term \(\bar{\mu }_2\) indicates the level of inflation that would lead to the same q as in the case where \(z=0\). The latter is simply given by \(q_{2,n}=xA\) (the second index, n, stands for “nonmonetary” equilibrium). To identify the relationship between \(\mu \) and q in an equilibrium where money is valued, set the derivative in (3.11) equal to zero, and evaluate it in steady state. This yields \(1+\mu =\beta u'(q_{2,m}),\) where \(q_{2,m}=z+xA\) (in the same spirit as above, the second index, m, stands for “monetary” equilibrium). Hence, \(\bar{\mu }_2\) solves \(\bar{\mu }_2 = \{\mu : q_{2,m}=q_{2,n}\}\), which implies that \(\bar{\mu }_2=\beta u'(xA)1\). The derivation of \(\bar{\mu }_4\) follows identical steps. \(\square \)
Proof of Proposition 4.5
Consider first equilibria in Region 4. From the definition of steady state equilibrium, we know that in this region \(b=A\). Also, from Proposition 4.3, we know that \(\psi =\psi _4(\mu )\). Hence, in this region the haircut is given by \({\varXi }=11/\psi _4\). Consequently, we have
and since we have established that \(\psi _4\) is increasing in both \(\mu \) and \(\alpha _\mathrm{_S}\), the same is true about \({\varXi }\).
In Region 3, it is impossible to obtain an expression for \(\partial {\varXi }/\partial \mu \), since being in this region requires \(\mu =\beta 1\). However, we can study how \({\varXi }\) depends on \(\alpha _\mathrm{_S}\). To that end, recall from the definition of equilibrium that in Region 3 we have \(b=\tilde{b}(z)=(1l)^{1}(q^*zxA)\). Also, from the discussion that follows Proposition 4.3, recall that the asset price is given by \(\psi =\psi _4(\beta 1)=\beta (1l+x)\). Hence, in Region 3 the haircut is given by
It is now straightforward to verify that
which is clearly strictly positive.\(\square \)
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Geromichalos, A., Lee, J., Lee, S. et al. Overthecounter trade and the value of assets as collateral. Econ Theory 62, 443–475 (2016). https://doi.org/10.1007/s0019901509049
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DOI: https://doi.org/10.1007/s0019901509049