A Search Model of Centralized and Decentralized Trade Ch 2

Proposition 9 For any parameter values (k, r, ρ0, θ, fB , fS ) satisfying the assumption in Proposition 7, ther e exist values ρ > 0 and ρ > 0 such that if ρ ³ ´ converges up (below) to ρ ρ from ρ0, then the centralized (decentralized) market is driven out of the economy. If ρ converges up further to infinity, then the limiting equilibrium becomes Walrasian. Propositions 8-9 establishes that if either the time cost of search or the contact cost of search is small enough, then all traders prefer to trade in the decentralized market and the centralized market is driven out of the economy. When either type of search cost vanishes, the limiting equilibrium becomes Walrasian. On the other hand, if search cost is large enough, then there is no gain from trading in the decentralized market and all traders prefer to go to the centralized market. Proposition 10 For any parameter values (k0, r, ρ, θ, fB , fS ) satisfying the assumption in Proposition 7, there exist values k > 0 and k > 0 such that if k converges below (up) to k (k) from k0, then the decentralized (centralized) market is driven out of the economy. If k converges below further to zero, then the limiting equilibrium becomes Walrasian. Proposition 10 establishes that if the transaction cost of market makers is small enough, or if the bid-ask spread is small enough, then all traders prefer to trade in the centralized market and the decentralized market is driven out of the economy. Moreover, when the bid ask spread converges to zero, the limiting equilibrium converges to the Walrasian equilibrium. On the other hand, when the bid-ask spread is large enough, no traders go to the centralized market and it is driven out of the economy. I now study comparative statics with respect to parameters (k, r, ρ, θ) . I will focus particularly on trading volume fB (1 − vm) in the centralized market and the expected price in the decentralized market. 12 One can interpret trading volume as a measure of liquidity or a measure of the market participation. Proposition 11 Let the assumption in Proposition 7 hold. Then the expected negotiated price in the decentralized market, EG [p (v)] , is increasing with k and decreasing with θ. Proposition 11 shows that the expected negotiated price in the decentralized market is positively related to the transaction cost in the centralized market. That is, it is negatively related to the efficiency in the centralized market. 12 It follows from (22) that the trading volume in the decentralized market is given by the difference between the inflows of traders and the trading volume in the centralized market. This result follows directly from Proposition 5 since the competitive bid-ask spread is equal to the transaction cost. Proposition 11 also shows that similar to Proposition 2, the average negotiated price is negatively related to buyers’ bargaining power. However, unlike Proposition 2, under competitive market-making, it does not depend on the contact rate ρ and the discount rate r. This is because there are two opposing effects in force. An increase in the contact rate raises a buyer’s payoff from trading in the decentralized market, and hence lowers the negotiated price (see (1)). However, it also discourages high valuation buyers from entering the centralized market, leaving more high valuation buyers trading in the decentralized market. This raises the average negotiated price. Proposition 11 implies that these two effects offset each other. A similar analysis applies to r. The following proposition establishes properties of the trading volume in the centralized market. Proposition 12 Let the assumption in Proposition 7 hold. Then the trading volume in the centralized market, fB (1 − vm ) , is decreasing with k, ρ, and increasing in r. It is increasing with θ if kρ (1 − θ)2 < 2Rr. Proposition 12 implies that trading volume in the centralized market is negatively related to the bid-ask spread (or transaction cost k), and positively related to the search cost reflected by the contact rate ρ and the discount rate r in the decentralized market. This result is intuitive and simply says that trading volume should reflect the relative competitive position of the two markets. One may expect that if the bargaining power of the buyers θ increases, then buyers can negotiate low prices as described in Proposition 11. As a result, they should find the decentralized market more attractive and there should be less trade in the centralized market. Surprisingly, Proposition 12 implies that in general there is no monotonic relation between trading volume in the centralized market and the bargaining power of the buyers θ. The intuition is as follows. In the short run, increasing the buyers’ relative bargaining power θ raises a buyer’s payoff and lowers a seller’s payoff when they trade in the decentralized market. However, as shown earlier, a seller’s steady state payoff must be equal to R, which does not depend on the bargaining power. To maintain this payoff, a seller must meet a buyer more often. That is, the proportion of buyers α must be higher as implied by equation (17). This imposes a negative externality to the buyers since it implies that the decentralized market becomes tighter. Thus, if this negative effect dominates the preceding positive effect, then the buyer’s payoff from trading in the decentralized market actually falls. This leads more buyers to entering the centralized market. This case happens if the decentralized market is relatively less efficient than the centralized market in the sense that r is high or ρ is small, or k is small, as required by the assumption. 3.6 Monopolistic Market Maker I now analyze the case with a monopolistic market maker. A search equilibrium with a monopolistic market maker is defined as the equilibrium described in Section 3.3 for which the ask price is selected by the market maker so as to maximize profits. Formally, since I focus on steady states, the ask price is determined by the following static problem max (a − b − k) fB (1 − vm ) , a (30) subject to (23), where b = R and vm is given in (25). The monopolistic market maker faces the following trade-off: Increasing the ask price raises the profits from a transaction. But it lowers the number of transactions since some buyers may find the decentralized market is more attractive. Since it follows from (25) that vm is a linear function of a, the profit function in (30) is quadratic. The following proposition establishes the existence and uniqueness of search equilibrium and characterizes the equilibrium ask price. Proposition 13 If the expression given in (31) satisfies condition (23), then there is a unique nondegenerate search equilibrium with a monopolistic market maker. The equilibrium ask price is given by a= r + 2Rθρ k Rr (1 + θ) + , + 2 2 (r + θρ) 2 (r + θρ) (1 − θ) (31) which is higher than the competitive ask price R + k. Proposition 13 demonstrates that the bid-ask spread is positively related to the transaction cost k of the market maker. The intuition is similar to that for the case with competitive market makers: A monopolistic market maker makes positive profits, and hence the bid-ask spread must cover the transaction cost. Importantly, (31) also reveals that there exists a positive bid-ask spread even though there is no transaction cost in the centralized market. This result is in line with Proposition 6. Proposition 13 also implies that the monopolistic market maker sets a wider bid-ask spread than competitive market makers. Consequently, it follows from Propositions 4 and 6 that the average negotiated price in the decentralized market is higher, and trading volume in the centralized market is lower. This result is intuitive. The high ask price set by the monopolistic market maker discourages some high valuation buyers from entering the centralized market and lowers its trading volume. These high valuation buyers raise the average negotiated price in the decentralized market. Under monopolistic market-making, one can use Propositions 4 and 13 to establish similar limiting results to Propositions 8-10. The proof is also similar. Hence, I omit a formal statement of the results and the proof. The general intuition should be clear: If one market is sufficiently efficient relative to the other market, it should attract all traders and drives out the other market from the economy. However, unlike competitive market-making, when the transaction cost k goes to zero, the decentralized market does not disappear under monopolistic market-making. This is because there still exists monopoly inefficiency in the centralized market. The following proposition establishes comparative static properties. Proposition 14 Let the assumption in Proposition 13 hold. Then: (i) the bid-ask spread a − b, increases with r, decreases with ρ, and increases with θ if 2R (r + θ2 ρ) > (1 − θ)2 ρ (1 − R); (ii) the expected negotiated price in the decentralized market increases with k and r, decreases with ρ, and decreases with θ if r2 + rρ (1 − R) + k (r + ρθ)2 > 3r2 R; and, (iii) the trading volume in the centralized market fB (1 − vm ) increases with r, decreases with k and ρ, and increases with θ if kρ (1 − θ)2 < 2Rr. Part (i) of Proposition 14 reveals that the bid-ask spread is positively related to the search frictions in the decentralized market represented by r and ρ. This is because when r is decreased or ρ is increased (search cost is lower), buyers get higher payoffs from trading in the decentralized market. To attract buyers to trade in the centralized market, the market maker must lower the bid-ask spread. Importantly, part (i) also shows that there is no general monotonic relation between the bid-ask spread and the bargaining power. The intuition is similar to that described after Proposition 11: An increase in θ raises a buyer’s payoff from trading in the decentralized market, and hence encourages him to trade in that market. On the other hand, more searching buyers make it more difficult for a buyer to find a seller, and thus lowers the buyer’s payoff. The overall effect on a buyer’s payoff is ambiguous, and hence the effect on a buyer’s participation decision is also ambiguous. Consequently, the impact on bid-ask spread is also ambiguous since the market maker adjusts the bid-ask spread so as to attract traders to trade in the centralized market. Part (ii) of Proposition 14 shows that the expected negotiated price in the decentralized market is positively related to k, as in the case of competitive market makers described in Proposition 11. However, unlike Proposition 11, the expected negotiated price is also negatively related to the contact rate ρ and positively related to the discount rate r. This is because there are two opposing effects in force as discussed after Proposition 11. Proposition 14 demonstrates the relative efficiency effect dominates. Finally, part (ii) also implies that there is no monotonic relation between the average negotiated price and the buyer’s bargaining power. The intuition is similar to that described above. Finally, part (iii) of Proposition 14 shows that the trading volume in the centralized market is negatively related to k and ρ, and positively related to r. Moreover, there is no monotonic relationship with θ. This result is similar to that for competitive market makers shown in Proposition 12. The intuition is also similar: Trading volume or liquidity should reflect the relative efficiency of the two markets. 4 Welfare An important question is what is the welfare implication of the market fragmentation. To answer this question, one has to adopt a welfare criterion. Since the Walrasian equilibrium of the benchmark frictionless economy is in the flow sense of Gale (1987), I also adopt the flow sense welfare criterion. The social welfare Ww in the Walrasian equilibrium is given by the total buyer and seller surplus: Z1 Ww = fB (1 − R) (v − R) R fS (1 + R) dv + fS R = . 1−R 2 (32) This is the first-best value. The social welfare Wc in the pure centralized market equilibrium described at the beginning of Section 3 is given by the total buyer and seller surplus plus market maker profits: Z1 Wc = fB (1 − a) µ = fS (v − a) a dv + fS (a − b − k) + fS b 1−a ¶ 1+R −k , 2 (33) where I use the fact that the market maker sets the ask price a = R. Note that this welfare criterion does not depend on the market structure in the centralized market. It is clearly less than the first best value due to the loss of transaction costs. Consider next the social welfare in the search equilibrium without market makers described in Section 2. It is given by  1 Z1  Z dv dv  + fS R = fS VB (v) + R .(34) VB (v) 1−R 1−R Wd = (1 − R) fB R R Clearly, Wd < Ww due to search frictions. The following proposition compares this welfare with that in the pure centralized market. , then Wc > (<) Wd . Proposition 15 If k < (>) 21 r 3Rθ−R+1−θ (1−θ)(r+ρθ) The intuition behind Proposition 15 is as follows. From (33) and (34), one can see that search frictions reduce buyer surplus in the pure decentralized market economy, but transactions costs reduce social welfare in the pure centralized market economy. Thus, which one is bigger depends on the relative efficiency in the two markets described by the parameters k, ρ, and r. Turn to the equilibrium where centralized and decentralized markets coexist. In this equilibrium, the social welfare Wf is equal to the buyer and seller surplus from trading in the centralized and decentralized markets plus market maker profits: Zvm Wf = (vm − R) fB R 1 Z dv dv + (1 − vm ) fB (v − a) UB (v) vm − R 1 − vm v m +fS US + (1 − vm ) fB (a − b − k) v Zm  = fS UB (v) R 1  Z dv 1 − vm  dv + (v − R) +R− k , 1−R v 1−R 1−R (35) m where I have used (21)-(22) and the fact that US = R = b to derive the equality. It is ready to analyze the question as to whether market fragmentation improves social welfare. The following intuition is natural: The introduction of a centralized market to the decentralized market facilitates immediacy of trade and hence should improve social welfare. On the other hand, the fragmentation of a centralized market should lower social welfare since there are search frictions in the decentralized market. However, I will show that both claims are generally not true. Consider first the question of whether the opening of a centralized market improves the social welfare in the pure decentralized market economy. Rewrite (34) as v Zm Wd = fS  VB (v) R 1  Z dv dv + VB (v) + R . 1−R v 1−R (36) m It is clear from (35) and the fact VB (v) < v −R for v ≥ vm that high valuation buyers v ≥ vm benefit from immediacy of trade. However, the centralized market imposes negative externality on the decentralized market in the sense that it makes the latter market tighter (see Proposition 5). Thus, buyers in the decentralized market are worse off, i.e., VB (v) > UB (v) for v ∈ [R, vm ]. Moreover, market making incurs transaction costs. Consequently, the social welfare in the economy where the two markets coexist may not be higher than that in the pure decentralized market economy. To illustrate this point, I consider some simple numerical examples. Set parameter values as in Table 1. Suppose market makers are competitive and denote by Wcm the associated social welfare. For the base case parameter values in Table 1, I find that Wd = 58.82 < Wcm = 59.07. That is, the presence of a competitive centralized market improves social welfare. However, if I set ρ = 20, I find Wd = 59.41 > Wcm = 59.38. That is, if the decentralized market is relatively efficient enough, then the opening of a competitive centralized market may lower social welfare. [Insert Table 1 Here] Consider next the question of whether the fragmentation of a centralized market improves social welfare in the pure centralized market economy. Rewrite Wc as v Zm  Wc = fS (v − R) R 1  Z dv dv + (v − R) + R − k . 1−R v 1−R (37) m Compared with (35), it is clear that the centralized market provides immediacy of trade and hence improve the welfare of traders with valuation v ∈ [R, vm ] as v − R > UB (v). However, since in a completely centralized market all traders trade with market makers, the total transaction costs fS k is bigger than that in the two markets economy fS λS k. The following proposition shows that if the bid-ask spread after the market fragmentation is small enough, then the second effect dominates. In particular, under competitive market making, the fragmentation of centralized markets always improves social welfare. Proposition 16 If the bid-ask spread (a − R) in the two markets economy satisfies (a − R) < (>) 2k, then Wf > (<) Wc . In particular, under competitive market-making, Wcm > Wc . I finally address the following questions: What is the constrained socially opti- mal bid-ask spread? How do the equilibria under competitive and monopolistic market-making compare with the constrained social optimum? Again the bid price must be equal to R such that sellers are indifferent between trading in the two markets. Thus, the social planner chooses the ask price a only so as to maximize Wf given in (35). This is equivalent to choosing the cutoff value vm since it follows from (25) that vm is positively related to a. ∗ Proposition 17 The socially optimal cutoff value vm satisfies ∗ Zvm ∗ ∗ vm − R − k = UB (vm )+ R ∂UB (v) dv. ∂vm (38) This value is bigger than the cutoff value under competitive market-making. Furthermore, the socially optimal bid-ask spread is wider than that under competitive market-making. This proposition implies that under competitive market-making, the bid-ask spread is too narrow and there is too much entry into the centralized market, compared with the constrained social optimum. The main reason is that the social planner internalizes the externality of the bid-ask prices on the decentralized market. This externality effect is captured by the second term on the right-hand side of (38). Specifically, when vm or a changes, it follows from (24) that the matching probability α changes. Thus, it follows from (19) that the buyer value function UB (v) also changes. By contrast, competitive market makers completely ignore this externality effect when setting the ask price a = R + k. I now consider the welfare implication of monopolistic market-making. When the monopolistic market maker chooses the profit maximizing ask price (31), he or she takes into account the impact of the bid-ask prices on the traders’ participation decisions. That is, unlike competitive market makers, the monopolistic market maker takes into account the externality effect, even though this externality is not fully internalized. Thus, one should expect that monopolistic market-making may improve social welfare. I now use a simple numerical example to illustrate this point. According to the parameter values in Table 1, the outcome of the constrained social optimum is described in Row 2 of Table 2. In particular, the constrained efficient social welfare is equal to 59.12, the bid-ask spread is equal to 0.0139, and trading volume in the centralized market is equal to 61.38. Rows 3-4 of Table 2 reveal that under monopolistic market-making, the social welfare is closer to the constrained social optimum. Moreover, monopolistic market-making implies a too wide bid-ask spread and two little trading volume in the centralized market. [Insert Table 2 Here] The above analysis has important policy implications. First, there can be welfare gain from increasing the bid-ask spread in a perfectly competitive centralized market. This suggests that taxing transactions in the competitive centralized market might improve social welfare. Another policy that might improve social welfare might be to give some monopoly power to market makers. This might explain why the specialists system on the New York Stock Exchange could be socially useful. 13 5 Conclusion This paper provides a search model of centralized and decentralized trade. The model has a number of testable implications. As mentioned in the introduction, some are consistent with empirical and experimental evidence. The main results of the paper can be summarized as follows. First, a positive bid-ask spread exists if the centralized and decentralized markets coexist in equilibrium. Moreover, the average negotiated price in the decentralized market is inside the bid-ask spread. Second, under monopolistic market-making, the bid-ask spread is positively related to the transaction cost, search frictions, and average negotiated price. Third, liquidity in the centralized market measured by trading volume is negatively related to the bid-ask spread and positively related to search frictions. Fourth, several limiting results and convergence to the Walrasian equilibrium are established. Finally, perhaps the most important and surprising result is about the welfare implications. Specifically, I show that the fragmentation of a centralized market improves social welfare if the bid-ask spread after the fragmentation is small enough. However, the opening of a centralized market in a decentralized market economy may not improve social welfare. More interestingly, compared to the competitive market-making, monopolistic market-making may improve social welfare, because it partially internalizes the externalities of bid-ask prices on the decentralized market. The model is highly stylized and may not describe perfectly any specific market in reality. It intends to capture in a simple manner some crucial elements of trades in the centralized and decentralized markets. As a result, the model can be extended in a number of dimensions. 13 In a model with asymmetric information, Glosten (1989) argues that the specialists system on the NYSE may improve social. The intuition is that the monopolistic specialist can set prices that on average maximize profits. Thus, the market breakdown problem as in the market-for-lemons problem does not arise. First, since the model is stylized and it is difficult to gather decentralized trade data, an experimental study is helpful for testing the model. Related experimental studies have been carried out by Campbell et al. (1991), Lamoureux and Schnitzlein (1997), and Neeman and Vulkan (2003b). Apparently, further work along this line is interesting. Second, in the model, the only benefit of the centralized market is its publicity of prices and immediacy of trade. Centralized markets have other important advantages such as economies of scale and network externalities. A simple way to capture these advantages is to assume that the transaction cost k decreases with the volume of trade. Finally, in order to keep the model tractable and to derive analytical results, I assume that sellers are homogenous, which seems reasonable when a single homogenous asset is traded. It would be interesting to consider the case where sellers are heterogeneous. For example, they may have different costs of holding the asset. A Appendix: Proofs Proof of Proposition 1: See the main text. Q.E.D. Proof of Proposition 2: Substituting the expression for γ in (13) into (11) yields p (v) = R + Simple algebra delivers ∂p(v) ∂ρ r (3θR − R + 1 − θ) (v − R) . (r + ρθ) (1 − R) < 0, and ∂p (v) θρ (3Rθ − R + 1 − θ) (v − R) , = ∂r (r + ρθ)2 (1 − R) ∂p (v) r + ρ − ρR − 3Rr (v − R) . = −r ∂θ (r + ρθ)2 (1 − R) By assumption (14), one can verify that ∂p(v) ∂r > 0 and ∂p(v) ∂θ < 0. Q.E.D. Proof of Proposition 3: Substituting (13) into (11) and letting r converges to 0, or letting ρ converges to infinity, one obtains that the limiting price is given by R for all v. Thus, all traders in the market trade at the Walrasian price. Thus, the limiting equilibrium is Walrasian. Q.E.D. Proof of Proposition 4: To derive the equilibrium, observe first that the bid price b = R as discussed in the main text. Next, substituting US = R and (19) into (15) yields the bargained price between buyer v and a seller given in (26). I now solve for the matching probability α and the cutoff value vm . To this end, I turn to the seller’s problem (17). Because of the cutoff nature of buyers’ choice, the distribution of buyers in the decentralized market G is uniform over [R, vm ] and its density is vmdv−R . Thus, substituting (26) into (17) yields Zvm rUS = ρα (1 − θ) [v − US − UB (v)] R dv . vm − R (A.1) Substituting US = R and UB given in (19) into the above equation yields ρα (1 − θ) R= r + ρ (1 − α) θ Zvm [v − R] R dv . vm − R (A.2) Use this equation to solve for vm , vm = R 2 (r + θρ (1 − α)) + ρα (1 − θ) . ρα (1 − θ) (A.3) On the other hand, use equation (20) to derive vm = 1 [(r + ρθ (1 − α)) a − ρθ (1 − α) R] . r (A.4) Equate the right-hand sides of equations (A.4) and (A.3) to deliver a quadratic equation for α. Solving this equation yields two roots. One root is given by r+ρθ , which must be ruled out since it is bigger than 1. The other root is given ρθ by (24). Substituting this root into (A.3) yields (25). Finally, once vm and α are obtained, one can use (21)-(22) to derive (NB , NS , λS ), and an equilibrium is constructed. Q.E.D. Proof of Proposition 5: In a nondegenerate search equilibrium with market makers quoting bid-ask prices (a, b) , the cutoff value satisfies vm ∈ (R, 1) . From (12) and (A.2), it is straightforward to verify α > γ. Q.E.D. Proof of Proposition 6: Since a buyer with valuation v in the support of G, [R, vm ], prefers to buy the asset at price p (v) , at some random time τb , rather than to buy the asset immediately at the price a in the centralized market, h i v − a ≤ E e−rτb (v − p (v)) . Since E [e−rτb ] < 1, it follows that p (v) < a for all v ∈ [R, vm ]. Taking expectation with respect to G yields a > EG [p (v)] . Similarly, one can show that EG [p (v)] > b. 14 Finally, substituting (24) into (26) and taking expectation, one can easily derive (27). Q.E.D. Proof of Proposition 7: See the main text. Q.E.D. Proof of Proposition 8: Taking other parameter values as given, view vm as a function of r. By assumption there is a value r0 > 0 such that vm (r0 ) ∈ (R, 1) . One can also show that vm (r) → ∞ as r → 0. Thus, there is a positive solution to the quadratic equation vm (r) = 1. Take r as the maximum solution. When r converges to r, vm converges to 1 and hence no buyers go to the centralized market. The model then reduces to that in section 2. By Proposition 2, the economy becomes Walrasian if r converges below to 0. 14 I thank an anonymous referee for suggesting this simpler proof. Now consider increasing r from r0 . By (28) and (29), when r is sufficiently large, either α will exceed 1 or vm will decrease below R. For both cases, all buyers prefer to go to the centralized market and the decentralized market disappears. Q.E.D. Proof of Proposition 9: The proof is similar to that of Proposition 8, and is omitted. Q.E.D. Proof of Proposition 10: By (28) and (29), when k is sufficiently small, either α tends to 1, or vm tends to R. Thus, all buyers enter the centralized market and there is no trade in the decentralized market. When k converges below further to zero, then there is no bid-ask spread; that is, a = b. Moreover, only buyers with valuation v ≥ a enter the centralized market. Thus, to maintain a steady state, the following flow condition must be satisfied fB (1 − a) = NB = NS = fS . (A.5) This implies that a = R and the limiting equilibrium becomes Walrasian. Finally, it follows from (29) that vm increases with k. Thus, there exists a value k > 0 such that vm converges to 1 when k tends to k. This implies that if k is large enough, then no traders go to the centralized market, and hence the centralized market disappears. Q.E.D. Proof of Proposition 11: Since a = R + k, it follows from Proposition 6 that EG [p (v)] = R + k (1 − θ) . 2 (A.6) Thus, EG [p (v)] is increasing with k and decreasing with θ. Q.E.D. Proof of Proposition 12: It follows from straightforward differentiation of vm given in (29). Q.E.D. Proof of Proposition 13: Substituting (25) into (30) yields a quadratic function of a. The first-order condition gives the maximizer given in (31). I denote it by am . If it satisfies condition (23), then by Proposition 4 there is a unique search equilibrium with a monopolistic market maker quoting the ask price given in (31). To show that this ask price is bigger than R + k, observe that there are two roots, denoted by a1 and a2 , for the profit function (30). The maximizer am is between these two roots. Note that one root a1 is such that vm = 1, and the other root a2 = R + k. The maximizer am must satisfies a2 < am < a1 . Otherwise, suppose a1 < am < a2 . Since (25) implies that vm increases with a, the value of vm at am must exceed 1. This leads to a contradiction. Q.E.D. Proof of Proposition 14: Let a be the expression given in (31). For part (i), simple algebra implies that ∂a (3Rθ − θ − R + 1) θρ = , ∂r 2 (r + θρ)2 (1 − θ) ∂a (3Rθ − θ − R + 1) rθ =− , ∂ρ 2 (r + θρ)2 (1 − θ) ³ ´ 2 2 ∂a r 2R (r + θ ρ) − (1 − θ) ρ (1 − R) = . ∂θ 2 (r + θρ)2 (1 − θ)2 By assumption, r r+ρθ + Rθ(2r+ρ(1−θ)) (1−θ)(r+ρθ) − a > 0. Thus, Rθ (2r + ρ (1 − θ)) r r + 2Rθρ Rr (1 + θ) + − − r + ρθ (1 − θ) (r + ρθ) 2 (r + θρ) 2 (r + θρ) (1 − θ) 1 = (r + θρ)−1 (1 − θ)−1 (3Rθ − θ − R + 1) r > 0. 2 The desired result then follows. Consider part (ii). By (27), the expected negotiated price EG [v (p)] increases with the ask price a. The desired comparative statics result for k, r, and ρ follows from part (i). Finally, the comparative statics result for θ follows from the assumption and the following equation, ∂ (a − R) (1 − θ) r2 (1 − 3R) + rρ (1 − R) + k (r + ρθ)2 =− . ∂θ 2 (r + ρθ)2 Finally, consider part (iii). Substituting the expression given in (31) into (25) yields an expression for vm . Differentiating this expression yields: ∂vm k ∂vm = −r−2 ρθk < 0, = θ > 0, ∂r ∂ρ 2r ∂vm 1 ∂vm kρ (1 − θ)2 − 2Rr . = (r + θρ) > 0, = ∂k 2r ∂θ r (1 − θ)2 The desired result then follows from the assumption. Proof of Proposition 15: Substituting (10) and (13) into (33)-(34), and simplifying yield µ Wc − Wd = fS  1 ¶  Z dv 1−R + R − k − fS  VB (v) + R 2 1−R R 1 3Rθ − R + 1 − θ = r − k. 2 (1 − θ) (r + ρθ) The desired result follows from the assumption. Q.E.D. Proof of Proposition 16: Use (35) and (33) to derive v Zm  Wf − Wc = fS UB (v) R v Zm −fS  (v − R) R = fS k  1 Z dv 1 − vm  dv + (v − R) +R− k 1−R v 1−R 1−R m  1 Z dv dv + (v − R) + R − k 1−R v 1−R m vm − R − fS 1−R Ã Zvm (v − R − UB (v)) R dv 1−R r vm − R vm − R fS k − = 1−R r + θρ (1 − α) 2 vm − R fS [k − (a − R) /2] . = 1−R ! where the third equality follows from substitution of (19), and the last equality follows from substitution of (24) and (25). Since for competitive market makers a − R = k, the desired result follows from the assumption. Q.E.D. Proof of Proposition 17: Taking first-order condition for (35) yields (38). Using (19), (24), and (25), one can easily show that ∂U∂vBm(v) > 0. Let the ∗ ∗ constrained social optimal ask price be a∗ . By (20), vm − a∗ = UB (vm ) . Thus, one can rewrite (38) as ∗ Zvm ∗ a −R−k = R ∂UB (v) dv > 0. ∂vm It follows that a∗ > R + k, the ask price under competitive market-making. Moreover, since the cutoff value vm increases with a by (25), one obtains the desired comparison result for the cutoff value. Q.E.D. Table 1. Parameter Values fB fS r k ρ θ 125 100 0.1 0.01 10 0.5 Table 2. Comparison of Social Welfare. Row 2 lists equilibrium outcome for the constrained social optimum. Row 3 lists equilibrium outcome under competitive market-making. Row 4 lists equilibrium outcome under monopolistic market-making. The parameter values are given in Table 1. a−b EG [p (v)] fB (1 − vm ) Welfare Social Optimum 0.0139 0.2035 61.38 59.12 Monopolistic MM 0.0167 0.2042 43.16 59.09 Competitive MM 0.0100 0.2025 86.25 59.07