(preliminary and incomplete)

(tentative title) Speed Acquisition∗

Shiyang Huang† Bart Zhou Yueshen‡

this version: November 17, 2016

∗ This paper tremendously benefits from discussions with Dion Bongaerts, James Brugler, Sarah Draus, John Kuong, Joël Peress, Neal Galpin, Naveen Gondhi, Vincent Gregoire, Bruce Grundy, Semyon Malamud, Lyndon Moore, Mark van Achter, and Zhong Zhuo. In addition, comments and feedback from participants at seminars hosted by The University of Melbourne and Rotterdam School of Management are greatly appreciated. There are no competing financial interests that might be perceived to influence the analysis, the discussion, and/or the results of this article. † Huang is at School of Economics and Finance, The University of Hong Kong; [email protected]; K.K. Leung Building, The University of Hong Kong, Pokfulam Road, Hong Kong. ‡ Yueshen is at INSEAD; [email protected]; INSEAD, 1 Ayer Rajah Avenue, Singapore 138676. (preliminary and incomplete)

(tentative title) Speed Acquisition

Abstract

Speed has become a signature of modern financial markets. This paper theoretically studies investors’ endogenous speed acquisition, alongside their investment in information. Investments in speed and in information are shown intrinsically linked via a novel tension between investors’ intra- and intertemporal competition. Due to this tension, advances in speed technology exert a non-monotone effect on investors’ information acquisition, aggregate price efficiency, as well as firms’ real investment decisions. On the flip side, advances in information technology might also hurt investors’ speed acquisition, slowing down the price discovery process. Other testable implications for market qualities are also discussed.

Keywords: speed, information, technology, price discovery, price efficiency JEL code: D40, D84, G12, G14

(There are no competing financial interests that might be perceived to influence the analysis, the discussion, and/or the results of this article.) 1 Introduction

Price discovery is a fundamental function of financial markets. It involves two steps: First, investors acquire information about the underlying security. Second, via trading, such information

is incorporated in price. The first step determines the amount of information that the price can

eventually reflect, i.e. the magnitude of price discovery. The second step determines how soon the

price reflects the acquired information, i.e. the speed of price discovery.

Speed is an indispensable dimension of the price discovery process. This is because trading takes time: Not all investors with information instantaneously gather together; neither do their trading orders. If the informed only slowly arrive in the market, the resulting price discovery process will also be slow. All else equal, the market is more efficient (in price) if the informed investors trade faster.1 To date, the literature has mainly emphasized information acquisition, i.e. the magnitude aspect

of price discovery, following the pioneering works by Grossman and Stiglitz (1980) and Verrecchia

(1982). This paper complements this canonical perspective by studying investors’ speed acquisition,

alongside their information acquisition.

Indeed, speed has risen as a focal point of modern financial securities markets. Trading companies buy real estate to co-locate with exchanges; they pay for extra wide cables to directly

“speak to” the servers; billions of dollars have been spent on infrastructures like transatlantic

fiber-optic cables and microwave towers. While such microstructure advancement has sped up trading, the decision process to implement a trading idea has been arguably slowed down by the tightening regulatory environment seen in recent years. The increased burden of bookkeeping and compliance, following regulations like Dodd-Frank Act and Volcker Rule, could have hindered, for example, the process for buy-side analysts to convince fund managers of new trading strategies.

1 Under the assumption of (semi-)strong efficient market (Fama, 1970), all information is incorporated in price instantaneously. This can be thought of as an extremely fast price discovery process, implicitly requiring all informed investors be present and trade at the same time to clear the market.

1 A set of questions arise together with these phenomena: How much speed technology should investors acquire? Is speed technology favored over information technology? How does speed acquisition affect investors’ demand for information, and vice versa? Most importantly, what are the implications for the overall quality of price discovery and market efficiency?

This paper develops a model to address these questions. The model considers an economy populated with atomless investors, who first endogenously invest in both speed and information technology and then trade a risky asset. The information technology allows an investor to receive a more precise private signal about the asset, improving his information rent. The speed technology allows him to trade ahead of his peers. The sooner he trades, the less price discovery has occurred and the more valuable is his private signal. These are the direct effects of the two technologies. To fix the idea, consider a hedge fund for example. The fund’s information acquisition involves investments in, for example, analyzing companies’ annual reports, sending analysts for firm visits, buying data feeds, or building up machines to perform quantitative analysis. Such expenditure will improve the precision of the fund’s private signal. The fund’s speed acquisition covers different aspects. It can spend on microstructure technology, like colocation or microwave, to speed up the implementation of every single trade by micro/milliseconds. It can also invest in technologies to streamline the compliance process, so that the overall execution of trading ideas can be expedited by hours or even days, ahead of its competitors.

The model reveals that the magnitude and the speed of price discovery are intrinsically connected via investors’ competition for information rent—the indirect effects. First, there is intratemporal competition: An investor acquires less (more) information when there are more (fewer) competitors trading at the same time with him. Second, there is a novel effect of intertemporal crowding out:

When fast investors in aggregate acquire more information, the price becomes more efficient in the short-run. This hurts slow investors as there is less information rent remaining. As a result, speed acquisition can crowd out information acquisition. By the same token, too much information in the short-run can reduce investors incentive to become fast, crowding out speed acquisition. The

2 equilibrium is found where investors optimally choose their technology, accounting for both the direct and indirect effects, together with the investment costs.

The model yields testable predictions on how technology shocks affect market quality. The key driver is the pair of countervailing effects, intratemporal competition v.s. intertemporal crowding out. For example, while an advancement in speed technology hastens the price discovery process, it non-monotonically affects the eventual magnitude: On the one hand, as more investors acquire speed and become fast, there is less intratemporal competition for the slow investors. Hence they acquire more information. On the other, because there are more fast investors, the improved short-run price efficiency intertemporally crowds out slow investors’ information acquisition.

The net effect on the eventual (long-run) magnitude of price discovery depends on the strength of these two effects. In particular, it is possible that the eventual magnitude of price discovery actually shrinks, even though the process speeds up in the short-run. The worrying result of faster price discovery at the cost of long-run efficiency echoes the empirical finding by Weller (2016), who shows that aggressive algorithmic trading deters information acquisition. Dugast and Foucault (2016) share a similar prediction based on time-consuming information acquisition, a feature differs with the current paper as in their model speed and information investments are mutually exclusive.

A model application is then developed to illustrate the intrinsic connection between speed and magnitude of price discovery, via the lens of a real-decision making firm. Specifically, the firm is trying to learn about an underlying factor—e.g., the economic condition—from the financial market to make real investment decisions. The novelty lies in that the firm can also time its investment:

While waiting is costly due to discounting, it allows the firm to learn more precisely about the factor. As the speed technology advances, the firm is more willing to invest early due to the faster price discovery process. However, because of the intertemporal crowding out effect, the magnitude of the long-run price discovery can be hurt by the speed technology. Consequently, while some firms rush to make investment decisions, those who wait might have to make less reliable decisions.

The overall real efficiency is thus non-monotonically affected by the speed technology.

3 Further predictions are derived on how speed and information technology affects differently other market quality metrics like return volatility, trading volume, market depth, and trading costs.

In particular, because of the new angle of speed, the model is able to separately examine the patterns of short-run market quality measures and the long-run ones.

This paper contributes to three strands of the literature. First, the vast literature on costly information acquisition largely focuses on the magnitude aspect of price discovery, following the seminal works by, for example, Grossman and Stiglitz (1980) and by Verrecchia (1982). Recent studies explore other dimensions. Peress (2004, 2011) studies the wealth effect on information acquisition. Van Nieuwerburgh and Veldkamp (2009, 2010) analyze how investors acquire different types of information. Goldstein and Yang (2015) explore the implication of information diversity. To compare, the aforementioned literature assumes that the market always clears with all investors at the same time. Introducing the speed technology, this paper allows the magnitude and the speed aspects of price discovery to meet and speak to each other.

Second, this paper lends support to the high-frequency trading literature with endogenous bundling of speed and information acquisition in equilibrium. Price discovery is a non-decreasing process in nature. As it accumulates over time, the remaining information rent diminishes. In- vestors’ incentive to acquire information, therefore, also increases with their speed: The model predicts that fast investors, who can trade sooner than their slow peers, always acquire more informa- tion. This insight justifies a popular connotation for fast traders, especially high-frequency traders (HFTs), that they are also more informed (see surveys by Biais and Foucault, 2014; O’Hara, 2015; and Menkveld, 2016 and evidence by Brogaard, Hendershott, and Riordan, 2014). Consistent with this connotation, the investment decision to become HFT is often modeled as a “bundle”—once invested, the investor is both informed and fast (Hoffmann, 2014; Biais, Foucault, and Moinas,

2015; Bongaerts and Achter, 2016). Third, recent studies on trading speed have focused on the welfare impact of socially wasteful

“arms race” in speed. For example, Budish, Cramton, and Shim (2015) suggest a market design

4 of frequent batch auctioning. Biais, Foucault, and Moinas (2015) proposes directly taxing speed technology. Du and Zhu (2016) models and calibrates the optimal trading frequency. This paper

adds to this policy debate the insight that investments in speed and information technologies are

intrinsically connected. Regulations on the speed arms race should account for the (unintended)

consequence on price discovery. The rest of the paper is organized as follows. Section 2 sets up the model and derives its

equilibrium. Section 3 then exploits the model implications on market quality. Section 4 studies

a model extension on firms’ real investments to illustrate the tension between the speed and the

magnitude aspects of price discovery. Discussions on model assumptions and robustness are

collated in Section 5. Section 6 then concludes. Appendix A presents a continuous version of the model. All proofs are collated in Appendix B.

2 Model

2.1 Model setup

Assets. There is a risky asset and a risk-free numéraire (money). Each unit of the risky asset will pay off a normally distributed random amount V units of the numéraire. Without loss of generality, E τ−1 > V is normalized to 0 for notation clarity. The unconditional variance of V is denoted by V ( 0).

Investors. There is a continuum of atomless investors, indexed by i on a unity continuum of i ∈

[0, 1]. These investors have constant absolute risk-aversion (CARA) preference with a common risk-aversion coefficient denoted by γ (> 0).

Technology. Each investor i can invest in two types of technology, speed and information. His

speed choice, denoted by ti ∈ T = {1, 2}, determines when his trade will be executed. Without

speed investment, any investor can trade late at time ti = 2. Instead, he can trade early at ti = 1 by investing 1/gt units of the numéraire in the speed technology, where gt (> 0) is an exogenous

5 parameter measuring how advanced this technology is. Appendix A extends the trading rounds to continuous time and studies the robustness of the results with a generalized speed cost function.

Before trading, each investor i will receive a private signal Si about V. Specifically, Si = V + εi,

where εi is independent of V, independent of any other ε j,i, and normally distributed with zero τ−1 > mean and variance i ( 0). The investor i can spend mi units of the numéraire on an information production technology to determine his private signal’s precision:

τi = gτ kτ (mi), where kτ (·) is twice-differentiable, (weakly) concave, and strictly monotone increasing; and gτ (> 0) is an exogenous parameter measuring how advanced this information technology is. Without investing in information, the investor gets no signal, i.e. τi = gτ kτ (0) → 0. Because of the monotonicity of kτ (·), the analysis below will refer to either τi or mi as investor i’s information acquisition.

Timeline. There are three phases in the model: t = −1, t = 0, and t ∈ T = {1, 2}, as illustrated in

Figure 1. At t = −1, the continuum of investors independently choose their technology investment ti

and mi. At t = 0, each investor i produces his private signals si via the information technology with the chosen precision τi. Based on si and his speed ti, each investor then submits a demand schedule of xi (p, si) to trade the risky asset. The submitted schedules are executed in the last phase, t ∈ T . Specifically, a submitted demand

2 schedule xi (p, si) arrives at and is cleared in the market only at round ti. The trading details are specified in the next paragraph. At the end of the trading phase, the risky asset liquidates at V and

all investors consume their terminal wealth.

2 An investor’s speed in this context is best thought of as the time gap between he conceives a trading idea and he fully implements this idea—the implementation delay, due, e.g., to the latencies inherent in the investor’s access to the market, the communication needed between the trader and his broker, or the time needed to convince the manager and to get compliance approval. The mechanisms behind such gaps are not explicitly modeled here but modeled as the investor’s costly speed acquisition. An alternative interpretation of speed is offered in Section 5.1.

6 Invest in speed Submit demand Consume ter- and information schedule to trade minal wealth t = 1 t = 2 t = −1 t = 0 Trading rounds, T

Figure 1: Time line of the game. There are three phases: t = −1 when all investors invest in technology; t = 0 when all investors submit their demand and supply; and t ∈ T := {1, 2} which are the possible trading rounds. After the last trading round, investors consume their terminal wealth and the game ends.

Trading. Denote by I(t) the (possibly empty) set of investors who choose speed t ∈ T = {1, 2}.

Then the aggregate demand schedule at round t is ∫

(1) L(p; t) = xi (p, si)di + ∆U(t), i∈I(t)

where {∆U(t)}t∈T , independent of all other random variables, are i.i.d. normally distributed with τ−1 zero mean and variance U . Section 5.4 discusses the consequence of allowing time-varying sizes of noise trading.

There is a competitive market maker, who clears the market at all times at the efficient price conditional on all historical public information. That is, ∀t ∈ T , the market clearing price is [ ] = E { · } (2) P(t) V L( ; r) r≤t,∀r∈T

Clearing the market via such a competitive market maker is seen in, among many others, Hirshleifer, Subrahmanyam, and Titman (1994), Vives (1995), and Cespa (2008). Section 5.2 discusses the

motivation for and the robustness of this choice.

Strategy space and equilibrium definition. To sum up, each investor maximizes his expected

2 utility from consuming the final wealth by optimizing his demand schedule xi (p, si) : R 7→ R at t = 0; and, backwardly, by choosing his technology investment (ti, τi) ∈ T × [0, ∞), at t = −1.

(Note that choosing the amount of information production τi is equivalent to choosing how much

7 to invest in the information technology mi.)

Denote by π(ti, τi) the investor i’s ex ante certainty equivalent (whose functional form will be

derived below). Define P := {(ti, τi)}i∈[0,1] as the collection of all investors’ investment policies. A

Nash equilibrium is a collection P, such that for any investor i, fixing P\(ti, τi), he has π(ti, τi) ≥ π(t, τ), ∀(t, τ) ∈ T × [0, ∞).

One-round special case. As a remark, it is useful to recognize the root of the above model setup

from the special case of only one trading round: If T = {1}, then the model degenerates to the one-

round trading game as in Section 1 of Vives (1995), but with endogenous information acquisition.

This one-round special case is also consistent with the seminal framework by Verrecchia (1982) (differing only in the market clearing mechanism). With only one trading round, however, speed

investment is not meaningful—no investor will ever need to invest in speed. Such one-round special

case is unable to speak to the speed aspect of price discovery, but only to the magnitude aspect.

The two-round model nests both aspects.

2.2 Equilibrium analysis

The equilibrium is analyzed backwards. First, an investor’s optimal trading demand is derived.

Then, with the certainty equivalent from trading, the optimal technology investment is solved. Fix all other investors’ strategies and consider an investor i ∈ I(t), t ∈ T = {1, 2}, who has invested in information technology τi = gτ kτ (mi). At t = 0, he chooses his demand schedule xi to maximize his CARA preference over final wealth: [ ]

1 −γ·(V−P(t))xi xi ∈ arg max E − e | Si = si, P(t) = p xi γ

where P(t) is given by the market maker’s efficient pricing as in equation (2). Standard analysis as

in Vives (1995) yields the following lemma.

8 Lemma 1. For any technology pair (ti, τi), an investor i has optimal linear demand schedule τ ( ) , = i − , xi (p si) γ si p

where τi = gτ kτ (mi). His certainty equivalent at the time of technology investment (t = −1) is ( ) 1 τi 2 − ti (3) π(ti, τi) = ln 1 + − mi − , 2γ τP(ti) gt

−1 where the price efficiency τP(t) := var[V | {P(r)}∀r≤t ] satisfies ( ) ∫ 2 τj (4) ∆τP(t) = τP(t) − τP(t − 1) = dj τU j∈I(t) γ −1 with the initial condition τP(0) = var[V] = τV. The equilibrium price P(t) satisfies

∆τ (t) * γ∆U(t) + ∆P(t) = P(t) − P(t − 1) = P .V + ∫ − P(t − 1)/ τP(t) τ j , j∈I(t) jd - with the initial condition P(0) = EV (= 0).

To maximize his certainty equivalent, an investor chooses his information precision τi according to the first-order condition:

1 gτ k˙τ (m ) (5) i − 1 = 0 2γ τP(ti) + gτ kτ (mi) which has a unique solution, satisfying the second-order condition, due to the concavity assumed for kτ (·); see the “Technology” paragraph in Section 2.1. Note that an individual investor is atomlessly small and, hence, his choice of information precision does not affect the aggregate price discovery τP(ti). By symmetry, therefore, all investors of the same speed t invest the same amount, m(t), in the information technology to acquire signals of the same precision τ(t) = gτ kτ (m(t)).

Proposition 1. In equilibrium, faster investors always acquire more information than slow

investors. That is, m(1) ≥ m(2) and τ(1) = gτ kτ (m(1)) ≥ τ(2) = gτ kτ (m(2)).

Proposition 1 follows directly the first-order condition above. It shows that there is endogenous bundling of information and speed acquisition: An investor who acquires speed to become fast

(e.g. a high-frequency trader) also has incentive to acquire more information than slow investors.

9 It follows that a speed-t investor’s certainty equivalent becomes ( ) 1 gτ kτ (m(t)) 2 − t π(t) = ln 1 + − m(t) − 2γ τP(t) gt and critically depends on τP(t), which in turn depends on investors’ population sizes at each trading round (equation 4). Denote the population sizes by ∫ µ(t) := di, t ∈ T = {1, 2}. i∈I(t) In equilibrium, µ(1) and µ(2) should be such that no investor of either type, fast or slow, has incentive to change his technology. Note that µ(1) is also the demand for the speed technology in this economy and the discussion below shall refer to “fast population size” and “demand for speed” interchangeably. The equilibrium is pinned down below.

Proposition 2 (The two-round equilibrium). There exists a unique equilibrium P that depends

on the speed technology gt relative to a threshold gˆt (> 0, given in the proof):

Case 1 (corner). When gt < gˆt, all investors invest in (ti, τi) = (2, gτ kτ (m(2))), where m(2),

together with τP(2), uniquely solves the first-order condition (5) and the recursion (4).

Case 2 (interior). When gt ≥ gˆt, a mass µ(1) of investors invest in (ti, τi) = (1, gτ kτ (m(1))),

while the rest µ(2) = 1 − µ(1) investors invest in (ti, τi) = (2, gτ kτ (m(2))), such that the equilibrium variables {m(1), m(2), µ(1), µ(2)} solve the following equation system:

1 gτ k˙τ (m(t)) Optimal information acquisition: − 1 = 0, ∀t ∈ {1, 2}; 2γ τP(t) + gτ kτ (m(t)) Indifference in speed: π(1) = π(2);

Population size identity: µ(1) + µ(2) = 1;

where π(t) and the recursion of τP(t) are given in Lemma 1.

Intratemporal competition v.s. Intertemporal crowding out. Proposition 2 describes two cases for the equilibrium. The corner case is intuitive: When the speed technology gt is too poor, the benefit of becoming fast is too small (i.e. the cost of speed is too high). Hence, all investors stay

10 “slow” to trade at t = 2. In the interior case, investors self select into two types, some fast and the others slow. In equilibrium, both types earn the same certainty equivalent.

Why would an investor be indifferent between fast and slow? Consider a slow investor for

example. The cost to become fast is clear: He needs to pay for the speed technology. The benefit

of switching to fast is a larger information rent—his private signal will be worth more at t = 1, the short-run, than at t = 2, the long-run. This is because at t = 1, not very much price discovery

has occurred and the information rent earned over the uninformed noise traders is relatively large.

By t = 2, the market has processed a lot of information from the trading at t = 1, and a same private

signal will have lost some of its value.

As the information rent of private signals decays over time through price discovery, investors naturally have incentive to become fast (and even faster), in attempt to offset the decay. Collectively,

however, a larger cohort of fast informed investors fuels the short-run price discovery, which

accelerates the decay of the information rent for the remaining slow investors. Such slow investors

thus have lower incentive to acquire information, but stronger incentive to acquire speed to become fast. A fierce, self-reinforcing intertemporal crowding out effect arises.

As more and more investors become fast, the competition in the short-run intensifies, hurting

their profit. This is the conventional intratemporal competition channel in the literature of costly

information acquisition (see, e.g., Verrecchia, 1982). The equilibrium is formed when fast investors’

extra information rent wanes down to the level that just breaks even with the speed cost. Putting the words in algebra, the equilibrium condition π(1, m(1)) = π(2, m(2)) becomes [ ( ) ] [ ( ) ] 1 + gτ kτ (m(1)) − − 1 + gτ kτ (m(2)) − = 1 − . (6) γ ln 1 τ m(1) γ ln 1 τ m(2) 0 |2 {zP(1 ) } |2 {zP(2 ) } |g t{z } information rent from t=1 information rent from t=2 incremental speed cost

The left-hand side is the difference in the information rent between the fast (ti = 1) and the slow

(ti = 2). The right-hand side is the incremental cost of the speed technology from slow to fast. The two sides equate each other in equilibrium. The equilibrium condition highlighted above reveals

an intrinsic tension between information (the left-hand side) and speed (the right-hand side). The

11 following section builds on this intrinsic tension to yield predictions of market quality.

3 Model implications

This section explores the model implications based on the unique equilibrium identified in Propo-

sition 2. The discussion below focuses on the more interesting interior equilibrium.

3.1 Speed technology, gt

This subsection studies how the equilibrium market condition responds to exogenous shocks that

affect the speed technology.

Demand for speed. A higher value of gt reduces investors’ cost to trade fast. The usual demand effect applies: A price reduction increases the demand.

Proposition 3 (Speed and investor population sizes). In the interior equilibrium, as the speed

technology advances, the population of fast investors increases: ∂µ(1)/∂gt > 0.

The increased demand for the speed technology is illustrated in Panel (a) of Figure 2. The horizontal

axis shows the speed technology level gt, while the population sizes of the fast (ovals) and the slow (crosses) investors are indicated on the vertical axis. To the right of the vertical dashed line, the equilibrium is interior—there are both fast and slow investors. As the speed technology progresses, more and more investors acquire speed and become fast. The line separating the two areas,

µ(1) + 2(µ(2) − 1), can be read as the average trading latency in this economy and it reduces with gt.

Fast investors in the short-run. Speed technology advancement has non-trivial effects on in-

vestors’ information acquisition. Consider the fast investors. As their population size grows in the

short-run (µ(1) increases), the intratemporal competition forces each of them to acquire less infor-

mation. In spite of the reduction in individual information acquisition, in aggregate, the short-run

12 (a)

Population size, µ(t) 1.0

µ(1), fast 0.8

0.6

0.4

0.2 µ(2), slow

Speed technology, gt 0.0 100 101 102 103 104

(b) (c) Individual information τ Price efficiency by round t, P(t) production, τ(t) = gτ kτ(m(t)) 3.5

0.5 . 3 0 Long-run

2.5 0.4 Fast 2.0

. Short-run 0 3 1.5 Slow . τ Speed technology, gt 1 0 V Speed technology, gt 0.2 100 101 102 103 104 100 101 102 103 104

Figure 2: Varying the speed technology. This figure illustrates how fast and slow investor’ population sizes (Panel a), their individual information production (Panel b) and the cumulative price efficiency (Panel c) change with the speed technology. In all panels, the horizontal axis indicates the speed technology param- 0 4 eter gt , ranging from 1.0 × 10 to 1.0 × 10 . A vertical dashed line indicates the threshold ofg ˆt , below which all investors stay slow (Proposition 2). The primitive√ parameters used in this numerical illustration are: τV = 1.0, τU = 4.0, γ = 0.3, gτ = 1.0, and kτ (m) = m.

13 −1 price efficiency τP(1) = var[V | P1 ] increases. This result shares the same intuition as in Gross- man and Stiglitz (1980): Investors have less incentive to acquire information as the population of informed investors increases (relative to the uninformed and to the noise demand). In the extreme, without noise trading, price becomes fully revealing and, yet, no investor ever wants to acquire information—the Grossman-Stiglitz paradox. The patterns ofτ(t) (= gτ kτ (m(1))) and τP(1) are illustrated in Panel (b) and (c), respectively. A formal statement is given below.

Proposition 4 (Speed technology and information in the short-run). In the interior equilib-

rium, as the speed technology advances, individual fast investors acquires less information but

the short-run price efficiency increases: ∂τ(1)/∂gt < 0, and ∂τP(1)/∂gt > 0.

Slow investors in the long-run. The implications for the slow investors’ information acquisi- tion τ(2) and the long-run price efficiency τP(2) are very different. The reason is that a slow investor, in choosing his information technology, not only takes into account the intratemporal competition from his slow peers, but also recognizes the intertemporal crowding out effect from his fast competitors. Specifically, the conventional intratemporal competition dictates that as more investors become fast, the fewer remaining slow investors in t = 2 each wants to invest more in information. However, as more information gets revealed in the short-run t = 1 (Proposition 4), there is less information that remains for the slow investors to further acquire. In other words, fast investors crowd out slow investors’ information acquisition in the long-run. Proposition 5 summarizes the net effect of the above channels.

Proposition 5 (Speed technology and information in the long-run). In the interior equilib- rium, speed technology has a non-monotone effect on individual slow investors’ information

acquisition τ(2) and on the long-run price efficiency τP(2): initially τ(2) increases but τP(2)

worsens; yet eventually τ(2) decreases but τP(2) improves.

Proposition 5 essentially sketches when and which of the two forces—intratemporal competi- tion or intertemporal crowding out—dominates. When the speed technology gt is low, the short-run

14 price efficiency τP(1) is relatively low, as only a small fraction of investors is fast (small µ(1)). Therefore, the effect of intertemporal crowding out, which discourages slow investors’ informa-

tion acquisition, is small. The dominating effect is from the intratemporal competition, which

encourages more information acquisition with fewer investors remaining slow.

On the other hand, when gt is high enough, it is the intertemporal crowding out effect that dominates: As a lot of investors become fast (large µ(1)), the short-run price efficiency is very

high and slow investors’ incentive to acquire information is significantly curbed. The long-run

non-monotone effects are contrasted with the short-run effects in Panel (b) and (c) of Figure 2.

3.2 Information technology, gτ

This subsection studies how the equilibrium market condition responds to exogenous shocks that affect the information production technology. The analysis begins with the following lemma.

Lemma 2. Fixing the speed technology gt, there exists a gˆτ such that the equilibrium is interior

if and only if gτ ≥ gˆτ.

The lemma is intuitive: If the information technology is too poor, the resulting information rent for

becoming fast will not be high enough to offset the cost of acquiring speed and, as such, all investors

will stay slow. The rest of this subsection focuses on the interior equilibrium with gτ ≥ gˆτ.

Demand for speed. As the information technology improves, will more or fewer investors become

fast? There are two effects. First, an improved information technology advances the amount of

price discovery in the short-run, and hence intertemporally crowds out slow investors’ information acquisition and reduces their rent. In equilibrium, therefore, more slow investors will want to invest

in the speed technology to trade early before the rent of his private signal decays. Second, as there

are more and more fast investors, the competition among them also intensifies—the intratemporal

competition effect—and this tends to reduce the information rent in the short-run, dampening the demand for speed. The following proposition summarizes the net effect.

15 Proposition 6 (Information and demand for speed). In the interior equilibrium, the informa-

tion technology gτ has a non-monotone effect on investors’ demand for speed: initially µ(1)

increases but eventually µ(1) decreases, as gτ increases.

Panel (a) of Figure 3 illustrates the pattern. As the information technology gτ advances beyond

the right of the vertical dashed line, the equilibrium is interior. Initially, the information and the

speed technology exhibit certain complementarity: When the per unit cost of information reduces

(higher gτ), the demand for speed increases. This holds true when the intertemporal crowding

out effect dominates. However, eventually the demand falls for speed and the complementarity disappears, as the intratemporal competition among fast investors becomes too fierce.

Individual information acquisition and aggregate price efficiency. A better information pro- duction technology effectively reduces the average cost for acquiring information. Both fast and

slow investors will tend to acquire more information because of this lower cost. This is shown in

Panel (b) of Figure 3. In addition, this productivity increase in the information technology is the dominating effect for the aggregate price discovery, in spite of the complication arising from the intratemporal competition and intertemporal crowding out effects.

Proposition 7 (Information technology and aggregate price efficiency). Suppose τV is suf- ficiently small. In the interior equilibrium, as the information technology advances, both the

short-run and the long-run price efficiency improve: ∂τP(2)/∂gτ > ∂τP(1)/∂gτ > 0.

Panel (c) of Figure 3 illustrates the above proposition.

3.3 Joint effects

The analysis so far treats the effects of the speed and the information technology separately.

In reality, it is arguable that shocks affecting both technologies can occur. For example, there might be common hardware—computation power, data processing—underlying both technologies.

Breakthroughs in such common fundamentals can boost both technologies together.

16 (a)

Population size, µ(t) 1.0

µ(1), fast 0.8

0.6

0.4 µ(2), slow 0.2

Information technology, gτ 0.0 10−1 100 101 102

(b) (c) Individual information t τ t Price efficiency by round , P ( ) 6.0 production, τ(t) = gτ kτ (m(t)) 25

20 4.5

Fast 15 Short-run

Long-run 3.0 Slow 10

1.5 5

Information technology, gτ τ Information technology, gτ 0.0 0 V 10−1 100 101 102 10−1 100 101 102

Figure 2: Varying the information technology. This figure illustrates how fast and slow investor’ population sizes (Panel a), their individual information production (Panel b) and the cumulative price efficiency (Panel c) change with the information technology. In all panels, the horizontal axis indicates the information technology − parameter gτ, ranging from 1.0 × 10 1 to 1.0 × 102. A vertical dashed line indicates the threshold ofg ˆτ, below which all investors stay slow (Lemma 2). The primitive√ parameters used in this numerical illustration are: τV = 1.0, τU = 4.0, γ = 0.3, gt = 10.0, and kτ (m) = m.

17 To address such joint technology shocks, one needs to assume a specific functional form of how

gt and gτ are related. This paper takes an agnostic approach to such joint effects. Instead, Figure 4 numerically shows the patterns in a set of contour plots.

Three panels illustrate how joint technology shocks might affect, respectively, (a) investors’

demand for speed, (b) individual investors’ information acquisition, and (c) the aggregate effect on price efficiency. A set of isoquants are plotted in each panel, where both the horizontal axis (speed)

and the vertical axis (information) are in log-scale. For Panel (b) and (c), the isoquants are shown

both for the fast investors in the short-run in the dashed line and for the slow in the long-run in the

solid line.

The patterns are consistent with Figure 2 and 3 but more revealing. For example, from Panel (a) it can be seen that when both speed and information technologies advance, moving from lower-

left to upper-right, investors will not necessarily become faster. In particular, it is possible, if

information technology advances relatively stronger than speed technology, the demand for speed

will be lower in the economy. In the same spirit, Panel (b) and (c) offer qualitative prediction on how a specific joint technol-

ogy shock would affect individual information acquisition and aggregate price efficiency. Notably,

Panel (c) suggests that improved information technology only unambiguously improve total mag-

nitude of price efficiency τP(2) when the level of speed is held constant. Too much advancement in speed might overshadow the positive effect of information technology improvement, curbing the price efficiency in the long-run. This effect clearly illustrates the interconnectedness of the two

aspects of price discovery—the speed and the magnitude.3

3 Note that as speed technology improves, fixing the same information technology, price efficiency τP(1) and τP(2) converge eventually. This is consistent with Proposition 3 that for sufficiently high level of speed technology, almost all investors become fast and there is price discovery only in the short-run (t = 1).

18 (a) Demand for speed, µ(1)

101

0.9 τ g 0.8

0.6 0.4

0.2

100 0.0 Information technology,

10−1 100 101 102 103 104 Speed technology, gt

(b) Individual information acquisition, τ(t) (c) Aggregate price efficiency, τP(t) 1.0 1.0

0.23 0.40 0.28 2.8 0.9 0.23 0.9 τ τ

g g 2.6 2.8

2.4 2.6 0.8 0.20 0.20 0.8 2.4 2.2

0.17 2.2 0.7 0.7 2.0 0.17 2.0 1.8 Information technology, 0.6 Information technology, 0.6 0.14 1.8 0.14

0.5 0.5 100 101 102 103 104 100 101 102 103 104 Speed technology, gt Speed technology, gt

Figure 4: Varying both technologies. This figure shows how investors’ speed acquisition (Panel a), information acquisition (Panel b), and the aggregate price efficiency (Panel c) are affected jointly by speed and information technology. In each panel, the horizontal axis correspond to the level of the speed technology, 4 gt ∈ [1.0, 10 ], in log-scale. To manifest the patterns, Panel (a) plots on the vertical axis the level of the − information technology gτ ∈ [10 1, 2 × 101] in log-scale, while Panel (b) and (c) plot gτ ∈ [0.5, 1.0] (not logged). Further, in Panel (b) and (c), the dashed isoquant lines are for the fast investors in the short-run = = (t 1); and the solid lines are for the slow investors in the long-run (t 2).√ The primitive parameters used in this numerical illustration are: τV = 1.0, τU = 4.0, γ = 0.3, and kτ (m) = m.

19 3.4 Other market quality measures

This subsection examines how technology shocks affect a set of other market quality measures:

trading volume, return volatility, market depth, and investors’ gains and losses. The results are

summarized in Figure 5 and 6. Brief discussions of the patterns are provided below.

Trading volume. The round t trading volume, excluding noise trading, is given by

µ(t) · |x(si, p(t))|.

It is jointly determined by the population size µ(t) and the magnitude of the investors’ demand

|x(si, p(t))|. When the speed technology advances, more investors become fast. In equilibrium, their aggregate trading becomes more aggressive and the short-run total trading volume increases.

The long-run cumulative volume on the contrary, is subject to the countervailing effects of the intratemporal competition and intertemporal crowding out. The patterns shown in Panel (a) of

Figure 5 demonstrate the above effects. When the information technology advances, investors in both periods can acquire more precise

information and hence will trade more aggressively. This turns out to be the dominant effect that

drives up the trading volume in both the short- and the long-run. Panel (a) of Figure 6 depicts the

trends.

Return volatility. Thanks to the competitive market maker, the market clearing price P(t) is

always efficient. This means that the price returns only reflect new information from the investors’

demand schedules. Indeed, as noted by Vives (1995) (Corollary 2.1), the unconditional price

volatility can be derived as

= τ−1 − τ −1, var[P(t)] V P(t)

which is a monotone function in τP(t). Indeed, since the amount of noise trading τU is held constant, prices become more volatile only when there is more information. As such, the return volatility

has a similar pattern with that of price efficiency τP(t); see Panel (b) in Figure 5 and in Figure 6.

20 (a) (b)

Cumulative trading volume 0.5 Unconditional volatility, var[P(t)] 1.6 (excluding noise trading) Long-run Long-run 0.4 1.2 0.3

0.8 Short-run 0.2

0.4 0.1 Short-run

Speed technology, gt Speed technology, gt 0.0 0.0 100 101 102 103 104 100 101 102 103 104

(c) (d)

λ −1 12 Market depth, (t) 0.14 Investors’ gain and loss Noise traders’ trading cost 10 0.12

8 0.10

Long-run 6 0.08 Informed traders’ rent 4 Short-run 0.06 Speed technology, gt Speed technology, gt 100 101 102 103 104 100 101 102 103 104

Figure 5: Speed technology and market quality. This figure illustrates how market qualities are affected by the speed technology. Panel (a) plots the short- and long-run trading volume. Panel (b) plots (unconditional) price return volatility. Panel (c) plots the market depth. Panel (d) plots investors’ gain and losses. In all panels, the horizontal axis indicates the speed technology parameter gt . A vertical dashed line indicates the threshold ofg ˆt , below which all investors stay slow (Proposition 2). The primitive√ parameters used in this numerical illustration are: τV = 1.0, τU = 4.0, γ = 0.3, gτ = 1.0 and kτ (m) = m.

21 (a) (b)

8 Cumulative trading volume 1.0 Unconditional volatility, var[P(t)] (excluding noise trading)

0.8 6

0.6 Long-run 4 Long-run 0.4 Short-run

2 0.2 Short-run Information technology, gτ Information technology, gτ 0 0.0 10−1 100 101 102 10−1 100 101 102

(c) (d)

λ −1 40 Market depth, (t) 0.15 Investors’ gain and loss

35 0.12 30

Noise traders’ 25 0.09 trading cost 20

0.06 traders’Informed rent 15 Long-run

10 Short-run 0.03 5 Information technology, gτ Information technology, gτ 0 0.00 10−1 100 101 102 10−1 100 101 102

Figure 6: Information technology and market quality. This figure illustrates how market qualities are affected by the information technology. Panel (a) plots the short- and long-run trading volume. Panel (b) plots (unconditional) price return volatility. Panel (c) plots the market depth. Panel (d) plots investors’ gains and losses. In all panels, the horizontal axis indicates the information technology parameter gτ. A vertical dashed line indicates the threshold ofg ˆτ, below which all investors stay slow (Lemma 2). The primitive√ parameters used in this numerical illustration are: τV = 1.0, τU = 4.0, γ = 0.3, gt = 10.0, and kτ (m) = m.

22 Note, in particular, in Panel (b) of Figure 6, when the information technology advances significantly enough, the return volatility in both the long-run and the short-run converge to the unconditional

variance of V, i.e. τV. This is because investors’ signals are extremely precise, they trade very aggressively, and through trading the price becomes almost fully revealing.

Market depth. Following the literature, market depth λ(t)−1 is defined as the inverse of the price sensitivity to an additional unit of (noise) trading (see, e.g., Kyle, 1985; Vives, 1995). Panel (d) of

Figure 5 and of Figure 6 illustrate how the short-run and the long-run market depth vary with the

speed and the information technology, respectively. While characterizing the exact shape of λ(t)−1

is difficult (depending on the specification of kτ (·)), it can be shown that, in the interior equilibrium, the long-run market is initially thinner but eventually deeper than the short-run market as either technology advances.

Intuitively, when the speed technology is low, more price discovery occurs in the long-run than in the short-run. As such, a trade will move the price less—a deeper market—in the short-run than in the long-run. When the speed technology is high enough, more price discovery occurs in the short-run and the market becomes deeper in the long-run. The same intuition holds true for information technology.

The evidence of how technology advancement affects order book depth is mixed. For example,

Hasbrouck and Saar (2013) find deeper order book is associated with large amount of low-latency activity. Yet, focusing on the algorithmic trading activity, Hendershott, Jones, and Menkveld

(2011) show that the quoted book depth reduces. Exchange speed upgrades seem to correlate with

thinner order books as evidenced by Riordan and Storkenmaier (2012), while Brogaard et al. (2013)

find that co-location deepens the order book. Using a regulatory event, Chakrabarty et al. (2016) show that a market-wide slow-down led to increased book depth. The qualitative prediction above

helps understand such disagreement in the literature and offers a new empirical angle to compare

short-run and long-run depth.

23 Investors’ gains and losses. There are three types of agents in this economy: The competitive market maker always breaks even in expectation. All (informed) investors—fast or slow—must have the same ex ante certainty equivalent (Proposition 2): ( ) ∗ 1 gτ kτ (m(t)) 2 − t π := ln 1 + − mt − , ∀t ∈ T = {1, 2}, 2γ τP(t) gt which is the per capita rent of these investors. In addition, there are noise traders who trade for exogenous private values. While such private values are unmodeled, it is possible to calculate their expected trading cost as:

λ + λ τ−1. ( (1) (2)) U

Panel (d) of the above figures plots how informed investors’ gains and noise traders’ losses from trade vary with the speed technology. It can be seen that noise traders’ trading cost is the highest for moderate cost of speed. This finding mirrors the market depth result—when the market is too thin, noise traders move price a lot and pay high premium for their private values. On the other hand, (informed) investors’ rent first increases with speed, until the speed technol- ogy becomes extremely advanced. When the speed technology is extremely poor (or advanced), all investors flock to the first (or the second) round, in which case the overall competition is the most fierce and the rent is the lowest. For moderate levels of speed technology, investors split into the two rounds and the overall competition is less fierce, yielding higher rents for all. The same intuition holds for the case of information technology. When the technology is too poor or too advanced, most of the investors stay slow in round t = 2 (Proposition 6) and they compete out the rent. Only for moderate levels of the information technology the investors evenly split and their rent is the highest.

24 4 Learning from price: the magnitude and the speed

The above analysis zooms in on the intrinsic connection between the two aspects of price discovery: the magnitude (how much) and the speed (how soon). These two aspects are important for the end

users of information from financial markets to make real decisions like production, real investments,

etc. As an application, this section focuses on the speed technology and shows that speed technology

advancement implies a non-trivial tradeoff in the two aspects. The intuition is first discussed and

then a formal model extension is presented to study this tradeoff.

4.1 The intuition

Suppose there is a firm who learns from the asset’s price to make real investment decisions and it

needs at least two units of price efficiency, i.e. τP(t) > 2, to deem the asset’s price reliable. When

the speed technology is low, roughly gt < 10, the firm will have to wait till the long-run (t = 2)

because τP(1) < 2 < τP(2) as shown in Panel (c) of Figure 2. If waiting is costly, a boost in the

speed technology, e.g. to gt > 10, can help the firm as it will no longer need to wait. That is, when the speed technology significantly advances, the price discovery speeds up and the short-run price efficiency τP(1) becomes high enough to lend confidence to the firm’s decisions. The real efficiency—the firm’s real investment—benefits from such speed technology.

There is a catch, however, due to the non-monotone effect on the magnitude of price discovery.

The (cumulative) long-run price efficiency τP(2) initially decreases (and eventually increases) with the speed technology. This could create problem for the firm in the example above, especially when

it requires a more informative asset price, e.g. four units. If the speed technology is still very low (gt

close tog ˆt), the firm will be able to make its decision eventually in the long-run (t = 2) as τP(2) > 4. However, as the speed technology (moderately) improves, the intertemporal crowding out effect

hurts the total magnitude of price discovery. Now even in the long-run, τP(2) < 4 and the firm is never able to rely on the asset price confidently. The real efficiency is hurt.

25 4.2 Formal analysis

A firm needs to choose both the level q and the timing t for a real project to maximize the present value of future cash flows: [ ( ) ] 1 max E ϕt−1 qV − q2 | {p(1), ..., p(t)} , q∈R,t∈{1,2} 2 where ϕ ∈ (0, 1] is the exogenous discount factor between t = 2 and t = 1 for the firm. The

investment level q ∈ R scales the firm’s profitability V at a constant marginal cost normalized to unity (negative q can be interpreted as, for example, reducing production). This choice is conventional in the literature. The novel timing choice t ∈ T = {1, 2} determines how much the firm can learn from the financial market, {p(1), ..., p(t)}, trading off the loss of project value

captured by the discount factor ϕ. Such loss can be due to competition with peer firms (e.g., the

first-mover advantage à la Stackelberg, 1934) or simply the time value of money.

It is assumed that the firm can invest once and only once at the chosen time t. While investors need to pay a cost to trade in the short-run at t = 1, the firm’s investment timing is not subject to such speed cost. This is because the firm can obtain real-time information of the financial market

(e.g. stock price), via internet search or newspapers, at (almost) zero cost.

Optimal investment. To solve the firm’s optimal investment level and timing, consider the firm’s choice at t = 1, having just observed p(1) from the financial market. If it implements the project now, the optimal investment level q is given by, following the first-order condition,

q = E[V | p(1) ] = p(1),

where the second equality follows the definition of p(1), thanks to the competitive market maker. The firm’s expected cash flow becomes p(1)2/2. If instead the firm waits to the next round, its

expected cash flow conditional on p(1) is ϕ [ ] ϕ ( ) E P(2)2| p(1) = p(1)2 + τ (1)−1 − τ (2)−1 . 2 2 P P

26 Therefore, the firm will choose to wait till t = 2 if and only if √ 1 ϕ ( ) τ (1)−1 − τ (2)−1 p(1)2 < p(1)2 + τ (1)−1 − τ (2)−1 ⇔ |p(1)| ≤ P P . 2 2 P P 1 − ϕ

Define by Pˆ(1) the round-1 trading price, scaled by its standard deviation, andp ˆ(1) its realization. (From the firm’s perspective, Pˆ(1) follows a standard normal distribution.) The above threshold

can be rewritten as √

|P(1)| 1 (1 − θ)τV (7) Pˆ(1) := √ ≤ =:p ¯. varP(1) 1 − ϕ ∆τP(1) + θτV

where θ := (τP(1) − τV)/(τP(2) − τV) is a measure for the speed of price discovery, the amount of price discovery that has occurred by t = 1 as a percentage of total price discovery. The higher is θ,

the faster is the price discovery process. The above thus leads to the following lemma.

Lemma 3 (The firm’s optimal investment and timing). The firm’s optimal timing is t∗ = + 1 ∗ = ∗ ˆ 1 {|Pˆ(1)|≤p¯} and the optimal investment level q p(t ), where P(1) and p¯ are defined in equation (7).

In equilibrium, both the investment level q∗ and the timing t∗ depend on the observed state from the financial market. In particular, Lemma 3 predicts to observe firms rushing for projects in the

short-run only after significant shocks in the financial market, i.e. |Pˆ(1)| > p¯. Otherwise, firms

prudently wait for more information from the long-run. The thresholdp ¯ consists of three parts. First, when the discount factor ϕ decreases (larger losses

for postponing the project), the threshold also decreases and, cateris paribus, it is more likely for

the firm to invest early. Second, the threshold also decreases with the short-run price discovery

magnitude ∆τP(1). Intuitively, if the price is already very efficient in the short-run, the firm will have no incentive to wait for more information. Third, the threshold is decreasing in the price

discovery speed θ. That is, if the price discovery is very fast, the threshold will be very low and

the firm is likely to invest early. While the first component ϕ is exogenous, the other two effects

are endogenous of both the speed and the information technology. The following proposition

27 characterizes how technologies affect the firm’s investment timing.

Proposition 8 (Technology and the firm’s timing). Both the speed and the informa- tion technology improvements make the firm more likely to invest early. Mathematically, ( ) ( ) ∂P ˆ > /∂ ≥ ∂P ˆ ≥ /∂ > P(1) p¯ gt 0 and P(1) p¯ gτ 0.

Intuitively, as either technology advances, the prices from the financial market become more reliable and the firm relies more confidently on the short-run observations, hence investing early.

The difference lies in the sources of such confidence: When the speed technology improves, both the magnitude ∆τP(1) and the speed θ of price discovery improves (see Panel (c) of Figure 2). When the information technology improves, it is the increase in the magnitude ∆τP(1) that dominates (the slowing down of θ is dominated).

Technology impact on cash flow. While both the speed and the information technology in the

financial market hastens the firm’s investment, the effect on the expected cash flow is not trivial. In particular, earlier investment does not imply more efficient investment.

One can think of the firm’s investment problem as an “option”: The firm can always exercise this investment option late at t = 2; and in addition, it can exercise it early at t = 1 if the condition is favorable (|Pˆ(1)| > p¯). Thus, the firm’s expected cash flow can be written as [ ] ϕ ( ) ( ) 2 ϕ ( ) τ−1 − τ −1 + P ˆ ≥ E P(1) − 2 + τ −1 − τ −1 ˆ ≥ (8) V P(2) P(1) p¯ P(1) P(1) P(2) P(1) p¯ |2 {z } | 2 2 {z } reservation value of investing late option value of investing early where the first term represents the firm’s reservation cash flow if it commits to investing late and the second term the incremental value of exercising the option early. These two components are affected differently by the technologies: The reservation value is determined by the ultimate magnitude of price discovery τP(2), while the option value depends more on the conditional volatility of P(1)—the higher is the volatility, the higher the option value.

Consider a speed technology advancement. First, as Proposition 5 points out, speed has non- monotone impact on τP(2), the long-run magnitude of price discovery. Second, speed prompts the

28 (a) Varying the speed technology, gt (b) Varying the information technology, gτ

Firm’s expected cash flow Firm’s expected cash flow 0.35 0.24

0.32 0.21 9 = 0. φ = 0.9 φ

0.29 0.18 0.8 .8 = 0 φ = φ = 0.7 7 . φ 0.26 0 0.15 =

φ

Speed technology, gt Information technology, gτ 0.23 0.12 100 101 102 103 104 0.50 0.55 0.60 0.65 0.70

0 Figure 7: The real impact of technology. Panel (a) plots how speed technology gt , ranging from 1.0 × 10 to 1.0 × 104, affects the firm’s expected cash flow. Panel (b) plots how information technology gτ, ranging . . from 0 5 to 0 7, affects the firm’s expected cash flow.√ The primitive parameters used in this numerical illustration are: τV = 1.0, τU = 4.0, γ = 0.3, kτ (m) = m, and gτ = 1.0 for Panel (a) and gt = 10.0 for Panel (b). Three curves are plotted with the discount factor ϕ ∈ {0.7, 0.8, 0.9}.

firm to invest in a rush (Proposition 8), hence reducing the option value for investing early. The net impact is non-monotonic in the level of the technology, as shown in Panel (a) of Figure 7. In particular, for relatively slow speed, long-run price discovery τP(2) is hurt and, combined with the lowering option value, the firm’s real cash flow decreases with the speed technology.

Similarly, an improvement in the information technology can also hurt the firm’s cash flow. This is shown in the initial dips in Panel (b) of Figure 7, when the level of information technology gτ just improves beyond the thresholdg ˆτ. A higher gτ always increases the eventual magnitude of price discovery τP(2), improving the reservation value (the first term of equation 8). However, the incremental option value of exercising early reduces (the second term of equation 8). The numerical illustration from Panel (b) suggests that the negative effect on the option value component dominates when the information technology gτ is relatively low. Eventually, the improved long-run

29 price efficiency τP(2) dominates.

A remark. The above analysis only considers the firm’s learning from the financial market, but not the “feedback effect” (Bond, Edmans, and Goldstein, 2012) of how the firm’s real investment decision affects the financial market. (This feedback effect between speed and real efficiency is studied separately in a companion paper.) As such, it is best to interpret the firm as very small and its own cash flow too immaterial to affect the aggregate economic condition V.4 The model does not attempt to draw conclusion on the welfare implications of speed technology. Such analysis requires to build a full model with feedback effect and to endogenize noise trading.

5 Discussion

This section provides some remarks to enrich the interpretation of and to examine the robustness of the model results.

5.1 Interpretations of speed

In the current setup, all investors submit demand schedules at the same time (t = 0) and, depending on their endogenous speed investment, the demand schedules arrive asynchronously (Figure 1).

Speed in this context reflects the implementation delay, i.e. the delay between when one conceives a trading idea and when the trade is realized. Such implementation delay has both a high-frequency and a low-frequency interpretation. Under the high-frequency interpretation, this delay can be due to the latency in the transmission of electronic signals from the trader’s side to the exchange market server. In current times, such latencies can range from seconds to micro- or even nanoseconds and technology investments in cables, colocation, or microwave can improve this high-frequency speed.

4 For example, the traded asset can be thought of as the broad market index, like DJIA, S&P 500, MSCI, etc., (partially) reflecting the economic health V, which is useful for inferring the firm’s expected cash flow but is unlikely affected by an individual small firm.

30 The delay under the low-frequency interpretation arises from the time needed for analysts to convince their clients to implement a trading strategy, for the compliance to approve, for managers

to confirm, and for trading desks to finally execute the orders. Such a process can take hours or days

and investments in, e.g., efficient meeting technology (conference calls), making more effective

communication, and streamlining compliance procedures can speed up trading in this sense. The above “implementation delay” is not the only way to read the “speed” in the model. An

alternative is to think of investor arrival, demand schedule submission, and market clearing all

occurring at the endogenously chosen time t. That is, the speed t can reflect how soon an investor is

able react to certain market events occurring at t = 0, e.g., market opening, macro news, earnings

announcement, etc.5 All results presented in this paper go through under this “reaction delay” interpretation of speed.

A technical point is perhaps worth elaborating. There is a difference in the investor’s information

set under these two interpretations. Specifically, under this “reaction delay” interpretation, one can

argue that when an investor arrives at the market at time t, he could have observed all the trading history up until time t. In contrast, under “implementation latency”, the investor sees no trading history at the time of demand schedule submission (i.e. at t = 0). This difference is inessential for

the equilibrium analysis, thanks to the presence of the competitive market maker, who always clears

the market at the efficient price. This means that the round-t market clearing price is a sufficient

summary static for all past information. From the investor’s perspective, therefore, conditioning on {p(t)} in his demand schedule (under “implementation delay”) is as good as conditioning on the

entire history of {p(r)}∀r≤t (under “reaction delay”). As such, investors’ optimal demand schedules remain the same under either interpretation. The equilibrium analysis is not altered.

5 Under the “reaction speed” interpretation, the triggering event occurring at t = 0 should not fully resolve uncertainty about future, so that there is room for further information acquisition. (Mathematically, conditional on = τ−1 > the event, it is required that varV V 0.) Such remaining uncertainty is evidenced by well-known empirical phenomena like post-earnings-announcement drift, IPO under/overpricing, etc.

31 5.2 The market clearing mechanism

At each trading round t ∈ T , investors’ demand is cleared by a competitive market maker, who can take any position at the efficient price. The key advantage of having such a market maker, persisting at all trading rounds, is that he helps pass on information. This way, price efficiency in later rounds is naturally higher than in earlier rounds, reducing slow investors’ information rent and motivating them to become fast. This is the key driver for the novel intertemporal crowding out effect.

A persistent market maker is not the only way to generate such an incentive for speed. An alternative is to assume a Walrasian auctioneer who sets the market clearing price p such that

L(p; t) = 0 (c.f. equation 1) and let the speed-t investors be able to condition on all market

6 clearing prices {p(r)}∀r≤t before submitting their demand schedules. Such an alternative setup is close to Grossman and Stiglitz (1980) and Verrecchia (1982), but with endogenous asynchronously arriving investors in multiple trading rounds. In this alternative setup, investors trade for two reasons: 1) their private information and 2) making the market for the noise traders. To compare, investors in the current model trade only for their information advantage, because the competitive market maker can take any position needed to clear the market. This difference makes sure that in the current model, the intertemporal crowding out effect is only driven by information rent seeking, not by liquidity provision motives.

In fact, as the alternative setup maintains the exact same incentive for investors to acquire speed, unsurprisingly, the new insight of investors’ intertemporal crowding out is not affected by the difference in the market clearing mechanism. In unreported extensive numerical illustrations, the model’s qualitative predictions have remained robust.

6 Under such an interpretation, a speed-t investor arrives at the market at time t, observes market history, and then submits his demand schedule. The time t reflects his reaction delay; see Section 5.1 for details of this interpretation.

32 5.3 The technology cost functions

The current model setup assumes that the two technology investments are independent of each other.

That is, acquired information technology does not affect the cost of acquiring speed technology, or the other way. This need not necessarily be the case. On the one hand, investments in the two technologies can complement each other. The complementarity can arise from the common hardware needed, e.g. processing capacity (CPUs), bandwidth (cables and optical fiber), etc. Thus, having invested in such facility for one technology can reduce the cost for the other. This feature is often seen in the literature (e.g., Hoffmann, 2014; Biais, Foucault, and Moinas, 2015; Bongaerts and

Achter, 2016), where an investor’s technology investment is “bundled” to give him both advantages in information and in speed. The underlying assumption for such a bundle seems to be that the two technology investments complement each other.

On the other hand, the two investments can exhibit certain substitutability. For example, Dugast and Foucault (2016) argue that because information processing is time consuming, the speed of more informed investors is limited and they can only trade after the less informed. In the current paper’s setting, this effectively means that acquiring better information implies a prohibitive cost for speed. Equivalently, investing either technology increases the (marginal) cost for the other.

Exactly how speed and information technologies interfere with each other is a question of engi- neering and computer science. The current model specification sets a benchmark with independent technologies—an agnostic view. The outcomes of the model (Proposition 3, 4, and 5; Figure 2) therefore offer a clean set of predictions on investors’ endogenous demand for the two technologies, as opposed to exogenous substitutability/complementarity built in the cost functions.

5.4 The amount of noise trading

The noise trading ∆U(t) in each round can be interpreted as market orders submitted for unmodeled reasons. The current setup assumes that the magnitude of noise trading in each round is the same:

33 = τ−1 ∈ T var[∆U(t)] U for all t . This need not be the case. It is straightforward to extend the model −1 to account for time-varying noise trading by assuming time-dependent var[∆U(t)] = τU(t) . Such an extension will quantitatively change the equilibrium, as the amount of noise trading determines the rate of price discovery (equation 4) and affects investors’ ex ante certainty equivalent. However,

the insight of intertemporal crowding out, due to speed technology, remains. For example, Figure 8 numerically illustrates that the qualitative prediction of Proposition 5

remains robust. Panel (a) varies τU(2) with fixed τU(1), while Panel (b) varies τU(1) with fixed τU(2).

In either case, the long-run price discovery τP(2) initially decreases but eventually increases, as the

speed technology gt improves. The different sizes of τU(t) only affect whether the τP(2) is higher

at gt = gˆt or at gt → ∞. Intuitively, in either extreme (gt ∈ {gˆt, ∞}), all investors herd into one

of the trading rounds t ∈ {1, 2} and the only difference is the amount of noise trading τU(t) in that round. As the key driver of results is the robust intuition of intertemporal crowding out, the results

presented in Section 3 are also robust to time-varying τU(t). To this extent, the focus on a constant amount of noise trading is without loss of generality.

6 Conclusion

There are two aspects of price discovery: the magnitude and the speed. The magnitude aspect is

linked to investors’ information acquisition, a key focus of the existing literature. This paper turns

to the speed aspect, i.e. how soon the acquired information is incorporated into price, by studying

a model with investors’ endogenous speed acquisition, alongside their information acquisition. The analysis reveals that these two aspects of price discovery are intrinsically connected via

investors’ competition. There are two key mechanisms at work: On the one hand, an investor

competes intratemporally with his peers who acquire the same speed. On the other, he faces the

intertemporal crowding out effect due to the price discovery by those who acquire faster speed. Based on the interaction of these two mechanisms, the model generates testable implications

34 (a) Varying short-run noise trading, τU(1) (b) Varying long-run noise trading, τU(2) τ τ Price discovery by round 2, P (2) Price discovery by round 2, P (2) τ 4.0 τ ( U (2) fixed at 4.0) ( U (1) fixed at 4.0) 3.7 . .0 3 7 . = 6 τ 3 4 U (1) (1) τU 3.4 = .0 6 = 4 . 3.1 (1) 0 τU τ . U (1) 3 1 = 4 . .0 2 8 .0 2 .0 2.8 = = 2 (1) τU(1) τU 2.5 Speed technology, gt Speed technology, gt 2.5 100 101 102 103 104 100 101 102 103 104

Figure 8: Time varying noise trading. This figure illustrates how different amount of short-run and long-run noise trading, τU(1) and τU(2), affect the long-run price discovery, τP(2). Three levels of noise trading are illustrated: τU(t) ∈ {2.0, 4.0, 6.0}. Panel (a) varies τU(1) while fixing τU(2) = 4.0. Panel (b) τ τ = . varies U(2) while fixing U(1) 4 0. The√ other primitive parameters used in this numerical illustration are: τV = 1.0, γ = 0.3, gτ = 1.0, and kτ (m) = m. on how advancement in speed technology would affect market quality. Most notably, while speed technology, by definition, always speeds up the price discovery process, it can hurt the magnitude— the investors’ incentive to acquire information. Following the same intuition, an advancement in information technology can hurt investors’ incentive to acquire speed, hence slowing down the price discovery process.

35 Appendix

A A continuous time model

The model setup in Section 2 allows only two trading rounds and, hence, investors’ speed investment is a binary choice. This appendix relaxes this restriction and allow investors to continuously invest in the speed technology.

Model setup. The continuous time version of the model directly builds on the two-round model presented in Section 2. There are three key departures. First, the trading phase is expanded to a

continuous time interval T = (0, t¯]. Second, an investor i can invest ni units of the numéraire in

the speed technology to trade early at time ti, such that ( ) ti = kt gt ni

where the speed production function kt (·) is twice differentiable, convexly decreasing on R+, and

satisfies kt (0) = t¯ and limn→∞ kt (n) = 0; and gt (> 0) is the exogenous parameter measuring how advanced the speed technology is. Without loss of generality, set t¯ = 1. Finally, in the continuous ∈ T time, the amount of noise trading at each round t shall be rewritten as dU(t), which∫ is i.i.d. t over time and normally distributed with zero mean and variance τ−1dt. Thus, U(t) = dU(t) √ U 0 becomes a Brownian motion scaled by 1/ τU. The rest of the setup remains the same. The continuum of investors choose their technology

investment (ti, τi) at t = −1 to maximize their ex ante certainty equivalent. At t = 0, they each submit a demand schedule based on their private signal. At each point t during the trading phase T = (0, 1], the speed-t investors’ demand schedules are pooled together with noise trading dU(t) and cleared by a competitive market maker at the efficient price.

Equilibrium analysis. As in the two-round model, the equilibrium is found by first solving investors’ optimal demand schedule at t = 0 and then the optimal technology investment at t = −1. Similar to Lemma 1, the following result follows the standard analysis in Vives (1995).

Lemma 4. For any technology pair (ti, τi), an investor i has optimal linear demand schedule τ ( ) , = i − , xi (p si) γ si p and his certainty equivalent at the time of technology investment (t = −1) is ( ) 1 τi π(ti, τi) = ln 1 + − kτ (τi) − kt (ti), 2γ τP(ti)

36 −1 where the price efficiency τP(t) := var[V | {P(r)}∀r≤t ] has the following dynamics: ∫ (∫ ) r2 τ 2 τ τ − τ = j U P(r2) P(r1) γ dj r1 j∈I(t) dt −1 for 0 ≤ r1 < r2 ≤ 1 =: t¯, with the initial condition τP(0) = var[V] = τV.

Compared to Lemma 1, the only difference is the dynamics of price efficiency τP(t), which, thanks to the following lemma, will be a differentiable function in time t in equilibrium. , Lemma 5. In equilibrium, the unity mass of investors distribute∫ on [t0 1] for some endoge- ∈ , µ = µ nous t0 [0 1) with strictly positive density (t) defined by i∈I(t) di (t)dt. Since the density µ(t) exists, the aggregate signal precision at time t can be rewritten as ∫

τjdj = τ¯(t)µ(t)dt, j∈I(t) whereτ ¯(t) is the speed-t investors’ average signal precision. Lemma 5 ensures that the price

efficiency τP(t) is differentiable and allows to rewrite its dynamics as τ (A.1) τ˙ (t)dt = τ (t + dt) − τ (t) = U µ(t)2τ¯(t)2dt, P P P γ2

It also follows thatτ ˙P(t) is strictly positive for t ∈ [t0, 1]. To maximize his certainty equivalent, investor i optimally chooses his information technology according to the same first-order condition as before:

1 gτ k˙τ (m ) (A.2) i − 1 = 0. 2γ τP + gτ kτ (mi) It follows that all speed-t investors will spend the same amount of money to acquire the same level of information technology τi = τ¯(t) = gτ kτ (m(t)) as the unique solution to the above first-order condition. Thus, equation (A.1) together with the initial condition τP(0) = τV yields ∫ ∫ τ t τ 2 t U 2 2 Ugτ 2 2 (A.3) τ (t) = τ + µ(t) τ(t) dt = τ + µ(t) kτ (m(t)) dt. P V γ2 V γ2 t0 t0

In particular, since there is no investor acquiring speed ti ∈ (0, t0), it follows that τP(t) = τV,

∀t ∈ (0, t0]. From the first-order condition (A.2), the optimal information investment by the fastest

investors (with speed t = t0) is some m0 up to primitive parameters τV, γ, gτ, and kτ (·):

gτ k˙τ (m ) (A.4) 0 = 2γ. τV + gτ kτ (m0)

Hence, their information acquisition τ0 = gτ kτ (m0) is also a constant, irrespective of the endoge-

nous t0 to be determined.

37 In equilibrium, all investors must have the same ex ante certainty equivalent, π(t); that is, ( ) 1 τ(t) (A.5) π(t) = ln 1 + − m(t) − n(t), ∀t ∈ [t0, 1], 2γ τP(t) = −1 where n(t) kt (tgt ) is the investment in the speed technology to trade at time t. It remains to solve for t0 and µ(t) (∀t ∈ [t0, 1]), such that the indifference condition (A.5) is satisfied via the

dynamics of τP(t) implied by equations (A.2) and (A.3). Proposition 9 (The continuous time equilibrium). There exists an equilibrium P in which, ∀t ∈

[t0, 1], a density of µ(t) of investors invest in ti = t = kt (n(t))/gt and τi = τ(t) = gτ kτ (m(t)),

such that {t0, µ(t), τ(t)} satisfy the following: ˙ 1 gτ kτ (mi) − = , ∀ ∈ , Optimal information acquisition: γ τ + 1 0 t [t0 1]; 2 P gτ kτ(mi) ) 1 τ Indifference in speed: π(t) = ln 1 + 0 − m − n(t ), ∀t ∈ [t , 1]; 2γ τ 0 0 0 ∫ V 1 Population size identity: µ(t)dt = 1; t0

where π(t) and τP(t) are given in equations (A.5) and (A.3), respectively.

Numerical illustration. The results shown in the two-round model remain robust in the continu- ous time. To avoid repetitiveness, these results are only numerically illustrated in Figure 9 and 10. For the purpose of this numerical illustration, the speed technology function is parametrized as √ ti = 1/(1 + gt ni) and the information technology as τi = gτ mi. Panel (a) of Figure 9 illustrates the equilibrium distribution of investors according to their speed choices t, under three levels of speed technology. As speed technology rises (from “slow” to “fast” and eventually to “ultra fast”), more and more investors acquire faster speed technology. This also

pushes the endogenous fastest speed t0 to 0, closer to “instantaneous”. Panel (b) shows the information acquisition by individual investors. While in the short-run (small t, e.g. t = 0.5), faster investors tend to acquire less information due to intratemporal competition, in the long-run (large t, e.g. t → 1.0) this is not necessarily the case. Toward the long- run (t → 1.0), investors acquire the least amount of information under “slow” technology, followed by “ultra fast”, and then “fast”. This is because of the intertemporal crowding out effect: As shown

in Panel (c), under “fast” and “ultra fast” speed technology, the price efficiency τP(t) shoots up very quickly in the short-run, but the increment loses its “momentum” toward the long-run. Indeed, investors with relatively slow speed (large t) find it less meaningful to further acquire information if τP(t) is already high. Instead, under the “slow” speed technology, the price efficiency τP(t) starts

38 (a) 1 t t Cumulative population, Rt µ( )d 1.0 0 Ultra fast 0.8 Fast Slow

0.6

0.4

0.2

Time, t 0.0 0.0 0.2 0.4 0.6 0.8 1.0

(b) (c)

τ t Price efficiency, τ (t) 0.9 Individual information acquisition, ( ) P 8

0.7 6 Ultra fast Ultra fast . 0 5 Fast 4 Fast Slow Slow

0.3 2 τ V Time, t Time, t 0.1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Figure 9: Dynamics in the continuous time model. This figure plots how investors are distributed along the speed dimension in Panel (a), their information acquisition in Panel (b), and the resulting price efficiency in Panel (c). Three different speed levels are plotted: dot-dash lines for “Ultra fast”, dashed lines for “Fast”, and solid lines for “Slow”. The numerical illustration√ is performed under the following parametrization: τV = 1.0, τU = 4.0, γ = 0.3, gτ = 1.0, kτ (m) = m, and kt (n) = 1/(1+n). The speed technology parameter 3 is set to gt = 1.0 × 10 for “Ultra fast”, gt = 10.0 for “Fast”, and gt = 1.0 for “Slow”.

39 (a) Population size (b) Individual information acquisition Population size 1.0 Individual information acquisition, τ t +t ( 0 1)/2 s s Rt µ( )d , fast 0.8 0 0.7

0.6 0.6 Fast

0.5 0.4

t . 1 s s 0 4 R t +t µ( )d , slow 0.2 ( 0 1)/2 Slow 0.3 Speed technology, gt 0.0 Speed technology, gt 100 101 102 103 100 101 102 103

(c) Price efficiency (d) The real impact of speed τ Price efficiency, P Firm’s expected cash flow

0.42 6

0.41 Long-run

4 0.40 1% = r 0.39 r Short-run = 2 5% r = 2% τ 0.38 V Speed technology, gt Speed technology, gt 0 0.37 100 101 102 103 100 101 102 103

Figure 10: Varying the speed technology (the continuous time model). This figure qualitatively replicates the patterns shown in Figure 2 under the continuous time model. The short-run (fast) and the long-run (slow) time distinction is defined as before and after the midpoint (t0 + 1)/2. That is, an investor is fast if ti ∈ [t0, (t0 + 1)/2] and is slow if ti ∈ ((t0 + 1)/2, 1]. Panel (a) shows the (cumulative) population sizes. Panel (b) shows an average fast and slow investor’s information acquisition. Panel (c) shows the average price discovery in the short-run (t ∈ [t0, (t0 + 1)/2]) and in the long-run (t ∈ [(t0 + 1)/2, 1]). Finally, Panel (d) shows the impact on the expected cash flow of a firm, whose discount rate varies as r ∈ {0.01, 0.02, 0.05}. τ = . τ = . γ = . The numerical illustration√ is performed under the following parametrization: V 1 0, U 4 0, 0 3, gτ = 1.0, kτ (m) = m, and kt (n) = 1/(1 + n). The speed technology parameter gt ranges from 1.0 to 1.0 × 103. 40 to grow relatively late (t0 ≈ 0.4), but its “momentum” remains throughout the entire price discovery process. This way, the eventual magnitude of price efficiency τP(1) turns out to be the highest. As a robustness check, Figure 10 “reproduces” Figure 2 and 7 by redefining the continuous = ∈ T = , + / time from t0 to t¯ 1 as two periods: a short-run for t fast : [t0 (t0 1) 2] and a[ long-run for] ∈ T = + / , E τ ∈ T t slow : ((t0 1) 2 1]. Thus, the average[ information] acquisition is defined as (t) t j E τ ∈ T ∈ { , } and the average price discovery as P(t) t j for j fast slow , where the conditional expectations are taken on the investor space. It can be seen from Panels (a) through (c) that the patterns shown in Figure 2 remain robust under the continuous time setting. Panel (d) qualitatively reproduces Figure 7. Specifically, the firm introduced in the extension of Section 4 can choose

between investing at t ∈ {(t0 + t¯)/2, t¯}. Additional results on other market quality measures (volatility, volume, market depth, investors’ gains and losses) also remain robust and are omitted for brevity.

B Proofs

Lemma 1 and 4

Proof. The proof follows Vives (1995) and is omitted here. □

Lemma 2

Proof. Consider the thresholdg ˆt, at which the benefit of investing in speed to trade at t = 1 is just small enough, so that the marginal investor is just willing to stay slow. Therefore, at this threshold µ(1) = 0 and µ(2) = 1, implying ( ) π = 1 + gτ kτ (m(1)) − − 1 (1) γ ln 1 τ m(1) ; 2 ( V ) gˆt 1 gτ kτ (m(2)) π(2) = ln 1 + − m(2), 2γ τP(2) 2 2 2 ∗ where τP(2) = τV + τUgτ kτ (m(2)) /γ . In equilibrium, it has to be such that π(1) = π(2) = π , which implies τ τ P(2) > V τP(2) + gτ kτ (m(2)) τV + gτ kτ (m(1))

41 because m(1) + 1/gˆt > m(1) > m(2). Subtract by 1 on both sides and rearrange to get

kτ (m(2)) < kτ (m(1)) . τP(2) + gτ kτ (m(2)) τV + gτ kτ (m(1)) Next, from the expression of π(1), by envelope theorem, ∂π∗ ∂ = 1 kτ (m(1)) + 1 gˆt . ∂ τ γ τ + τ τ 2 ∂ τ g 2 V g k (m(1)) gˆt g Similarly, from the expression of π(2), ( ) ∗ 2 2 ∂π 1 1 2τgτ kτ (m(2)) = 1 − kτ (m(2)) 2 ∂gτ 2γ τP(2) + gτ kτ (m(2)) γ τP(2) ∂π∗ ∂ < 1 kτ (m(2)) < 1 kτ (m(1)) = − 1 gˆt . γ τ + τ τ γ τ + τ τ ∂ τ 2 ∂ τ 2 P(2) g k (m(2)) 2 V g k (m(1)) g gˆt g

Therefore, ∂gˆt/∂gτ < 0. Further, consider the extremes of gτ ↓ 0 and gτ ↑ ∞. Toward the lower bound 0, from the expression of π(1) it can be seen that the first term in π(1) drops down to zero. Since an investor always has the option not to trade, π(1) is bounded below by zero. This leads to m(1) ↓ 0 and / ↓ = ∞ π 1 gˆt 0, implying limgτ↓0 gˆt . On the other hand, the first-order condition (5) applied to (1) implies ( ) k˙τ (m(1)) τV 1 0 = − kτ (m(1)) − < − m(1) k˙τ (m(1)), 2γ gτ 2γ

where the inequality follows because τV/gτ > 0 and because kτ (m) ≥ k˙τ (m)m by concavity.

Hence, m(1) is always bounded from above by 1/(2γ). From the first-order condition, with τP(1)

fixed at τV, it follows the concavity of kτ (·) that m(1) monotone increases in gτ, and so does kτ (m(1)). Taken together, ( ) 1 gτ kτ (m(1)) 1 1 lim π(1) > lim ln 1 + − − lim . gτ↑∞ 2γ gτ↑∞ τV 2γ gτ↑∞ gˆt > π If limgτ↑∞ gˆt 0, then the above limit of (1) shoots to infinity. In that case, the assumed equilibrium will not hold, however, because all slow investors will have incentive to move acquire / = speed by paying 1 gˆt to earn inifinite profit. Therefore, it has to be the case that limgτ↑∞ gˆt 0.

Finally, the above concludes thatg ˆt is a strictly decreasing function in gτ, withg ˆt (0) → ∞ and

gˆt (∞) → 0. As the strict monotonicity implies invertibility, there existsg ˆτ (gt ) for all gt ∈ (0, ∞)

such that the equilibrium is interior if and only if gτ ≥ gˆτ (gt ). □

42 Lemma 3

Proof. The lemma directly follows in the paragraphs proceeding its statement. □

Lemma 5

Proof. The proof needs to establish two results. First, there are no “holes” on the support [t0, 1]. A

“hole” is defined as a non-empty interval h ⊂ (t0, 1] such that 1) ∀t ∈ h, the investor set I(t) = ∅; and 2) I(t) , ∅ for t ∈ {inf h, sup h} (where I(t) denotes the set of all investors who acquire the

same speed t). If such h exists, then the marginal investor at ti = inf h will be strictly better off if = −1 / − −1 / he reduces his speed to ti sup h to save the speed cost of kt (inf h) gt kt (sup h) gt. This is

because i) without investors, there is no price discovery over a hole, τP(inf h) = τP(sup h); because

ii) the optimal choice of mi, as a function of τP determined by the first-order condition (A.2), is also

a constant if τP does not vary; and because iii) the investor’s expected profit (3) is strictly increasing

in ti when τP is held fixed. This no hole result implies that the density µ(t) is strictly positive on

[t0, 1]. Second, there is no “block” of investors, either. A block of investors is defined as a time point b ∈ (t0, 1] if the investor set I(b) has strictly positive mass (not density). Such a block b generates a discontinuous jump in the price discovery by t = b: By Lemma 4, τP(b) −limt↑b τP(t) >

0. Such a jump hurts all investors in that block (ti = b) as investors’ expected profit (3) is

strictly monotone decreasing in τP. Because the speed technology is assumed to be continuous, −1 / − −1 / = π − π > limt↑b kt (t) gt kt (b) gt 0 and limt↑b (t) (b) 0. That is, all investors in that block have incentive to speed up to just before b. Note that this no block result immediately implies t0 < 1, as otherwise a block is formed at t = 1. This completes the proof. □

Proposition 1

Proof. In equilibrium, all investors of the same speed t acquire the same amount of information m(t). By concavity and monotonicity of kτ (·), the first-order condition implies that m(t) must be mono-

tone decreasing in the round-t price efficiency τP(t), which by construction is non-decreasing over

time, i.e. τP(1) ≥ τP(2). Therefore, m(1) ≥ m(2) and also gτ kτ (m(1)) ≥ gτ kτ (m(2)). □

Proposition 2

Proof. The proof begins by writing investors’ certainty equivalent π(1) and π(2) as functions of a the fast population size µ(1)in[0, 1]. To do this, first note that from the first-order condition (5),

43 investors’ endogenous choice of m(t) can be written as a monotone function of τP(t). By Lemma 1, 2 2 2 2 τP(1) = τV +∆τP(1) and τP(2) = τV +∆τP(2), and ∆τP(t) = τU µ(t) gτ kτ (m(t)) /γ . Hence, τP(1) is

effectively a function of µ(1), while τP(2) of both µ(1) and µ(2). Finally, note that µ(1) + µ(2) = 1. As such, investors’ certainty equivalent π(t) are functions of µ(1). Then, depending on µ, there are three cases.

Case 1: First, suppose µ(1) = 1 and µ(2) = 0; i.e. all investors pay the speed technology cost 1/gt and become fast. Suppose this were the case, then in equilibrium it must be the case that π(1) ≥ π(2). Consider an investor i’s unilateral deviation to not investing in the speed

technology, saving the cost of 1/gt and becomes slow. By equation (4), the price efficiency

must remain the same, τP(1) = τP(2), in this case because a single investor’s deviation has

zero population measure (µ(2) = 0 unaffected). Then i’s optimal technology investment τi,

by the first-order condition (5), remains the same as if he were fast: τi = τ(τP(2)) =

τ(τP(1)) = τ(1). As a result, his certainty equivalent π(2) = π(1) + 1/gt > π(1) and he will deviate. Case2: Second, consider the case of µ(1) = 0 and µ(2) = 1. This will correspond to the corner equilibrium stated in the proposition. If this is an equilibrium, it has to be the case that π(1) ≤ π(2), i.e. all investors stay slow. The argument below shows that fixing all other

exogenous parameters, π(1) ≤ π(2) holds if and only if gt < gˆt for some thresholdg ˆt.

At µ(1) = 0, τP(1) = τV < τP(2) and thus a slow investor’s unilateral deviation to becoming fast yields ( ) 1 τ(1) 1 π(1)|µ(1)=0 = ln 1 + − m(1) − 2γ τV gt

where τ(1) = gτ kτ (m(1)) and m(1) is the unique solution implied by the first-order τ = τ ∂π /∂ = / 2 > condition (5) with P V. By envelope theorem, (1) gt 1 gt 0. Therefore, π µ = π = −∞ < < π < (1; (1) 0) is monotone increasing in gt with limits limgt ↓0 (1) 0 (2) π π | limgt ↑∞ (1). (Note that (2) µ(1)=0 is a finite number unaffected by gt.) By continuity, therefore, there exists a uniqueg ˆt such that π(1) = π(2) when µ(1) = 0. As such,

π(1) ≤ π(2), supporting µ(1) = 0 and µ(2) = 1, if and only if gt ≤ gˆt. When instead

gt > gˆt, this corner equilibrium does not exist. Case3: Third, consider the interior case of µ(1) ∈ (0, 1), implying π(1) = π(2). The key is to show the following result: The difference π(1) − π(2) strictly decreases in µ(1). Evaluate the partial derivative of π(1) − π(2) with respect to µ(1) and after some simplification, [ ] ∂(π(1) − π(2)) τ(2)/τ (2) τ(1)/τ (1) ∂τ (1) τ(2)/τ (2) ∂∆τ (2) · 2γ = P − P P + P P . ∂µ(1) τP(2) + τ(2) τP(1) + τ(1) ∂µ(1) τP(2) + τ(2) ∂µ(1)

44 Note that the term in the square-brackets is non-positive, because τP(2) ≥ τP(1) by

construction and because τ(t) = τ(τP(t)) decreases in τP(t) as implied by the first-order condition (5).

One still needs to sign both ∂τP(1)/∂µ(1) and ∂∆τP(2)/∂µ(1). To do so, rearrange the first-order condition (5) for t = 1 as

1 ( ) gτ τV + ∆τP(1) + gτ kτ (m(1)) = k˙τ (m(1)) 2γ 2 2 2 2 with ∆τP(1) = µ(1) gτ kτ (m(1)) τU/γ following equation (4). It can then immediately be concluded that m(1) strictly decreases in µ(1), as otherwise the left-hand side of the above equation is always increasing in µ(1), unable to maintain the equality. (Recall that kτ (·)

is concavely increasing.) Similarly, it is also known that τP(1) (= τV + ∆τP(1)) decreases

in m(1). Hence, τP(1) (and ∆τP(1)) increases in µ(1). For t = 2, 1 gτ (τV + ∆τP(1) + ∆τP(2) + τ(2)) = k˙τ (m(2)) 2γ 2 2 2 2 with ∆τP(2) = (1 − µ(1)) gτ kτ (m(2)) τU/γ . Note that ( ) ∂∆τ (2) ∂τ(2) τ P = −2(1 − µ(1))τ(2)2 + 2(2 − µ(1))2τ(2) U . ∂µ(1) ∂µ(1) γ2

As such, if ∆τP(2) increases in µ(1), then it has to be the case that ∂τ(2)/∂µ(1) > 0. Because τ(2) = gτ kτ (m(2)), m(2) is also increasing in µ(1). It then follows that the

left-hand side of the above equation strictly increases in µ(1)—∆τP(1), ∆τP(2), and m(2)

all increase with µ(1), invalidating the equality. Therefore, it must be ∆τP(2) decreases in µ(1).

As τP(1) increases in µ(1) but ∆τP(2) decreases in µ(1), one can conclude from the above partial derivative that the difference π(1) − π(2) indeed strictly decreases in µ(1). To sum up, recall from the first cases that at µ(1) = 1, π(1) < π(2). From the second case,

at µ(1) = 0, π(1) > π(2) if and only if gt > gˆt. Hence, when gt ≤ gˆt, the equilibrium with interior µ(1) does not exist due to the above monotonicity of π(1) − π(2) in µ(1).

When gt > gˆt, there exists a unique µ(1) ∈ (0, 1) such that π(1) = π(2), sustaining this equilibrium. This completes the proof of this proposition. □

45 Proposition 3

Proof. In the interior equilibrium, π(1) − π(2) = 0 and, following the proof of Proposition 2, the equality implies an implicit function of µ(1) in terms of the speed technology gt, which implies: µ ∂π /∂ d (1) = − (1) gt dgt ∂(π(1) − π(2))/∂µ(1) where the denominator of the fraction is negative as shown in Case 3 of the proof for Proposition 2. / 2 > µ □ The numerator equals 1 gt 0 by envelope theorem. Therefore, (1) increases in gt.

Proposition 4

Proof. In the interior equilibrium, the first-order condition (5) for t = 1, together with τP(1) = 2 2 2 τV + τUτ(1) µ(1) /γ , implies an implicit function of τ(1) = gτ kτ (m(1)) and µ(1), from which ∂τ(1) 2τ µ(1)τ(1)2/γ2 = − U < 0, ∂µ 2 2 (1) 2τU µ(1) τ(1)/γ + 1 − k¨τ (m(1))/k˙τ (m(1))

where the inequality follows because kτ (·) is concave. From Proposition 3, ∂µ(1)/∂gt > 0.

Therefore, by chain rule, ∂τ(1)/∂gt < 0. The first-order condition (5) also implies that τP(1)

decreases with m(1); hence, also with τ(1), yielding ∂τP(1)/∂gt > 0. □

Proposition 5

Proof. In the interior equilibrium, the first-order condition (5) for t = 2 always holds. Recall 2 2 2 2 2 2 τP(2) = τV + τUτ(1) µ(1) /γ + τUτ(2) µ(2) /γ . By implicit function theorem, it implies ∂τ ∂τ τ µ(2)τ(2)2 − µ(1)τ(1)2 − µ(1)2τ(1) (1) (2) = −4 U ∂µ(1) . ∂µ γ ¨τ 4τ (2) − k (m(2)) + 2γ + U µ(2)2τ(2) k˙τ (m(2)) γ As done in the proof of Proposition 4, the idea is to first sign the above partial derivative and then

sign ∂τ(2)/∂gt using chain rule: ∂τ ∂τ ∂µ ∂µ (2) = (2) (2) (1), ∂gt ∂µ(2) ∂µ(1) ∂gt

where ∂µ(2)/∂µ(1) = −1 following the identity µ(1) + µ(2) = 1 and ∂µ(1)/∂gt > 0 following Proposition 3.

Consider the limits of ∂τ(2)/∂µ(2) as gt ↑ ∞ and gt ↓ gˆt, respectively. To evaluate of these limits, one needs to show that τ(1), τ(2), and ∂τ(1)/∂µ(1) are have finite bounds. The finite bounds for τ(t) can be easily established by noting from the first-order condition (5)

46 that τ(t) in equilibrium is monotone decreasing in τP(t). From Lemma 1, it is known that τP(t) has strictly positive lower bound τV. Therefore, both τ(1) and τ(2) have finite upper bounds. (They also have lower bounds of zero by construction.) Finally, from the expression of ∂τ(1)/∂µ(1) derived in the proof of Proposition 4, it can be seen that ∂τ(1) 2τ µ(1)2τ(1)/γ2 µ(1) = − U τ(1) > −τ(1) ∂µ 2 2 (1) 2τU µ(1) τ(1)/γ + 1 − k¨τ (m(1))/k˙τ (m(1)) is also bounded.

Now the limits can be evaluated. When speed technology gt ↑ ∞, almost all investors become fast and µ(2) ↓ 0 and −µ τ 2 − µ 2τ ∂τ(1) ∂τ(2) 4τ (1) (1) (1) (1) ∂µ(1) lim = − U > 0. µ ↓ ∂µ γ − ¨τ (2) 0 (2) k (m(2)) + 2γ k˙τ (m(2))

Similarly, when speed technology gt ↓ gˆt, almost all investors stay slow and µ(1) ↓ 0 and ∂τ(2) 4τ µ(2)τ(2)2 lim = − U < 0. µ ↓ ∂µ γ ¨τ 4τ (1) 0 (2) − k (m(2)) + 2γ + U µ(2)2τ(2) k˙τ (m(2)) γ

As the above shows, for sufficiently large (low) gt, τ(2) increases (decreases) in µ(2) and hence

decreases (increases) in gt by the chain rule expression above. The first-order condition (5) implies

that τP(2) decreases with τ(2) and the stated results are proved. □

Proposition 6

Proof. Lemma 2 establishes the existence of a lower boundg ˆτ for gτ, such that the equilibrium is interior if and only if gτ ≥ gˆτ. In particular, when gτ ↓ gˆτ, the marginal investor is just indifferent between becoming fast or not, implying µ(1) = 0. When gτ is slightly aboveg ˆτ, the equilibrium is interior with µ(1) > 0. By continuity, therefore, as the information technology improves fromg ˆτ, initially the population size µ(1) is increasing. To complete the proof, the argument below establishes that in the limit of gτ → ∞, µ(1) ↓ 0 again and, therefore, by continuity, eventually the µ(1) decreases with gτ. Conjecture that in this limit, instead, µ(1) > 0 and then this conjecture will be falsified by showing it will not support the interior equilibrium. Note from fast investors’ first-order condition (equation 5) for information acquisition: 1 1 1 k˙τ (m(1)) > k˙τ (m(1)) − τP(1) = kτ (m(1)) ≥ kτ (0) + m(1)k˙τ (m(1)) = m(1)k˙τ (m(1)) 2γ 2γ gτ

where the first inequality holds because τP(1) ≥ τV > 0 and the last inequality holds by concavity

47 of kτ (·) and by kτ (0) = 0. Therefore, there exists an upper bound for m(1) ≤ 1/(2γ), an upper bound for kτ (m(1)) ≤ kτ (1/(2γ)), and a lower bound for k˙τ (m(1)) ≥ k˙τ (1/(2γ)) > 0. Rearrange the first-order condition:

gτ k˙τ (m(1)) k˙τ (1/(2γ)) τ (1) = − τ(1) ≥ gτ − kτ (1/(2γ)) P 2γ 2γ where the inequality follows the lower and the upper bounds above. Take limit on both sides of the inequality to get

k˙τ (1/(2γ)) lim τP(1) ≥ lim gτ − kτ (1/(2γ)) = ∞. gτ↑∞ 2γ gτ↑∞ τ τ = τ + U µ 2τ 2 τ = ∞ Given tyat P(1) V γ2 (1) (1) , the above limit implies that limgτ↑∞ (1) . The ratio of

τ(1)/τP(1) thus converges to zero as τP(1) grows quadratically in τ(1), when gτ ↑ ∞. Now take limit on π(1): ( ) 1 τ(1) 1 1 lim π(1) = lim ln 1 + − lim m(1) − ≤ − < 0 gτ↑∞ 2γ gτ↑∞ τP(1) τ↑∞ gt gt where the first inequality holds because, as shown above, the log term converges to zero and because m(1) ≥ 0. Consequently, this cannot be an equilibrium because fast investors are strictly better off if they refrain from participating in the market, in which case they will at least get π = 0. The conjecture of µ(1) > 0 is thus falsified, leaving the alternative of µ(1) = 0 in the limit of gτ ↑ ∞. This completes the proof. □

Proposition 7

Proof. For notation simplicity, the time argument t in this proof will be indicated only as a subscript.

The first-order condition (5) can be written as k˙τ (mt )/(2γ) − kτ (mt ) = τPt/gτ which uniquely solves mt. (Second order condition clearly holds.) For later use, the following comparative statics will be useful. Holding gτ (and γ) constant, ( )−1 ∂mt 1 k¨τ (mt ) = − k˙τ (mt ) ≤ 0 ∂τPt gτ 2γ

where the inequality follows the concavity of kτ (m). Note that this also implies that m1 ≥ m2 because τP2 ≥ τP1 by construction. In addition, ( )−1 ∂mt τPt k¨τ (mt ) τPt ∂mt = − − k˙τ (m ) = − ≥ 0. 2 t ∂gτ gτ 2γ gτ ∂τPt

By construction, τP1 = τV + ∆τP1 and τP2 = τV + ∆τP1 + ∆τP2. From the first-order condition,

48 one can write m1 and m2 as implicit functions of m1(∆τP1) and m2(∆τP1, ∆τP2). Further, ∆τPt = τ 2 2 µ2/γ2 Ugτ kτ (mt ) t , which implies √ γ ∆τPt µt = √ . τU gττ(mt )

Therefore, the equilibrium can be pinned down by a two-equation, two-unknown system: π1−π2 = 0

and µ1 + µ2 − 1 = 0. or, equivalently, defining a vector function F(∆τP1, ∆τP2; gτ), ( ( ) ) ( ( ) )  1 gτ kτ (m1) 1 1 gτ kτ (m2)     γ ln 1 + τ − m1 − − γ ln 1 + τ − m2     2 P1 gt 2 P2  0 F = √ √ √ = .  τ τ τ     ∆ P1 ∆ P2 U     + − gτ  0  kτ (m1) kτ (m2) γ 

Take total derivatives with respect to gτ on the equilibrium condition F = 0 to get            τ      F11 F12 d∆ P1 F1g 0     +  dgτ =      τ      F21 F22 d∆ P2 F2g 0 One can easily evaluate, using envelope theorem, ( ) = 1 kτ (m1) − 1 kτ (m2) = 1 kτ (m1) − kτ (m2) > , F1g γ τ + γ τ + 0 2 P1 gτ kτ (m1) 2 P2 gτ kτ (m2) gτ k˙τ (m1) k˙τ (m2) where the last equality follows the first-order condition (5), while the inequality follows the concavity

of kτ (m), knowing that m1 > m2. Also, √ √ √ τ ∆τP1 ∂m1 ∆τP2 ∂m2 U F = − k˙τ (m ) − k˙τ (m ) − < 0, 2g 2 1 2 2 kτ (m1) ∂gτ kτ (m2) ∂gτ γ

where ∂mt/∂gτ is derived earlier to be positive. The elements in the Jacobian matrix can also be evaluated. Using envelope theorem,

kτ (m1) kτ (m2) F11 = − + ≤ 0 k˙τ (m1)τP1 k˙τ (m2)τP2

where the inequality holds because kτ (m1)/k˙τ (m1) ≥ kτ (m2)/k˙τ (m2) (concavity) and τP1 ≤ τP2. Similarly,

kτ (m2) F12 = > 0. k˙τ (m2)τP2

Now consider the partial derivatives with respect to F2: √ √ 1 ∆τP1 ∂m1 ∂τP1 ∆τP2 ∂m2 τP2 F21 = √ − k˙τ (m1) − k˙τ (m2) τ kτ m 2 ∂τ ∂∆τ kτ m 2 ∂τ ∆τ 2 ∆ P1kτ (m1) √( 1) P1 √P1 ( 2) P2 P1 1 ∆τP1 ∂m1 ∆τP2 ∂m2 = √ − k˙τ (m ) − k˙τ (m ) > 0 2 1 ∂τ 2 2 ∂τ 2 ∆τP1kτ (m1) kτ (m1) P1 kτ (m2) P2

49 because ∂mt/∂τPt ≤ 0 as shown before. Similarly, √

1 ∆τP2 ∂m2 ∂τP 2 F = √ − k˙τ (m ) > 0. 22 2 2 ∂τ ∂ τ 2 ∆τP2kτ (m2) kτ (m2) P2 ∆ P2

Now we can examine the signs of ∂τPt/∂gτ. By Cramer’s rule,

−F g F F −F g 1 12 11 1 −F F F −F ∂∆τP1 2g 22 ∂∆τP2 21 2g = and = . ∂gτ ∂g F11 F12 F11 F12

F21 F22 F21 F22

The sign of the denominator is easy to shown: F11F22 − F12F21 < 0. It remains to examine the

numerators. For τP1, this is also easy: −F1gF22 + F12F2g < 0; hence ∂∆τP1/∂gτ > 0.

Expand the numerator of ∂∆τP2/∂gτ to get (to simplify the notation further, the arguments of kτ (·) and k˙τ (·) are replaced by subscripts): ( ) ( ) ( ) ( ) 1 kτ1 kτ2 kτ1 kτ2 kτ1 F2g F21 kτ2 F2g F21 −F11F2g + F1gF21 = − F21 + − F2g = + − + gτ ˙ ˙ ˙ τ ˙ τ ˙ τ gτ ˙ τ gτ (kτ1 kτ2 ) k(τ1 P1 kτ)2 P2 ( kτ1 )( P1 ) kτ2 P2 F F F ≤ kτ1 2g + F21 − kτ2 2g + F21 = kτ1 − kτ2 2g + F21 τ τ τ k˙τ1 P1 gτ k˙τ2 P1 gτ k˙τ1 k˙τ2 P1 gτ ˙ ˙ where the last equality holds because F2g < 0 and τP2 ≥ τP1. By concavity, kτ1/kτ1 > kτ2/kτ2.

Hence, it remains to sign F2g/τP1 + F21/gτ: ( ) √ √ √ τ ∂ τ ∂ τ F2g F21 * ∆ P1 m1 ∆ P2 m2 U + + τ = − k˙τ − k˙τ − τ P1 2 1 ∂ 2 2 ∂ γ P1 gτ , kτ gτ kτ gτ - √1 √ 2 √ τ τ ∂ τ τ ∂ τ * P1 ∆ P1 m1 P1 ∆ P2 m2 P1 + + √ − k˙τ1 − k˙τ2 . τ τ 2 ∂τ gτ 2 ∂τ gτ ,2 ∆ P1 1 kτ1 P1 kτ2 P2 - Rearrange to get ( ) √ ( ) F2g F21 ∆τP1 ∂m1 ∂m1 ∆τP1 + τ = − k˙τ · + P1 2 1 τP1 gτ kτ ∂gτ ∂τP1 gτ √ 1 ( ) ( √ √ ) ( √ √ ) τ τ τ τ ∆τP2 ∂m2 ∂m2 ∆τP1 P1 U P1 U − k˙τ2 · + + √ − ≤ √ − , 2 ∂gτ ∂τ gτ τ τ γ τ τ γ kτ2 P2 2 ∆ P1 1 2 ∆ P1 1 τ ∂mt Pt ∂mt where the first two terms equate zero because previously it is shown that ∂ = − ∂τ . At last, gτ gτ Pt ( ) ( √ √ ) √ ( ) F F τ τ τ µ τ 2g + 21 τ ≤ √ P1 − U = U 1 P1 − . τ P1 γ γ τ 1 P1 gτ 2 ∆τP1τ1 2 ∆ P1

For sufficiently small τV, τP1/∆τP1 ↓ 1 and the right-hand side above yields an upper bound of zero.

50 That is, the second numerator is also negative for sufficiently small τV. Hence, ∂∆τP2/∂gτ > 0.

Taken together, ∂τP1/∂gτ = ∂∆τP1/∂gτ > 0 and, for sufficiently small τV, ∂τP2/∂gτ =

∂∆τP1/∂gτ + ∂∆τP2/∂>0. □

Proposition 8

Proof. Denote by N (·) the c.d.f. of standard normal distribution. Then the probability of investing early P(|Pˆ(1)| ≥ p¯) = 2(1 − N (p¯)), where the thresholdp ¯ can be rewritten as √ 1 τ ∆τ (2) p¯ := V P 1 − ϕ ∆τP(1) τV + ∆τP(1) + ∆τP(2)

The proof of Proposition 2 shows that ∂∆τP(1)/∂gt > 0 and ∂∆τP(2)/∂gt < 0. Therefore,

∂p¯/∂gt < 0 and, hence, P(|Pˆ(1)| ≥ p¯) is increasing with gt. Similarly, because Proposition 7

shows that both ∆τP(1) and ∆τP(2) are strictly increasing in gτ, the fractions in the square-root term above both decrease with gτ. Therefore, ∂p¯/∂gτ < 0 and P(|Pˆ(1)| ≥ p¯) also increases with gτ. □

Proposition 9

Proof. An equilibrium P is characterized by some t0 ∈ (0, 1) and functions µ(t) and τ(t) which satisfy the conditions stated in the proposition. To prove the existence of such an equilibrium,

consider the following algorithm to find {t0, µ(·), τ(·)}:

1. Conjecture an equilibrium t0 ∈ (0, 1). ∗ 2. Evaluate investors’ equilibrium certainty equivalent π at this conjectured t0. This is possible

as at t = t0, π(·) only depends on the choice of t0, following the second line of equation (A.5).

3. The investors’ indifference condition implies a solution for all τP(t), ∀t ∈ [t0, 1], as a function of t and π∗: ( ) τ π∗ = 1 + (t) − 1 τ − 1 γ ln 1 τ kτ ( (t)) kt (t) 2 ( P(t) )gτ gt 1 τ(τP(t)) 1 1 = ln 1 + − kτ (τ(τP(t))) − kt (t) 2γ τP(t) gτ gt

where the second line replaces τ(t) with the implicit function τ(τP(t)) implied by the first-

order condition (A.2). The implied solution τP(t) is unique, thanks to the convexity and

monotonicity assumptions on kt (·) and kτ (·).

4. The unique τP(t) implies the dynamicsτ ˙P(t).

51 τ τ = U µ 2τ 2 5. From equations (A.1) and (A.3), ˙P(t) γ2 (t) (t) , or √ √ γ τ˙ (t) γ τ˙ (t) µ(t) = √ P = √ P . 2 2 τU τ(t) τU τ(τP(t))

where the second equality uses the implicit function τ(τP) implied by the first-order condi- tion (A.2). ∫ 1 6. Evaluate the cumulative population sizeµ ˆ := µ(t)dt. and compare it to unity (the total t0 population size). An equilibrium is found ifµ ˆ = 1. If not, go back to the first step and try

another t0.

In the above algorithm, both τ(t) and µ(t) are parametrized by the initial guess t0: τ(t) is a unique ∗ function of τP(t), which is parametrized via π , which in turns is a function of t0; and similarly,

µ(t) is a function ofτ ˙P(t) and τ(t). As such, so is the cumulative population size:µ ˆ = µˆ(t0).

It remains to show that the above algorithm can always find a t0 ∈ (0, 1), such that the population

size identityµ ˆ(t0) = 1 holds. Observe that by construction,µ ˆ(1) = 0. Due to the continuity ofµ ˆ(·), µ > the proof reduces to show that if the left limit satisfies limt0↓0 ˆ(t0) 1. To do this, the steps below first construct a lower bound forµ ˆ(t0) and then show that the limit of the lower bound, as t0 ↓ 0, is strictly larger than 1. To begin with, observe that ∫ ∫ √ √∫ 1 γ 1 τ˙ (t) γ 1 τ˙ (t) µˆ(t ) = µ(t)dt = √ P dt ≥ √ P dt, 0 τ τ 2 τ τ 2 t0 U t0 (t) U t0 (t) where the inequality follows the fact that the square root of a sum is weakly larger than the sum √ of square roots (the subadditivity of ·). Recall from the first-order condition (A.2) that τ as an implicit function of τP is strictly decreasing in τP (due to the assumed convexity in kτ (·)). By construction, τP(t) is non-decreasing over time. Therefore, τ0 := τ(t0) ≥ τ(t), ∀t ∈ [t0, 1]. (Note that τ0 is independent of the choice of t0.) The above inequality therefore continues as √∫ γ 1 1 γ 1 √ µ ≥ √ τ = √ τ − τ , ˆ(t0) τ τ ˙P(t)dt τ τ P(1) V U 0 t0 U 0

where the last equality uses the initial condition τP(t0) = τV.

The above lower bound ofµ ˆ(t0) depends on t0 only via τP(1): At t = 1, kt (1) = 0 and ( ) ( ) ∗ 1 τ(τP(1)) 1 1 τ0 1 1 π = ln 1 + − kτ (τ(τP(1))) = ln 1 + − kτ (τ0) − kt (t0) 2γ τP(1) gτ 2γ τV gτ gt where the second equality uses the definition of π∗. Taking total derivative (and using the first-order

52 condition A.2) yields ( ) 2 ∂τP(1) 2γ τP(1) = k˙t (t0) + τP(1) < 0. ∂t0 gt τ(τP(1))

Therefore, the terminal price discovery τP(1) monotonically decreases in t0. From the equality ∗ above, in the limit of t0 ↓ 0, π ↓ 0 and it has to be that τP(1) ↑ ∞. Therefore, to sum up, γ 1 √ lim µˆ(t ) ≥ lim √ τ (1) − τ = ∞ > 1. ↓ 0 ↓ P V t0 0 t0 0 τU τ0

As such, by continuity, there exists at least one t0 ∈ (0, 1) such thatµ ˆ(t0) = 1, solving the algorithm described above. This completes the proof. □

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List of Figures

1 Time line of the game ...... 7 2 Varying the speed technology ...... 13 3 Varying the information technology ...... 17 4 Varying both technologies ...... 19 5 Speed technology and market quality ...... 21 6 Information technology and market quality ...... 22 7 The real impact of speed technology ...... 29 8 Time varying noise trading ...... 35 9 Dynamics in the continuous time model ...... 39 10 Varying the speed technology (the continuous time model) ...... 40

55