## Abstract

Much financial activity is zero-sum. While providing transactional and diversification services to others, participants also prey upon each other. High-ability predators trade opportunistically with less-able prey. In our dynamic model these features amplify real shocks. The presence of more low-ability traders reduces expected losses to high-ability traders, leading to equilibria with high levels of financial activity and employment. Shocks to profits can motivate exit by low-ability traders, rendering those of intermediate skill more vulnerable. Thus, our relatively simple model generates boom-bust dynamics suggestive of Wall Street. (JEL G00, G20, E44)

The set of markets and intermediaries known collectively as “Wall Street” exist to match borrowers and lenders, buyers and sellers, or investors and producers. It is not surprising, then, that we see changes in employment and volume in response to variation in the real economy, as the needs of borrowers and lenders change through time. The point of this paper is that the nature of competition between financial intermediaries may lead to fluctuations in employment and volume beyond their direct response to changes in the real economy.

Traders in financial markets not only create value by providing transactional services that improve risk sharing and reallocate capital to high-return activities, but they also extract value from each other through trades based on information or by exploiting temporary anomalies in prices. The securities industry is unusual relative to other service sectors in that so much of the activity is zero-sum in nature. Our model characterizes some of the consequences of this fact for levels and dynamics of participation and employment.

In our model traders differ in ability. New entrants to the industry learn their type through experience. If they learn that they have above average ability, they become predators, extracting from each trading encounter an unequal fraction of the surplus created whenever they are paired with lower ability types. When a larger fraction of the population of surviving traders are high-ability, other things equal, exit by incumbent low-ability traders increases. This further increases the vulnerability of any remaining low-ability traders, encouraging more exit, and so forth. In this way, the model generates predator-prey dynamics similar to those that have been long used to describe populations of interdependent species. We explore the conditions under which these forces can generate collapses in the population and trading activity in the industry, and the extent to which swings in employment or volume exaggerate the underlying dynamics in the shared surplus from trade.

We interpret the model in terms of employment cycles, but it could apply generally to other measures of trading activity and volume. If the trading that goes on is largely or predominantly zero-sum in nature, then our model would predict cascade-like crashes. We show through simulation that in some cases these crashes are followed by periods of gradual rebuilding. In others they lead to a permanently lower level of activity.

That much of the competition in financial market trading is zero-sum is evident in the high volume of trade in financial securities relative to the exogenous demand from outside the sector for transactional services. For example, volume in the foreign exchange markets vastly exceeds the demand for foreign exchange transactions generated by international trade of goods and services. According to the “FX Volume Survey” of the New York Fed, average annual volume in foreign exchange is between 140 and 200 trillion USD.1 Gross international trade flows are close to 20 trillion USD per year. Similarly, the volume of commodity contracts traded on exchanges grew by nearly 100% in the three years from 2005–2008, and by nearly 50% in the three-year period 2008–2010. This is beyond any conceivable growth in the underlying demand for transactional or hedging services by producers and consumers of the underlying physical commodities. Indeed, the total volume of silver mined throughout history is estimated to be slightly less than 47 billion ounces, whereas the turnover of exchange traded silver in 2010 was approximately 90 billion ounces.2 The high-frequency trading programs that have been so controversial in recent years are an obvious, if extreme, example of trading that is zero-sum in nature.

Our setting allows us to consider the effects of such behavior in a dynamic model. Each period surviving traders, knowing their ability level, decide whether to exit. These decisions are conditioned on knowledge of the aggregate surplus available through trade in the coming period. New entrants, who are uncertain about their ability, join the industry. Given the resulting mix of populations, traders are randomly matched. Traders of the same type who encounter each other equally share the surplus their trade generates. Traders of unequal ability asymmetrically share the gains to trade. Thus, for a given total population and aggregate surplus available, high-ability types expect to do much better if the low-ability types and new entrants comprise a larger proportion of the population.

We formulate the dynamic programming problem faced by the different types and solve for steady-state outcomes. There are often both high-population equililbria and low-population equilibria, because of the positive participation externalities created by entry and retention of low-ability traders. As with fish that form schools or birds that form flocks, the presence of many other similar traders reduces the chances any one of them will encounter a predator, and thus increases their continuation values. Thus, if a low-ability incumbent anticipates the survival of a large number of his peers, he in turn will choose not to exit. If he anticipates exit by his peers, he optimally does likewise.

Our model illustrates how small shocks to fundamentals can trigger a chain reaction in employment and trading activity. A shock that triggers exit by the lowest ability traders leaves those with intermediate ability exposed to greater risk of predation, even when the shock leaves their profits positive, holding fixed the composition of the population. We analytically characterize situations in which effects on the composition of the population amplify real shocks, and through simulation illustrate the cycles that can result.

We argue that the outcomes the model produces can help us understand the volatility of employment, participation, and trading volume in the securities industry. Financial activity shows dramatic contractions over the business cycle. Figure 1 shows employment over the last decade in three service sectors, all of which involve high levels of education and compensation; the data cover the two most recent recessions.3 The drop in employment in the securities business is quite dramatic in both of the past two downturns. Table 1 lists the percentage drop in employment from peak to trough, and the number of months from peak to trough, in each of the last two recessions for a range of financial services, business services, all service providers, and all goods providers. Services show less dramatic contractions than goods providers. Securities trading stands out among the highly professionalized services as having had dramatic contractions in both downturns, and these contractions, despite their severity, were completed relatively quickly. Figure 2 plots demeaned growth rates for employment in securities and manufacturing for New York State since 1990. In the early portion of the time period, the growth rates for manufacturing show more volatility than for the securities industry. The peaks and troughs for manufacturing tend to lie outside of those for securities. In the later decade, this relative volatility appears to be reversed.

Figure 1

Employment, in thousands of employees

Graph is based on monthly data from January 2001 to February 2011. Industry classification and employment statistics are taken from the U.S. Bureau of Labor Statistics.

Figure 1

Employment, in thousands of employees

Graph is based on monthly data from January 2001 to February 2011. Industry classification and employment statistics are taken from the U.S. Bureau of Labor Statistics.

Figure 2

Monthly growth rates in employment less their time-series average for manufacturing and the securities industry in New York State, January 1990 to August 2011

Figure 2

Monthly growth rates in employment less their time-series average for manufacturing and the securities industry in New York State, January 1990 to August 2011

Table 1

Employment change and months passed in different sectors from employment peak to trough. Legal services for 2001, listed “na,” are not available

% Decrease

Months

2001 2008 2001 2008
Securities, commodities, and investment 10.96 8.67 32 22
Commercial banking 0.51 4.41 22
Insurance 0.55 4.25 35 21
Funds, trusts, and others 7.13 5.36 20 26
Legal services na 5.95 na 38
Accounting and bookkeeping 11.34 10.07 22 35
Architecture and engineering 4.70 12.34 24 31
Computer systems 18.36 2.15 30 13
Management and technical consulting 3.10 2.35 17 20
All service providers 0.45 3.87 24
All goods producers 11.64 21.24 35 38
% Decrease

Months

2001 2008 2001 2008
Securities, commodities, and investment 10.96 8.67 32 22
Commercial banking 0.51 4.41 22
Insurance 0.55 4.25 35 21
Funds, trusts, and others 7.13 5.36 20 26
Legal services na 5.95 na 38
Accounting and bookkeeping 11.34 10.07 22 35
Architecture and engineering 4.70 12.34 24 31
Computer systems 18.36 2.15 30 13
Management and technical consulting 3.10 2.35 17 20
All service providers 0.45 3.87 24
All goods producers 11.64 21.24 35 38

The model is also consistent with patterns of entry in the industry. Even in periods of contraction, however, Wall Street continues to hire “new blood.” This is evident from the employment statistics published on the Web sites of leading business schools. For example, in 2008 and 2009, just under 50% of Wharton’s graduating MBA students took jobs in financial services. This actually exceeded the percentage entering that industry in 2005 and 2006. Layoffs have resumed more recently, as firms anticipate the effects of increased regulation on trading profits. Yet in November of 2011 the Wall Street Journal reported4: “With many firms trying to reduce pay by cutting highly paid staff, business students intent on a Wall Street career are continuing to find opportunities, although some schools are reporting a slowdown in interviews and less robust hiring than before the 2008 crisis.” In our model, new entrants are undifferentiated in ability, and learn about their suitability once they are on the job. Because the payoff for successful workers is very high, the real option motivates entry even if average ability is relatively low.

These outcomes in our model are examples of the sort of multiplicity of equilibria that often arise in the presence of what Chatterjee and Cooper (1989) call “positive participation externalities.” Such externalities play an important role in models of networks (Katz and Shapiro 1985), search in employment (Diamond 1982), and market microstructure (Pagano 1989). Similar mechanisms have been shown to help explain the evolution and persistence of norms of behavior as, for example, in Bond’s (2008) model of corruption.

We first solve for the steady-state equilibria analytically in a deterministic setting. This allows us to illustrate intuitively the basic qualitative results. The elasticity of the number of traders can become extreme through the participation externality. There are also generally multiple equilibria. The high-participation equilibria can be sustained only with a high level of (exogenous) industry profitability. A drop in profitability leads to cascade-like effects on the level of participation. Lower ability traders exit, rendering the environment less profitable to those of intermediate ability, who in turn are forced to exit. This unraveling of the high-participation outcome is reminiscent of the collapse of pooling equilibria due to adverse selection in settings such as insurance markets.

We then simulate the model under uncertainty about the level of aggregate surplus. Our simulations reveal a wide range of possible dynamic behaviors. With relatively low volatility to the fundamental shock, and a high initial proportion of low-ability types, the population is likely to grow steadily. With more volatility, shocks will occur that trigger exit by the incumbent low-ability traders, and the industry then remains stuck in the low-employment equilibrium. In other cases, in response to a negative shock, the population can contract to a point at which it becomes attractive for low-ability traders to remain active, followed by population growth. The population can then collapse when a sufficiently negative shock to profitability occurs. A period of low employment then follows, until a recovery in aggregate profits causes the population to begin rebuilding, at first quickly and then more gradually, towards the high-employment state. This growth may at any point be interrupted by a negative shock sufficient to cause a sudden return to the low-employment state. Thus, our relatively simple model generates boom-bust cycles that are quite suggestive of employment patterns in financial services.

We assume a particularly simple matching mechanism to describe the trading process, to illustrate the basic forces at work. Traders are randomly matched with others based on the (endogenous) distribution of ability levels in the population, and gains to trade are then split based on relative ability. Our results are qualitatively robust, however, to generalizations such as allowing traders to refuse a particular match. What matters is that when faced with a set of potential counterparties with higher average ability, lower ability traders anticipate losses on average.

## 1. Model Structure

Consider an environment in which overlapping generations participate in the financial sector. We will hereafter refer to these agents as “traders.” Traders act as intermediaries, providing transactional or advisory services to outside customers, but they also trade with each other. Individual traders in any given cohort are infinitesimal in size. A new generation of traders of measure $$M$$ is born each period. Older traders die off between periods at rate $$1-\beta,$$ and can also decide to exit. Alternatively, and perhaps more realistically, we can view their employers as deciding to terminate their employment.

Traders are endowed with differences in ability or aptitude for the job, denoted θ. Higher-ability traders have an advantage in trading with lower-ability traders. For any entering cohort, a mass $$f_0(\theta)$$ of the population is θ-ability, but individual traders do not learn their type until after they have experience on the job. Let $$N_t$$ denote the total (measure of the) population of traders at date $$t,$$ after exit choices have been made.

We denote as $$f_t(\theta)$$ the density of the population at date $$t$$ with ability level $$\theta.$$ Through the services they provide to outside customers, industry participants produce an aggregate surplus of $$\Pi(\text z_t,N_t,f_t).$$ Aggregate profits depend, in general, on the population size and distribution of ability, both endogenous outcomes, as well as an exogenous shock, $$z_t,$$ to demand for the financial services traders provide. For example, a larger population may increase aggregate profits because there will be more individuals soliciting business and servicing customers, and higher average ability could have similar effects. For the moment, we take the form of the aggregate profit function as given, because our focus is on how competition over the division of this surplus affects population dynamics. In Section 1.6, however, we describe a simple model for the interaction with customers that produces the functional forms we employ in our examples and simulations.

Along with their interaction with outside customers, traders must interact with each other: to rebalance inventories, locate end customers, obtain or provide liquidity, and exploit pricing anomalies or informational advantages. We assume the industry surplus is shared across traders as follows. At each date, each trader is randomly matched with a counterparty, and they divide a surplus

(1)
$$\pi(\text z_t,N_t,f_t)=\frac{\Pi(\text z_t,N_t,f_t)}{(\frac{N_t}{2})}.$$
The division of the gains to trade in any trading encounter depends on the ability of the two traders. If a type $$\theta_i$$ meets a type $$\theta_j,$$ the i-ability trader receives a share $$\frac{1}{2}+(\theta_i-\theta_j)$$ of the surplus. The case of symmetric ability, where $$\theta_i=\theta_j=\theta$$ provides a natural benchmark against which to compare the outcomes of interest.

Each trader also faces a per-period fixed cost of operating, denoted as $$c.$$ Note that all traders are the same “size” or capacity in this model, so that the operating cost is fixed. The aggregated cost of operations, $$N_tc,$$ varies linearly with the population. Ours is a model of industry equilibrium.5 Also, we allow $$\theta_i-\theta_j\gt \frac{1}{2}.$$ The lower-ability traders may anticipate being net losers when encountering a counterparty of superior skill, but if the higher-ability counter-parties are a sufficiently small portion of the population, or are expected to be in the future, it may still pay for the low-ability agents to participate.

We are assuming that once they have entered and gained experience in the industry, traders know their own type. Thus, the “learning” in the model is very simple. They also know (i.e., have rational expectations about) the distribution of ability in the rest of the population. Once randomly matched with a counterparty, we assume trade takes place despite the fact that one of the counterparties may be at a disadvantage. This can be interpreted as describing an institutional setting in which trade is intermediated by third parties, although we will show that the results are robust to alternative specifications for which traders can reject a given match. Such a structure might be viewed as closer to the institutional settings in which some financial intermediaries operate.

### 1.1 Payoffs

In this section we specify the periodic payoffs to the continuing traders, and to the new entrants. We then formulate their value functions.

Traders are randomly matched with counter-parties in any given period. The per-period expected payoff to a continuing type-θ trader, then, is:

(2)
$$\int^{\theta_H}_{\theta_L}f_t(q)\pi(z_t,N_t,f_t)\left[\frac{1}{2}+(\theta-q)\right]dq-c=\pi(z_t,N_t,f_t)\left[\frac{1}{2}+(\theta-\bar\theta_t)\right]-c,$$
where $$\bar\theta_t=\int^{\theta_H}_{\theta_L}f_t(\theta)\theta\,\,\,d\theta.$$ A new entrant’s expected payoff is as in Equation (2), with $$\theta=\bar\theta_0=\int_{\theta_L}^{\theta_H}f_0(\theta)\theta\,\,\,d\theta.$$

These expressions make clear that expected payoff to any one trader is decreasing in the average ability of the population. The lowest ability types are the most disadvantaged in competition with other traders, yet their exit, by raising average quality, makes things worse for the next lowest type and so on. This can lead to collapses in trading activity in response to an adverse shock, $$z_t.$$

### 1.2 Exit

Three state variables determine the decisions to continue or exit: the shock to per-period profits, $$z_t,$$ the population carried forward from the previous period, $$N_{t-1},$$ and distribution of types, $$f_{t-1},$$ the latter being an infinite-dimensional object. The shock is exogenous, and the population and distribution of types are endogenous. Each incumbent trader $$h$$ chooses either to stay or exit the market. In making these choices they take as given the exit decisions of others, and thus, because each individual is of zero measure, they treat the aggregate quantities $$N_t$$ and $$f_t$$ as given.

An incumbent with ability θ stays in the market if her periodic profit and continuation value exceed the outside alternative, which is normalized to be zero. The value function is thus given by:

(3)
$$\mathcal{v}(\theta;z_t,N_{t-1},f_{t-1})=\text{max}\left\{\pi(z_t,N_t,f_t)\left[\frac{1}{2}+(\theta-\bar\theta_t)\right]-c+\beta E_t[\mathcal{v}(\theta;z_{t+1},N_t,f_t)],0\right\}.$$

In the results reported below, we treat the arrival of a new generation of $$M$$ workers as exogenous and deterministic to simplify the analysis. Although this simplification allows us to focus on the phenomena of most interest, it could be easily generalized to allow for endogenous entry, or stochastic amounts of new entry.6 The model is trivially robust to endogenous entry if the measure of potential entrants is sufficiently small.

The policy choice for an agent of type θ is the probability of continuing, $$p_t(\theta).$$ When the distribution of types is discrete—or degenerate, as under symmetric ability—there may be mixed strategy equilibria, where $$0\lt p_t(\theta)\lt 1.$$ In this case, traders of a particular type will be indifferent to continuation or exit, but the fraction of the population continuing is determinate.

### 1.3 Laws of motion

The endogenous state variables evolve as follows. The population of traders combines the new entrants with the incumbents who survive exogenous attrition at rate $$\beta,$$ and who decide not to exit:

(4)
$$N_t=M+\int^{\theta_H}_{\theta_L}p_t(\theta)\beta f_{t-1}(\theta)N_{t-1}\,\,d\theta.$$
The density of traders with ability level θ, in turn, is
(5)
$$f_t(\theta)=\frac{Mf_0(\theta)+p_t(\theta)\beta f_{t-1}(\theta)N_{t-1}}{N_t}.$$

### 1.4 Equilibrium

All agents behave competitively, and treat the state variables as independent of their own decisions. Equilibrium requires that each of the policy functions is a best response to the others, given rational expectations about the evolution of the state variables.

### 1.5 Remarks

We would generally expect that the profitability per match, $$\pi(z_t,N_t,f_t),$$ would be declining in the size of the population, simply as a result of standard competitive forces and a downward-sloping demand for financial services. Aggregate trading profits, $$N_t\pi(z_t,N_t,f_t)/2,$$ may be increasing in $$N_t.$$ We would interpret this as positive network externalities, or as traders “drumming up more business.” If there is persistence in shocks to profitability, then, positive shocks should increase the size of the industry, and negative shocks should reduce it. The size of $$M$$ and the state of the incumbent population, $$N_t$$ and $$f_t,$$ control the speed with which the system adjusts to unexpected shocks to profitability. If the measure of new entrants is small, it might take multiple periods for new entry to eliminate rents to incumbent traders as a result of a positive and persistent shock to $$z_t.$$ Similarly, if the existing population is large and has a large fraction of low-ability types, large-scale exit will take place in response to a negative and persistent shock, leading to a sudden drop in population. With continued bad news this decrease will slow, because fewer low-ability types will be present and adjustment must occur through natural attrition at the rate of $$(1-\beta).$$

Finally, because new entrants do not know their type, we can certainly have situations where the expected benefit to new entry is positive, whereas negative payoffs for the low-ability types are leading to exit. Hence, we observe periods with simultaneous entry and exit.

Some things are simple to demonstrate through recursive argument. Because the periodic payoffs for the higher-ability types dominate those of the lower-ability types, the following must hold.

Lemma 1

For any set of the state variables, if $$\theta_i\gt\theta_j$$ and the value functions exist,

(6)
$$\mathcal{v}(\theta_i;z_t,N_{t-1},f_{t-1})\ge \mathcal{v}(\theta_j;z_t,N_{t-1},f_{t-1}).$$

Accordingly, if any lower types remain, then no higher types will exit.

### 1.6 Customers trading with intermediaries

Here, we describe a simple model for the interaction between traders and outside customers or clients that produces the functional forms for the trading profits we employ in our subsequent examples and simulations.

The mass of customers and their average ability to negotiate or trade advantageously with intermediaries in financial transactions is assumed to be constant through time. Denote this ability level as $$\theta^c.$$ Each customer is willing to pay up to $$z_t$$ for the financial services offered by brokers/traders. Because of search and informational frictions, a given customer will be served with probability $$g(\frac{N_t}{C}),$$ where $$N_t$$ is the mass of traders and $$c$$ is the mass of customers. Then expected total profit of traders from dealing with customers is:

(7)
$$\Pi(z_t,N_t,f_t)=\int^{\theta^H}_{\theta_L}z_t g\left(\frac{N_t}{C}\right)f_t(q)\text{max}\left[1,\frac{1}{2}+q-\theta^c\right]\,\,\,dq.$$
That is, every trader of ability $$q$$ has a chance of $$g\left(\frac{N_t}{C}\right)$$ of finding a customer and then splitting surplus $$z_t.$$ The maximum in the sharing rule limits the customers’ from suffering losses beyond the total surplus from the transaction. Given the aggregate profit above, the per-match surplus to be divided among traders can be written:
(8)
$$\pi(z_t,N_t,f_t)=\frac{\Pi(z_t,N_t,f_t)}{N/2}=\frac{z_t}{\underbrace {N_t}_I}\underbrace {g\left(\frac{N_t}{C}\right)}_{II}\underbrace{2\int^{\theta^H}_{\theta_L}f_t(q)\text{max}\left[1,\frac{1}{2}+q-\theta^c\right]\,\,\,dq.}_{III}$$

The first term $$\frac{z_t}{N_t}$$ captures the crowding typical of models of competition in an industry. The second term reflects capacity constraints for the industry. The market is larger if there are more traders to deal with customers. The third term captures the idea that ability of traders might matter for the overall profit generated. Because the reservation value for customers is zero, the maximum traders can get in a single interaction is the trading surplus. The last term is thus less than two.

In our examples and simulations we set $$c$$ = 1, and assume the aggregate profits do not depend directly on $$f_t.$$ We parameterize the function $$g(x)=x^{1-\gamma}.$$ Thus, the profits per match are $$\pi(z_t,N_t)=\frac{z_t}{N_t^\gamma}.$$ As γ increases to unity, aggregate profits grow more slowly with the population. Thus, γ controls the speed with which profits are dissipated through increased competition of the traditional sort. Our focus here is, instead, on how the composition of ability within the population affects outcomes.

## 2. The Deterministic Case

To gain some intuition regarding the behavior of the model, we start by considering as a functional form for the per-period profit $$\pi(z_t,N_t)=zN_{t}^{-\gamma}$$ where $$N_t$$ is the total population and $$z$$ is constant. Because profits fall with more competition, $$\gamma\gt 0.$$ We further assume that

(9)
$$\frac{1}{2}\frac{z}{M^\gamma}-c\gt 0,$$
so that if all incumbents exit, the new generation earns positive rents. They will also have a positive valuation because they can always exit costlessly after one period.

### 2.1 Population dynamics with symmetric ability

We now characterize a steady-state equilibrium in which ability levels are the same for all traders. This provides a natural benchmark for comparison with the consequences of predator-prey interactions.

With $$\theta_i=\theta_j$$ for all $$i$$ and $$j,$$ and $$\pi_t=\frac{z}{N^\gamma_t},$$ the periodic payoffs are simply:

(10)
$$\frac{1}{2}\frac{Z}{N^\gamma_t}-C$$
for all agents. The value function for incumbents will satisfy:
(11)
$$\mathcal{v}(N_{t-1})=\text{max}\left\{0,\frac{1}{2}\frac{Z}{N^\gamma_t}-c+\beta \mathcal{v}(N_t)\right\},$$
and the one state variable, $$N_t,$$ is governed by
(12)
$$N_t=p_t\beta N_{t-1}+M,$$
where $$p_t$$ is the (endogenous) probability of exit.

In any steady state, the endogenous variables are independent of time: $$p_t=p^*$$ and $$N_t=N^*.$$ Thus,

(13)
$$N^*=p^*\beta N^*+M.$$

The first possible steady state to consider is with all incumbents continuing: $$p^*=1,$$ and $$N^*=\frac{M}{1-\beta},$$ the maximum possible sustainable population. If profits are nonnegative at this level of population, or

(14)
$$\frac{z(1-\beta)^\gamma}{2M^\gamma}\ge C,$$
then this is an equilibrium.7 The positive profits are sustainable because the measure of available entrants is insufficient $$(M\lt (\frac{z}{2c})^{1/\gamma}(1-\beta))$$ to exhaust available rents, given natural attrition.

If profits are negative at $$N_t=\frac{M}{1-\beta},$$ then the steady-state population will be at the level at which, with partial exit and the entry of the new generation, profits are zero:

(15)
$$\frac{1}{2}\frac{z}{(N^*)^\gamma}-c=0,$$
or,
(16)
$$N^*=\left(\frac{z}{2c}\right)^{1/\gamma}.$$
Evidently, $$N_t=N^*$$ is a steady-state equilibrium if there is a solution, $$p^*\in[0,1],$$ to
(17)
$$N^*=p^*\beta N^*+M,$$
or
(18)
$$p^*=\frac{\left(\frac{z}{2c}\right)^{1/\gamma}-M}{\beta\left(\frac{z}{2c}\right)^{1/\gamma}}\in[0,1].$$

In this case, at a population level that simply replaces natural attrition with new entry, with $$p^*=1$$ in (17), profits will be negative or zero. If they are negative, then there will be exit by incumbents in the steady state. In such an equilibrium, competition eliminates all rents. New entrants are indifferent between participating in this industry and their alternative. Incumbents are indifferent between exiting and continuing.

Changes in the demand for their services affect the population of traders in a very simple way in the symmetric case. If we consider the elasticity of the population with respect to the demand for services we obtain:

(19)
$$\frac{\frac{dN^*}{N^*}}{\frac{dz}{z}}=\frac{1}{\gamma}.$$
With $$\gamma=1,$$ an increase in population, holding $$z$$ constant, has no effect on aggregate trading profits for the industry. For this case, the steady-state population moves one-to-one with $$z.$$ When $$\gamma\lt1,$$ increases in the mass of traders raises the surplus available, magnifying the effect of a change in $$z$$ on the population. In the next section we will see that this effect can be amplified by the changes in the distribution of ability across the population.

The following lemma also provides a contrast with the case in which there are differences in ability. When there are such differences, there will often be multiple steady-state equilibria, one with high population and others with low population.

Lemma 2

If it exists, the steady state at $$N_t=N^*$$ and $$p_t=p^*$$ is unique.

The population dynamics for this system are very simple. If, initially the population is too large, there is a sudden downward adjustment to the steady state. When the population is initially too small relative to the surplus available, there is only exogenous attrition until, though new entry every year, the population grows to the point at which profits are driven to zero. Along this path incumbents enjoy rents every period because the supply of appropriate new entrants is limited. As we will see in what follows, the dynamics become more interesting with asymmetric ability, because increased numbers of low-ability traders not only provide more profitable opportunities for predatory, high-ability traders. They also shield their fellows from predation, providing positive congestion externalities for each other.

### 2.2 Effects of differences in ability

With differences in ability, participation externalities or composition effects can amplify the effects of shocks to the external demand for financial services, and also lead to multiple equilibria. The steady states in the deterministic case illustrate the central intuitions.

#### 2.2.1 High-population equilibrium

If there exists an equilibrium steady state in which the endogenous quantities are independent of time, then $$N_t=N$$ and $$f_t(\theta)=f(\theta).$$ If traders exit only due to exogenous attrition, which affects all ability levels in the same way, then clearly $$f(\theta)=f_0(\theta).$$ The new entrants each period simply replace those dying off, and $$M=(1-\beta)N.$$ This determines the steady-state population:

(20)
$$N=\frac{M}{1-\beta}.$$
If this is an equilibrium, it will clearly be the maximum (constant) population the industry can support.

The value functions for each type will be simply the discounted infinite stream of periodic payoffs. If all types remain in the industry in equilibrium, the value of continuing for each type must be positive, or equivalently, their periodic profits must be positive. If they are positive for the lowest-ability traders ($$\theta=\theta_L$$), they must necessarily be so for the higher-ability traders by Lemma 1. Evaluating the profit for the lowest ability level, at $$N=\frac{M}{1-\beta}$$ and $$f(\theta)=f_0(\theta)$$ for all $$\theta,$$ implies the following restriction on the parameters:

(21)
$$\frac{z(1-\beta)^\gamma}{M^\gamma}\left(\frac{1}{2}+(\theta_L-\bar\theta_0)\right)-c\ge0.$$
Notice in this expression that it will be more difficult to support a high population of traders the greater the portion of the population endowed with high ability, which implies higher $$\bar\theta_0,$$ and the more dramatic the asymmetry in ability, $$\theta_L-\bar\theta_0.$$

#### 2.2.2 Lower-employment equilibria

An alternative steady-state equilibrium might be one in which all traders with ability below a certain level, $$\theta^*$$ exit after experience in the industry. If the population and the distribution of types are constant, $$N_t=N$$ and $$f_t(\theta)=f(\theta),$$ then the number of new entrants must just replace the lower-ability types who exit and the higher-ability types who die. Define the following:

(22)
$$F(\theta^*)=\int^{\theta^*}_{\theta_L}f(\theta)\,\,d\theta$$
and
(23)
$$F_0(\theta^*)=\int^{\theta^*}_{\theta_L}f_0(\theta)\,\,d\theta.$$
The measure of the survivors must be $$(1-F(\theta^*))\beta N,$$ so the new entrants are
(24)
$$M=N-(1-F(\theta^*))\beta N.$$
The density of the population with ability θ, for $$\theta\ge\theta^*$$ must be
(25)
$$f(\theta)=\frac{f_0(\theta)M+f(\theta)\beta N}{N},$$
whereas for $$\theta\lt\theta^*,$$ we have:
(26)
$$f(\theta)=\frac{f_0(\theta)M}{N}.$$

The expressions above specify the functional equation for $$f(\theta)$$ and $$N.$$ On simplification we obtain the population in terms of exogenous variables:

(27)
$$N(\theta^*)=\frac{M(1-\beta F_0(\theta^*))}{1-\beta}.$$
The densities of each type are
(28)
$$f(\theta)=\frac{f_0(\theta)}{1-\beta F_0(\theta^*)},$$
for $$\theta\ge\theta^*,$$ whereas for the lower-ability types exiting each period the densities are
(29)
$$f(\theta)=\frac{f_0(\theta)(1-\beta)}{1-\beta F_0(\theta^*)}.$$

It is evident from inspection that the population when all ability levels continue, $$\frac{M}{1-\beta},$$ is higher than the population given in Equation (27). Because $$F_0(\theta^*)$$ and β are both positive fractions, $$1-\beta F_0(\theta^*)\lt1.$$ Similarly, (28) and (29) imply $$f(\theta)\gt f_0(\theta)$$ for the higher-ability traders, whereas for those of lower ability their share of employment will be lower than their share of the entering cohorts. As in any steady-state equilibrium, the valuations for the traders of each type must simply be their (constant) periodic expected payoffs, discounted as a perpetual stream using probability of survival,

(30)
$$\frac{1}{1-\beta}\left[\frac{z}{N(\theta^*)^\gamma}\left(\frac{1}{2}+(\theta-\bar\theta(\theta^*))\right)-c\right],$$
where $$\bar\theta(\theta^*)=\int^{\theta_H}_{\theta_L}\,f(\theta)\theta\,\,d\theta$$ is the average ability in the population, calculated using the densities in (28) and (29). Accordingly, for our conjectured equilibrium to be an equilibrium steady-state, we must have the above expression nonnegative for $$\theta\ge \theta^*$$ and negative for $$\theta\lt \theta^*.$$

Because the average ability level $$\bar\theta(\theta^*)$$ will be increasing in the cutoff level, $$\theta^*,$$ for a given set of parameters, it is quite possible to have these conditions holding for various levels of $$\theta^*,$$ including the high-population case in which $$\theta^*=\theta_L.$$ In such cases, the model will have multiple equilibria.

### 2.3 Composition effects

A fall in aggregate profits, $$z,$$ generally leads to a fall in the population. It also causes changes in the distribution of ability within the population, which in turn can feed back to affect the size of the population. We illustrate these effects in two ways. First, we perform comparative statics on an interior equilibrium, and show how changes in ability alter the elasticity of the population with respect to $$z.$$ Second, we consider situations in which a fall in $$z$$ can render the high-population equilibrium infeasible. Because the average ability rises when low-ability traders exit, the drop in $$z$$ can lead to a chain reaction, where even though the fall in profits leaves traders with moderate ability in a position to make profits when others remain, by driving out the weakest participants it renders those with intermediate ability more vulnerable to predation and thus causes them to exit as well.

#### 2.3.1 Population elasticity

Suppose $$\theta^*$$ is an interior cutoff point at an equilibrium steady state. Traders with ability less than $$\theta^*$$ will exit. The next lemma consolidates some results that illustrate how the population responds to a change in the industry’s aggregate profitability.

Lemma 3

The population elasticity with respect to $$z$$ is

(31)
$$\frac{\frac{dN}{N}}{\frac{dz}{z}}=\frac{1}{\gamma}\left(1+\frac{\partial[\theta^*-\bar\theta(\theta^*)]}{\partial\theta^*}\frac{z^2}{cN^\gamma}\frac{d\theta^*}{dz}\right).$$
This elasticity will exceed $$1/\gamma,$$ the elasticity when ability is symmetric, if
(32)
$$\frac{\partial[\theta^*-\bar\theta(\theta^*)]}{\partial\theta^*}\lt 0,$$
which in turn can be expressed in terms of the distribution of types in entering cohorts:
(33)
$$\frac{\partial[\theta^*-\bar\theta(\theta^*)]}{\partial\theta^*}=1-(\bar\theta-\theta^*)\frac{\beta f_0(\theta^*)}{1-\beta F_0(\theta^*)}$$

In the stochastic case, $$z$$ will be an exogenous shock to the external demand for the services the industry provides. For example, as the economy generally expands, savers have more funds to invest and businesses issue more securities to raise funds. This increases the demand for the services of intermediaries. The elasticity in Equation (31) tells us that the response of the population to changes in the exogenous surplus consists of two terms. The first simply reflects the amount of direct competition. As we saw in Section 2.1, $$1/\gamma$$ is the elasticity of the population when ability is symmetric. As $$z$$ falls, $$N$$ must also fall in an amount that reflects the parameter that governs the effect of population size on growth in the market. The second term amplifies or mollifies this effect depending on whether

(34)
$$\frac{\partial\left[\theta^*-\bar\theta(\theta^*)\right]}{\partial\theta^*}\gt 0,$$
because $$\frac{z^2}{cN^\gamma}\gt0,$$ and the cutoff ability rises as profits fall, $$\frac{d\theta^*}{dz}\lt0.$$

Although it may not be the case if the surviving population is small relative to new entry, we would expect that generally $$\theta^*-\bar {\theta}(\theta^*)\lt0$$—average ability must exceed the ability of the worst surviving trader. In such a case, if

(35)
$$\frac{\partial\left[\theta^*-\bar\theta(\theta^*)\right]}{\partial\theta^*}\gt0,$$
then the difference is becoming less negative: the pool is changing so that the worst trader is closer to the average trader. This will be the case when the equilibrium distribution of ability is relatively flat and uniform in shape. As more of the distribution is dropped from the pool, the surviving traders all have roughly equal density, but they are concentrated within a smaller portion of the original interval over which the population is distributed. The marginal trader is then less far away from the average as ability rises. The second term in the elasticity is negative, and the composition effects in this case tend to mitigate the drop in the surplus.

The response of average ability will amplify the effects of a drop in profitability if the distance between the ability of the worst survivor and the average grows (becomes more negative) as lower ability traders drop out. That is, when

(36)
$$\frac{\partial\left[\theta^*-\bar\theta(\theta^*)\right]}{\partial\theta^*}\lt0.$$
This will occur when the exiting traders have relatively high density, and thus a big effect on the mean. This will be the case when the distribution of types is discrete, or multimodal, but it can also occur if the distribution of ability is skewed. Then there are a small number of highly skilled traders who have a big impact on the mean, and large numbers of less-able traders.

We see this analytically in expression (33). Recall that the changes in average ability amplify the effects of a fall in surplus on the population if this expression is negative. If only very high-ability traders survive in the market, so that the cutoff level of ability is above the average ability level supported by new entry, then (33) is clearly positive. The only setting in which it could be negative is when the cutoff ability level is significatly below the mean. Moreover, the distribution should be such that the hazard ratio $$\frac{f}{1-F}$$ at the cutoff level is high. This means that the mass of the marginal trader should be substantial in the mass of surviving agents. If the original distribution of types is highly skewed to the right, as with a lognormal or an exponential distribution, this will be the case. In that situation a small number of very effective predators will be extracting most of the surplus from the broader population of market participants.

Figure 3 illustrates the set of equilibria for a specific example, where there are two interior equilibria. In this example, $$\gamma=1,$$ so industry-wide profits do not depend on the population. The left-hand top diagram of the figure shows the density of incoming agents’ ability. The right-hand top diagram represents the relationship between $$\theta^*,$$ the cutoff level of ability, average ability $$\bar\theta,$$ and the losses of marginal trader due to the difference in ability $$\theta^*-\bar\theta.$$ At low levels of $$\theta^*,$$ the losses of the marginal trader increase with cutoff level of ability, so that in this range the effects of a drop in the surplus are amplified by the effect on average ability. With a drop in the level of $$z,$$ the marginal trader would require a higher per-person share of overall surplus to cover those extra losses, meaning $$Z/N$$ goes up, so $$N$$ must fall more than $$z.$$

Figure 3

Multiple equilibria with long-normal distribution of ability level

In each period $$M$$ = 1 agents enter with ability level $$\theta\,\,\in[0,\,\infty]$$ distributed lognormally with mean $$\mu=2.3$$ and standard deviation $$\sigma=1.5.$$ Per-period operating costs are 0.5 and 5% of agents exit the market for exogenous reasons. The top left-hand diagram represents the density of incoming agents, $$f_0(\theta).$$ The upper right-hand diagram shows the relationship between the cutoff level $$\theta^*,$$ the average ability level $$\bar\theta,$$ and the losses of marginal investor due to differences in ability, $$\theta^*-\bar\theta.$$ The effects of a shock are amplified where $$\theta^*-\bar\theta$$ is decreasing. The second row of plots represents the profits of the marginal trader (curved line) and an indicator function for when those profits are positive or negative. For the system to be in a steady-state equilibrium, $$H(\theta^*)=0;$$ that is, two curves must intersect. The left-hand figure magnifies the portion of the right-hand figure close to zero. There are two possible equilibria—one with cutoff level of ability next to zero (big population equilibrium) and one with $$\theta^*$$ close to 0.7 (small population equilibrium). In the latter case, only very skilled agents find it profitable to stay in the market.

Figure 3

Multiple equilibria with long-normal distribution of ability level

In each period $$M$$ = 1 agents enter with ability level $$\theta\,\,\in[0,\,\infty]$$ distributed lognormally with mean $$\mu=2.3$$ and standard deviation $$\sigma=1.5.$$ Per-period operating costs are 0.5 and 5% of agents exit the market for exogenous reasons. The top left-hand diagram represents the density of incoming agents, $$f_0(\theta).$$ The upper right-hand diagram shows the relationship between the cutoff level $$\theta^*,$$ the average ability level $$\bar\theta,$$ and the losses of marginal investor due to differences in ability, $$\theta^*-\bar\theta.$$ The effects of a shock are amplified where $$\theta^*-\bar\theta$$ is decreasing. The second row of plots represents the profits of the marginal trader (curved line) and an indicator function for when those profits are positive or negative. For the system to be in a steady-state equilibrium, $$H(\theta^*)=0;$$ that is, two curves must intersect. The left-hand figure magnifies the portion of the right-hand figure close to zero. There are two possible equilibria—one with cutoff level of ability next to zero (big population equilibrium) and one with $$\theta^*$$ close to 0.7 (small population equilibrium). In the latter case, only very skilled agents find it profitable to stay in the market.

The plot at the bottom right shows the equilibria. The plot on the bottom left magnifies the area close to zero to detail the behavior there. The curves in the plot represent the profits of the marginal trader for different levels of $$\theta^*.$$ The curved line are the profits, and the other line is the indicator function for when the profits are positive. The two intersect where the profits of the marginal trader are zero. The outer two of these intersections represent equilibria. Traders with ability below $$\theta^*$$ all would lose money if they entered, whereas traders with higher ability all make money. (At the third, middle intersection, this is reversed, so it is not an equilibrium.) The intersection that occurs with a high value of $$\theta^*$$ is an equilibrium with a small population with very high ability. Here, the losses to the marginal trader, $$\theta^*-\bar\theta,$$ are reduced as the cutoff ability rises. The worst traders become more like the average survivor as the population contracts.

At the high-population equilibrium, the slope of $$\theta^*-\bar\theta$$ is negative, and as $$z$$ falls, the marginal survivor moves further from the average because of the relatively high density of the departing marginal traders. This renders them more vulnerable to predation, leading more agent’s to exit. This is the behavior we refer to as a “cascade” in the population. Specifically, in this example, a 1.17% drop in income (z) would result in a permanent drop in employment by 9.34% if the system was in large-population equilibrium (cutoff level of ability $$\theta^*$$ is close to $$\theta_L$$), and only in a drop of 0.11% if the system was in low-population equilibrium, when only high-ability agents are present in the market. This illustrates the difference in elasticity of employment with respect to the income shock in two possible steady-state equilibria. The cascade is generated by an income drop of 4.26%—the large-employment equilibrium is not sustainable after the shock, and the system that started in the large employment equilibrium would experience a drop in employment of 85.57% by settling in the low-employment equilibrium.

### 2.4 Coexistence of steady-state equilibria: Positive and negative congestion

The case of two ability levels, $$\theta\,\in\{\theta_H,\theta_L\},$$ is sufficient to illustrate some of the tensions at work in the model. Figure 4 shows the combinations of $$z,$$ which governs profitability, and asymmetry in ability given by $$(\theta_H-\theta_L),$$ that support the different steady-state equilibria. Other parameters are held fixed. The figure labels the areas in which the low- and high-population steady states exist. The area labeled “both” can sustain either steady state. In the unlabeled areas, neither equilibrium exists.

Figure 4

Sustainable steady-state equilibria with two ability types

The graph represents possible sustainable steady-state equilibria. The fraction of new entrants with superior ability is $$f^H_0=0.4.$$ The survival probability for older generations is $$\beta=0.9,$$ the periodic operating cost is set to $$c=0.5,\,\gamma=1,$$ and the measure of the new generation is $$M=0.25.$$ “High Population” area corresponds to the pairs of $$(\theta_H-\theta_{L},z)$$ that support steady state-equilibrium at which both types of traders survive. In the “low-population” area, there is a steady state in which the low-ability traders all exit. The intersection, labeled “Both,” contains parameter values for which both steady-state equilibria are sustainable.

Figure 4

Sustainable steady-state equilibria with two ability types

The graph represents possible sustainable steady-state equilibria. The fraction of new entrants with superior ability is $$f^H_0=0.4.$$ The survival probability for older generations is $$\beta=0.9,$$ the periodic operating cost is set to $$c=0.5,\,\gamma=1,$$ and the measure of the new generation is $$M=0.25.$$ “High Population” area corresponds to the pairs of $$(\theta_H-\theta_{L},z)$$ that support steady state-equilibrium at which both types of traders survive. In the “low-population” area, there is a steady state in which the low-ability traders all exit. The intersection, labeled “Both,” contains parameter values for which both steady-state equilibria are sustainable.

To sustain the high-population equilibrium, lower values of $$\theta_H-\theta_L$$ and higher values of $$z$$ are required. This illustrates the offsetting effects of the two externalities in the model that affect the survival of low-ability traders.

• There is a negative externality due to crowding.

• There is a positive externality due to the dilution of predators in the population.

The first effect, associated with increased competition, is a standard negative congestion externality. Each new entrant or surviving low-ability trader raises $$N,$$ which reduces the profits available per individual trade for everyone. Because the agents have zero measure in this model, this takes the extreme form of each agent acting as if he has no effect at all on the aggregate. This standard externality does not depend on differential ability. It is evident in the model with only one type. The negative effect of more traders on profits per trade may be offset by a positive externality that low-ability traders confer on each other (and on new entrants). The expected profit per person may be positively related to the number of agents staying in the market, despite the negative dependence of $$\pi(z-\,N)$$ on the population size, $$N.$$

The high-ability agents also benefit from the presence of a larger proportion of low-ability traders. Their expected profit depends negatively on the probability, $$f^H_t,$$ that they are matched with a counterparty of similar ability. If asymmetric ability can be viewed as the result of investments made by traders in education and technology, then these investments can be self-defeating. Greater asymmetry of ability benefits more talented traders, given the population of counterparties available, but eventually drives out potential prey from the market. For a sufficient difference in ability, $$\theta_H-\theta_L,$$ as is clear from Figure 4, the high-participation equilibrium becomes unsustainable. This argument bears some similarity to Glode, Green, and Lowery (2012), where financial expertise is viewed as an arms race. There, however, counterparties refuse to trade, despite gains to doing so, because of adverse selection.

Which effect dominates for the high-ability traders—the larger population overall depressing profits through greater competition, or the greater availability of potential prey—depends on the speed with which profits fall as the population grows. This is governed by the parameter γ in the model. As $$\gamma\rightarrow0,$$ the increases in population have less and less effect on the unconditional expected profits per match (that is, averaging across possible types), which are just $$\frac{z}{2N\gamma}.$$ In this case, the beneficial effect of a lower $$f^H_t$$ in the high-participation equilibrium dominates for the high-ability types. On the other hand, as γ becomes very large, the effect of the larger population dominates. The following lemma formalizes these arguments.

Lemma 4

If both steady states exist, then for γ sufficiently close to zero the high-ability traders earn more in the high-population equilibria, while for sufficiently large γ they earn less.

### 2.5 Population collapses

Because the payoff to any one type of trader depends on the average ability of the population, as well as on its size, exogenous changes to aggregate profits, $$z_t,$$ can trigger a domino effect. Exit by the lowest ability traders shifts the average ability of the population. It can render continuation untenable for higher ability types, by reducing the opportunities they have to gain at the expense of others and exposing them to more predation by the most-able and experienced traders in the population.

Table 2 illustrates this using the steady state-equilibria in a numerical example with three types. In this example, $$\beta=0.95,\,c=0.25,\,\gamma=1,$$ and $$M=0.25.$$ The proportions of types in the entering cohorts are $$f^L_0=0.4,$$$$f_0^M=0.3,$$ and $$f^H_0=0.3,$$ for low to high ability, respectively. Ability levels are $$\theta_L=0.6,\,\theta_M=0.75,$$ and $$\theta_H=1.7.$$ The first two columns of the body of the table show the average ability, $$\bar\theta(i^*),$$ and population $$N(i^*),$$ assuming the steady-state equilibria exist, for equilibria in which all types participate, only the high- and middle-ability types participate, and only the high-ability types participate. The remaining columns show the periodic payoffs to each type of trader.

Table 2

Profits for high-(H), medium-(M), and low-(L) ability traders at different levels of participation

Types participating  Population Average ability Profits for types

$$N$$ $$\bar\theta$$ $$H$$ $$M$$ $$L$$
$$\{H,\,M,\,L\}$$ 5.00 0.975 2.20 0.30 0.00
$$\{H,\,M\}$$ 3.10 1.205 2.96 −0.10 −0.59
$$\{H\}$$ 1.68 1.592 3.38 −2.29 −3.19
Types participating  Population Average ability Profits for types

$$N$$ $$\bar\theta$$ $$H$$ $$M$$ $$L$$
$$\{H,\,M,\,L\}$$ 5.00 0.975 2.20 0.30 0.00
$$\{H,\,M\}$$ 3.10 1.205 2.96 −0.10 −0.59
$$\{H\}$$ 1.68 1.592 3.38 −2.29 −3.19

The parameter controlling the level of profits, $$z$$ = 10, is set in this example so that the profit of the lowest ability traders is exactly zero in the full participation steady state. As the first row of the table shows, both the higher ability types are also making profits in this situation, so full participation is an equilibrium. The low-employment outcome is also an equilibrium. The bottom row shows that if only the high-ability types participate, they make positive profits, but neither the low- nor mid-ability levels have positive payoffs. The middle row shows that it is not an equilibrium for the high and middle types to participate, if the low types do not. In this situation the midability traders have negative profits, despite the fact that the top row shows at the same level of $$z$$ they continue to have positive payoffs as long the lowest-ability traders participate, and despite the lower overall level of competition ($$N$$ falls from 5.0 to 3.1).

Lowering $$z$$ slightly below ten then has the effect of forcing out the lowest-ability traders, so that full-participation is no longer an equilibrium. Although the drop in $$z$$ does not in itself lead to negative payoffs for the mid-ability traders, the response of the lowest ability traders does, and thus forces their exit as well. In this way, a relatively small real shock can be amplified in its effect on the employment of traders.

Figure 5 illustrates the range of steady-state outcomes that can be supported in this example. The shaded areas in the plot correspond to combinations of $$\theta_H$$ and $$z$$ for which there are steady-state equilibria, holding the other parameters at the values in the example above. Notice that in some cases, a fall in $$z$$ would move the industry from high employment to middle levels of employment. In other cases, a lower $$z$$ is only consistent with the lowest employment steady state. The star on that diagram, on the boundary of the high-population area in the middle of the plot, indicates the combination of $$\theta_H$$ and $$z$$ used in the numerical example in Table 2.

Figure 5

Sustainable steady-state equilibria for three types of ability level

For $$f_0=(0.4,0.3,0.3),\,\,\beta=0.95,\,\,c=0.25,\,\,M=0.25,\,\theta_L=0.6,$$ and $$\theta_M=0.75,$$ the graph represents possible sustainable steady-state equilibria. The “High population” area corresponds to the pairs of $$(\theta_H,z)$$ that support the steady-state equilibrium at which all types of traders survive. In the “Low-population” area, there is a steady state in which the low- and middle-ability traders all exit. “Middle size population” refers to the case in which only low-ability traders exit, whereas middle- and high-ability traders find it optimal to continue. Darker areas represent parameter values that support multiple equilibria. The central darkest areas contain parameter values that support all three possible equilibria. The top areas and the area immediately below on the right are where only low-population and high-population equilibria are possible. These are of interest to our analysis as even smallest changes in z can trigger the change from high to low population, without passing through midlevel equilibria. White areas correspond to situations in which none of the steady-state equilibria are sustainable.

Figure 5

Sustainable steady-state equilibria for three types of ability level

For $$f_0=(0.4,0.3,0.3),\,\,\beta=0.95,\,\,c=0.25,\,\,M=0.25,\,\theta_L=0.6,$$ and $$\theta_M=0.75,$$ the graph represents possible sustainable steady-state equilibria. The “High population” area corresponds to the pairs of $$(\theta_H,z)$$ that support the steady-state equilibrium at which all types of traders survive. In the “Low-population” area, there is a steady state in which the low- and middle-ability traders all exit. “Middle size population” refers to the case in which only low-ability traders exit, whereas middle- and high-ability traders find it optimal to continue. Darker areas represent parameter values that support multiple equilibria. The central darkest areas contain parameter values that support all three possible equilibria. The top areas and the area immediately below on the right are where only low-population and high-population equilibria are possible. These are of interest to our analysis as even smallest changes in z can trigger the change from high to low population, without passing through midlevel equilibria. White areas correspond to situations in which none of the steady-state equilibria are sustainable.

### 2.6 Matching technology

Our model is a reduced-form representation of the trading process. It abstracts from asymmetric information and many details of the search process that may be important in any given institutional setting. In particular, the model assumes that once two counterparties are randomly matched, surplus is distributed based on relative ability with no opportunity to withdraw. This may be descriptive of a setting in which trading is anonymous, such as on an exchange or in some over-the-counter markets, where trading is routinely intermediated by third parties. In other over-the-counter settings, however, traders know the identity of their counterparties, and this may lead to an unwillingness to trade, or to bargaining over the terms.

Here, we demonstrate that our model is qualitatively robust to allowing traders to decline or withdraw from a particular match, when they know their counterparty is of superior ability. Giving agents in the model this option complicates the calculation of the distribution of types, but what is most important to ensuring the intuition of the model goes through is that when confronted with a population with a sufficiently large proportion that has superior ability, a given agent expects to lose money on average. All that is required for this, is that failure to execute trade in a given period is costly because of idle resources or wasted effort. In the model, this is a consequence of our assumption that each trader bears a fixed periodic cost of operating, $$c.$$

Suppose trader $$i$$ is matched with trader $$j$$ and either party can refuse the match on learning the identity of their counterparty. Agent $$i$$ will withdraw if

(37)
$$\pi(z_t,N_t)\left[\frac{1}{2}+(\theta_i-\theta_j)\right]-c\lt-c$$
or if
(38)
$$\theta_j\gt\theta_i+\frac{1}{2}.$$
Similarly, agent $$j$$ will withdraw if $$\frac{1}{2}+(\theta_j-\theta_i)\lt0,$$ or if
(39)
$$\theta_j\lt\theta_i-\frac{1}{2}.$$
Putting these conditions together, trade will occur if
(40)
$$\theta_i-\frac{1}{2}\le\theta_j\le\theta_i+\frac{1}{2}.$$

At the point at which agent $$i$$ makes a decision to exit or continue, she does not yet know the identity of her next counterparty. Her expected payoff is therefore,

(41)
$$-c+\int^{\theta_i+\frac{1}{2}}_{\theta_i-\frac{1}{2}}\pi(z_t,\,N_t)\left[\frac{1}{2}+(\theta_i-\theta)\right]f(\theta)d\theta$$
or
(42)
$$-c+\text {Pr}\left\{\theta_i-\frac{1}{2}\le\theta\le\theta_i+\frac{1}{2}\right\}\pi(z_t,N_t)\left[\frac{1}{2}+\theta_i-E\left(\theta|\theta_i-\frac{1}{2}\le\theta\le\theta_i+\frac{1}{2}\right)\right],$$
where $$f(\theta)$$ is the endogenous distribution of types in the surviving population.

From the above expressions it is clear that the same forces will be at work. The option to break a match truncates the losses of the low types and the gains of the high types, but the low types still expect to lose, and lose more, when it is more likely that they will be facing counterparties with superior ability. A higher proportion of high-ability counterparties lowers the probability of trade (i.e., the probability $$\theta_j\in[\theta_i-\frac{1}{2},\theta_i+\frac{1}{2}]$$). It also raises the expectation of $$\theta_j$$ conditional on trade occurring. In the extreme case, suppose there are two types, high ability and low ability, and that $$\theta_H\gt\theta_L+\frac{1}{2}.$$ If a low-ability trader anticipates that all the other low-ability traders will exit, then it is very unlikely she will be able to trade in any given period, and she must simply absorb the operating loss. She will therefore exit. Alternatively, if she anticipates all low-ability types will continue, then trade is likely to occur and to occur on equal terms. She will also continue.

Figure 6 illustrates that this intuition also applies to cases in which the distribution of types is continuous. The figure plots the expected periodic profit, $$H(\theta^*),$$ for a trader with ability $$\theta^*,$$ assuming that $$\theta^*$$ is the marginal ability level, that is, assuming all traders with ability above $$\theta^*$$ continue and all those below exit. The plot shows the expected profits, assuming (1) unfavorable matches cannot be broken (the curve with the continuously increasing slope) and (2) they can be broken (the flatter curve). Any point at which the profit curve $$H(\theta^*)$$ crosses zero from below will be an equilibrium. For these parameter values, there is only one equilibrium when the match cannot be broken, whereas there are two equilibria when they can be broken: one at a high population level and one at a low population level.

Figure 6

Profit of the marginal trader when agents can break the match

The plot shows the profit at the cutoff level of ability for the two cases in which the match can be broken and where it cannot. Ability of entering cohorts is distributed lognormally with $$\mu=-2.8$$ and $$\sigma=1.8.$$ This distribution is truncated at a maximum value of 2.0 with the remaining mass at 2.0. Other parameters are $$\beta=0.95,$$$$c=1.0,$$$$M=1.0,$$ and $$z$$ = 25. The profit of the marginal trader crosses zero from below at two points, each of which is an equilibrium.

Figure 6

Profit of the marginal trader when agents can break the match

The plot shows the profit at the cutoff level of ability for the two cases in which the match can be broken and where it cannot. Ability of entering cohorts is distributed lognormally with $$\mu=-2.8$$ and $$\sigma=1.8.$$ This distribution is truncated at a maximum value of 2.0 with the remaining mass at 2.0. Other parameters are $$\beta=0.95,$$$$c=1.0,$$$$M=1.0,$$ and $$z$$ = 25. The profit of the marginal trader crosses zero from below at two points, each of which is an equilibrium.

These arguments also suggest an additional channel through which shocks to real activity are amplified in the financial industry. As a sector matures, and market participants become more familiar with each other, mismatches in ability are more easily identified, suppressing volume and profits. Innovations and new business not only bring in more traders with lower average ability than the incumbent population but they also bring in traders who are less clearly differentiated, also encouraging trade.

## 3. Dynamics with Stochastic Profits

In this section we consider the evolution of the population when the shock to profits per match, $$z_t,$$ evolves stochastically. We assume $$z_t$$ follows an AR(1) process with autocorrelation ρ and conditional variance σ. In particular, we assume that $$z_t=\mu e^{\varepsilon_t}$$ and $$\varepsilon_t=\rho\varepsilon_{t-1}+e_t,$$ where $$e_t$$ is iid normal with zero mean and variance σ, and μ plays the role of a scaling parameter.8

Our purpose here is to illustrate, first, that the qualitative features of the deterministic case in the previous sections survive in a dynamic, stochastic setting. Often, multiple equilibria arise that support either high or low population levels.

Second, we show that a shock to the system that in one situation may have little effect on the path of population growth for the industry can, in another situation cause the population to fall dramatically. Not only does the high-population equilibrium suddenly cease to be viable but the drop in expected profits triggers exit by lower-ability incumbents, which in turn leaves traders of intermediate ability facing a higher probability of predation. The result is a collapse in population where all exit, except the traders with the highest ability.

Consideration of the stochastic case also reveals a range of behaviors that are not apparent from the deterministic model.

When the fraction of new entrants with high ability is relatively large, the industry can become “trapped” in the low-population equilibrium. The percentage of high-ability traders in the industry jumps when low-ability traders leave in response to a negative shock. Even when aggregate profitability recovers, the population may remain stuck at a lower level because the threat of predation deters continuation by the relatively small number of new low-ability entrants who arrive each period. It can be difficult to accumulate the critical mass of prey needed to make continuation viable simply through the replenishment provided by routine new entry. This may help to explain why employment growth and subsequent contractions in financial services tend to be triggered by financial innovation. The new businesses lead to an increase in employment that refreshes the population by bringing in large numbers of new and undifferentiated workers. Even though they soon become differentiated by ability, the high population can be sustained unless and until a negative shock to fundamentals causes a dramatic “shake out.” The associated activities then continue as a mature business with much lower employment even when profitability returns.

For other sets of parameters (when $$f_0^H$$ is relatively low), the industry can alternate between high- and low-population outcomes. Following a favorable shock, the population begins to build towards the high-population steady state. This growth then can be interrupted by a downturn that triggers a drop to the low-population outcome, where the economy remains until a sufficiently favorable shock again initiates a period of growth. Thus, our relatively simple setting can replicate the phenomenon of steady growth interrupted by periodic busts with sudden contractions in industry employment. Varying the operating cost parameter, $$c,$$ leads to variation in the average time in the boom and bust phases.

In the Appendix we provide detailed descriptions of the numerical procedures used to solve and simulate the model. As explained in more detail there, solving for an equilibrium along a sample path away from the steady states requires an assumption about which equilibrium the incumbent agents think they will be in going forward, because there might be multiple equilibria for any set of parameters and state variables. In a given period $$t,$$ the expected profit of a low-ability agent can be positive, given that all low-ability agents continue, and negative if all low-ability agents are exiting the system. In that situation we favor the high-population outcome—if low-ability agents can stay, they remain in the industry.

We allow for three levels of ability in the model, to keep the number of state variables manageable. The model does not generate realistic short-term dynamics, because all the transitory fluctuations in $$z_t$$ are simply absorbed by the profits. This could be remedied, without increasing the number of state variables, by introducing iid noise to the attrition rate or the population. We choose not to do this, in order to illustrate our central points more starkly. In the benchmark case, with symmetric ability, given the anticipated path of $$z_t,$$ the population simply grows or declines smoothly towards a steady state, while profits absorb the transitory variation in $$z_t.$$ When there is asymmetric ability, the population fluctuates more dramatically in response to small changes in fundamentals. There are chain reactions in which a shock pushes the lowest ability types out of the population, leaving those of intermediate ability exposed to greater predation by the most-talented competitors and in turn pushes them out.

Table 3 lists the parameter values we employ in the various scenarios for which we simulate the model. Scenarios I and III differ only in the scaling parameter that governs the expected exogenous growth in profits. Scenario II gives a bigger advantage to the high-ability types. Scenario IV will be used to illustrate the effects of variation in the continuation cost.

Table 3

Parameter values for simulations

Parameter Scenario I Scenario II Scenario III Scenario IV
$\begin{pmatrix}\theta_H\\\theta_M\\\theta_L\end{pmatrix}$

$\begin{pmatrix}1.3\\0.65\\0.6\end{pmatrix}$

$\begin{pmatrix}1.5\\0.65\\0.6\end{pmatrix}$

$\begin{pmatrix}1.3\\0.65\\0.6\end{pmatrix}$

$\begin{pmatrix}1.3\\--\\0.6\end{pmatrix}$

Impact of population on profits (γ) 1.00 1.00 1.00 1.00
Continuation cost ($$c$$1.15 1.15 1.15
$\begin{pmatrix}1.00\\1.30\\1.15\end{pmatrix}$

Measure of new entrants $$(M)$$ 0.25 0.25 0.25 0.25
Survival probability (β) 0.85 0.85 0.85 0.85
Conditional volatility of shock (σ) 0.20 0.20 0.20 0.20
Autocorrelation of shock (ρ) 0.60 0.60 0.60 0.60
Proportion of types by ability among entrants
$\begin{pmatrix}f_0^H\\f_0^M\\f_0^L\end{pmatrix}$

$\begin{pmatrix}0.3\\0.2\\0.5\end{pmatrix}$

$\begin{pmatrix}0.3\\0.2\\0.5\end{pmatrix}$

$\begin{pmatrix}0.3\\0.2\\0.5\end{pmatrix}$

$\begin{pmatrix}0.25\\0.0\\0.75\end{pmatrix}$

Scaling parameter for shock (μ) 10.00 7.00 5.00 5.00
Parameter Scenario I Scenario II Scenario III Scenario IV
$\begin{pmatrix}\theta_H\\\theta_M\\\theta_L\end{pmatrix}$

$\begin{pmatrix}1.3\\0.65\\0.6\end{pmatrix}$

$\begin{pmatrix}1.5\\0.65\\0.6\end{pmatrix}$

$\begin{pmatrix}1.3\\0.65\\0.6\end{pmatrix}$

$\begin{pmatrix}1.3\\--\\0.6\end{pmatrix}$

Impact of population on profits (γ) 1.00 1.00 1.00 1.00
Continuation cost ($$c$$1.15 1.15 1.15
$\begin{pmatrix}1.00\\1.30\\1.15\end{pmatrix}$

Measure of new entrants $$(M)$$ 0.25 0.25 0.25 0.25
Survival probability (β) 0.85 0.85 0.85 0.85
Conditional volatility of shock (σ) 0.20 0.20 0.20 0.20
Autocorrelation of shock (ρ) 0.60 0.60 0.60 0.60
Proportion of types by ability among entrants
$\begin{pmatrix}f_0^H\\f_0^M\\f_0^L\end{pmatrix}$

$\begin{pmatrix}0.3\\0.2\\0.5\end{pmatrix}$

$\begin{pmatrix}0.3\\0.2\\0.5\end{pmatrix}$

$\begin{pmatrix}0.3\\0.2\\0.5\end{pmatrix}$

$\begin{pmatrix}0.25\\0.0\\0.75\end{pmatrix}$

Scaling parameter for shock (μ) 10.00 7.00 5.00 5.00

Figures 7 and 8 illustrate the range of behaviors the model can generate. In Figure 7 the three plots in each row show, on the left, the population and the profits to the low- and mid-ability types along the sample path for $$z_t,$$ which is plotted on the right. The middle panel plots the population and profits for the same parameters and starting values, but for the benchmark of symmetric ability $$\theta_H=\theta_M=\theta_L.$$

Figure 7

Simulated dynamics of the system with three ability types

Each row of subfigures above corresponds to a specification from Table 3. The last column plots the sample path of income shock $$z_t.$$ The middle column represents the equilibrium path of the system with symmetric ability of all traders ($$(\theta_H=\theta_M=\theta_L)$$). The first column illustrates the evolution of the system in which some agents are more skilled than others. In each plot, the profit is on the right-hand vertical axis, and the number of traders $$N_t$$ is on the left-hand vertical axis.

Figure 7

Simulated dynamics of the system with three ability types

Each row of subfigures above corresponds to a specification from Table 3. The last column plots the sample path of income shock $$z_t.$$ The middle column represents the equilibrium path of the system with symmetric ability of all traders ($$(\theta_H=\theta_M=\theta_L)$$). The first column illustrates the evolution of the system in which some agents are more skilled than others. In each plot, the profit is on the right-hand vertical axis, and the number of traders $$N_t$$ is on the left-hand vertical axis.

Figure 8

Variation in time spent in low-population equilibria: The effects of varying the continuation cost, $$c,$$ and the dispersion of abilities, and population effect on surplus, γ

Other parameters are at the values in scenario IV in Table 3. In the first row the cost, $$c$$ is 1.0, 1.15, and 1.3, left to right, respectively. In the second row, the three values of ability, θ are (0.6,0.65,1.3), (0.65,0.65,1.3), and (0.6,0.6,1.3), left to right. In the bottom row, γ and μ are set to (1.0,10), (0.75,10), and (0.75, 7.46), left to right.

Figure 8

Variation in time spent in low-population equilibria: The effects of varying the continuation cost, $$c,$$ and the dispersion of abilities, and population effect on surplus, γ

Other parameters are at the values in scenario IV in Table 3. In the first row the cost, $$c$$ is 1.0, 1.15, and 1.3, left to right, respectively. In the second row, the three values of ability, θ are (0.6,0.65,1.3), (0.65,0.65,1.3), and (0.6,0.6,1.3), left to right. In the bottom row, γ and μ are set to (1.0,10), (0.75,10), and (0.75, 7.46), left to right.

In the first row, the scaling parameter μ is relatively high, so that for a given population, expected future profits are high. This is evident in the sample paths for $$z_t$$ on the right-hand side of the figures: the average of simulated profit in scenario I is larger than that of scenarios II and III. As a result of this, the population in this case converges smoothly to a high-population steady state. In scenarios II and III, where there are lower unconditional expected aggregate profits, the population along some sample paths can collapse in response to a negative shock, when suddenly the higher-population equilibrium ceases to be viable. In the model with symmetric ability convergence to the steady state is smooth, because in that case there is only one steady state. With asymmetric ability, the percentage drop in the population can be much bigger than the associated exogenous shock to profits, as is evident after comparing the behavior of the two series 20 periods out in the left and right subfigures in the middle row. In this case, where the advantage to ability for the highest types is large, the industry remains stuck in the low-population equilibrium even when the fundamentals recover. The refreshment coming from the new entry of low-ability agents is never sufficient to make continuing in the industry viable for the low-ability traders. For the parameters we considered in scenario II, the low-ability agents’ payoffs are proportional to $$\frac{1}{2}+(\theta_L-\bar\theta),$$ which is negative in the low-employment steady state regardless of $$z_t.$$ Thus, no matter how large the realization of the positive shock, $$z_t,$$ the low-population equilibrium thus becomes absorbing.

In the industry simulated in scenario III in Figure 7, the advantage to high ability is lower than in scenario II, and the expected exogenous growth in trading profits is also lower. The industry now shows cycles of growth and crashes. The growth phase is at first rapid, and then more gradual as $$N_t$$ approaches the high-population steady state. These periods of expansion are interrupted by crashes of short duration until the aggregate shock, $$z_t,$$ recovers. For these parameters, $$\frac{1}{2}+(\theta_L-\bar\theta)\gt0,$$ so for high values of $$z_t,$$ expected trading profits will exceed the continuation costs, encouraging the low-ability agents to continue in the industry.

We saw through comparative statics in the deterministic case that a fall in profits that drives out the lowest-ability traders can then render continuation for the middle no longer viable, even when, ignoring the composition effects, their profits would remain positive. Figure 7 illustrates how this can occur in the dynamic setting. In the middle row, a negative shock drives out both the low- and middle-ability types, because profitability for the mid-ability traders positively depends on the presence of lower-ability traders. The bottom row shows that in some cases a negative shock drops the population to an intermediate level, whereas in others it can cause a collapse in which only the highest ability agents survive in the industry.

Figure 8 illustrates the effects of several critical parameters that govern the time spent on average in the low-population, or “bust,” states. Other parameters are held fixed at the values in scenario IV in Table 3, and in each row the sample path for $$z_t$$ is held fixed.

The top row varies the fixed cost of operations, $$c.$$ As it increases across the panels from left to right, the low-population outcomes become more persistent and, the industry recovers much less quickly. For example, regulatory changes that increased administrative overhead could be interpreted as increasing $$c.$$

The middle row of Figure 8 illustrates the impact of changing the dispersion of ability. The left-hand panel simulates the industry with three types of traders as in Figure 7: $$\theta_L=0.6,$$$$\theta_M=0.65,$$ and $$\theta_H=1.3.$$ In the middle panel the lowest ability types are eliminated by raising the value of $$\theta_L$$ to $$\theta_M=0.65.$$ This could be viewed as barriers to entry, such as educational requirements, to screen out the lowest ability types. The effect is to reduce the frequency of crashes, but not their intensity. In the panel on the right this change is reversed, and $$\theta_M$$ is reduced to 0.6, so that the number of types is reduced, but the spread between ability levels is maintained. The resulting behavior is an intermediate case.

The final row of Figure 8 illustrates the effects of changing γ, the parameter that governs how the industry-wide trading profits grow with the population. We adjusted the scaling parameter μ so that $$N_t\gt1,$$ ensuring that a smaller value of γ leads to a higher level of aggregate profits, which are proportional to $$z_tN^{1-\gamma}_t,$$ along the whole simulated path. The left-hand panel provides a benchmark: with $$\gamma =1$$ aggregate trading profits are unaffected by $$N_t.$$ In the middle panel, γ falls to 0.75, so that profits grow with population, whereas μ, the average level of income shock $$z_t,$$ is held fixed. Not surprisingly, the consequence of this change is similar to the effect of raising the scaling parameter μ, which we observed in Figure 7. With sufficient optimism about future profitability, the population grows smoothly to the high steady state. In the right-hand panel, we leave $$\gamma=0.75$$ while lowering the scaling parameter from 10 to 7.46. This value was chosen so that at the high-population steady state, profits are the same as in the left-hand panel with $$\gamma=1.$$ Evidently, in this situation, the low-population states become more persistent through time. This is not surprising, because in on the left aggregate profits recover exogenously, whereas on the right for them to recover requires that the population grow, thus aggravating the coordination problem.

## 4. Conclusion

The zero-sum nature of competition in the financial sector may contribute to fluctuations in employment and volume in that industry. In our model, traders of low ability benefit from the presence of other low-ability traders. It reduces the chances they will encounter predators, who are able to extract from them most of or more than the surplus available in any one trading encounter. This can increase the elasticity of financial activity with respect to the underlying surplus. It also gives rise to multiple steady-state equilibria, with high and low employment. Negative shocks to fundamentals, which would leave traders with intermediate levels of skill profitable, holding fixed the composition of the population, will push those traders out of the industry if others of lower ability exit. Thus, even our very simple setting, with finite types and uncomplicated dynamics for fundamentals, can generate the types of behavior typical of the “boom and bust” employment and liquidity cycles on Wall Street.

We greatly appreciate the comments and advice of Jonathan Berk, Vincent Glode, Nicolas Petrosky-Nadeau, Yaroslav Kryukov, Christian Laux, an anonymous referee, and the editor, Wayne Ferson. Seminar participants at Carnegie Mellon, Frankfurt School of Finance and Management, Norwegian Business School (BI), University of Stavanger, and the University of Vienna also provided valuable comments and suggestions.

### Appendix A. Proofs

Proof of Lemma 1.

Suppose there is a terminal date for the industry, $$t,$$ and for $$t\lt T,$$$$i\gt j,$$$$\mathcal{v}(\theta_i;z_{t+1},N_t,f_t)\ge \mathcal{v}(\theta_j;z_{t+1},N_t,f_t)$$ for all possible values of the state variables. Then $$E_t[\mathcal{v}(\theta_i;z_{t+1},N_t,f_t)]\ge E_t[\mathcal{v}(\theta_j;z_{t+1},N_t,f_t)],$$ and

(A1)
$$\pi(z_t,N_t)\left[\frac{1}{2}+(\theta_i-\bar\theta_t)\right]-c+\beta E_t[\mathcal{v}(\theta_i;z_{t+1},N_t,f_t)]\\ \ge\pi(z_t,N_t)\left[\frac{1}{2}+(\theta_j-\bar\theta_t)\right]-c+\beta E_t[\mathcal{v}(\theta_j;z_{t+1},N_t,f_t)]$$
Because $$\theta_i\gt\theta_j.$$ If the left- and right-hand sides are both positive, then both types will survive and it follows that $$\mathcal{v}^i_t\ge \mathcal{v}^j_t.$$ If the left-hand side is positive and the right-hand side is zero or negative, then type $$\theta_i$$ survives, while type $$\theta_j$$ exits, and the same result follows. If both sides are negative, then both types exit, and both value functions equal zero.

One period prior to the terminal date $$t,$$ the value functions clearly satisfy the inequality of interest, because they consist only of the final, periodic payoff. The value functions for the infinite horizon case, as limits to the value functions as the horizon recedes, then satisfy the inequalities as long as these limits exist.▪

Proof of Lemma 2.

For the case in which profits are positive at the maximum steady-state population, the arguments in the text show that at any $$p_t\lt p^*=1.$$ All incumbents would benefit by choosing $$p^i_t=1.$$ There is no possible population level that can be sustained that is higher than $$\frac{M}{1-\beta},$$ because at this level new entry exactly offsets natural attrition.

When

(A2)
$$\frac{z(1-\beta)^\gamma}{2M^\gamma}-c\lt0,$$
a steady state will be at that population level at which periodic profits are zero: $$N^*$$ as given by Equation 16. The profit function is monotically declining and continuous at in $$N_t.$$ At $$N_t=0,$$ profits are positive by assumption (see (9)). With $$\gamma\gt0,$$ profits must be negative for sufficiently large populations. It follows that $$N^*$$ is unique. Suppose there is a steady-state equilibrium population lower than $$N^*.$$ Profits will be positive at this population level. Because exit is costless in this model, the value function must be nonnegative. Thus, any and all incumbents will prefer to continue, and set $$p_t=1,$$ which implies the population will grow and the conjectured population is not a steady state. Suppose the population level exceeds $$N^*.$$ Then periodic profits will be negative. If this is a steady state, the value function must simply be the discounted value of this infinite stream of negative profits. Thus, all incumbents would prefer to exit—this is a contradiction.▪

Proof of Lemma 3.

The value function for the marginal type, $$\theta^*$$ is zero, so in steady state the periodic profit must be zero, and we know that

(A3)
$$\frac{z}{N(\theta^*)^\gamma}\left(\frac{1}{2}+\left(\theta-\bar\theta(\theta^*)\right)\right)=c.$$
Taking the total differential of this expression yields
(A4)
$$\frac{1}{N^\gamma}\left(\frac{1}{2}+[\theta^*-\bar\theta(\theta^*)]\right)dz-\frac{\gamma z}{N^{\gamma+1}}\left(\frac{1}{2}+[\theta^*-\bar\theta(\theta^*)]\right)dN$$

(A5)
$$+\frac{z}{N^\gamma}\frac{\partial[\theta^*-\bar\theta(\theta^*)]}{\partial\theta^*}d\theta^*=0.$$
Multiplying both sides by $$\frac{N\gamma}{z}$$ and rearranging yields:
(A6)
$$\frac{\frac{dN}{N}}{\frac{dz}{z}}=\frac{1}{\gamma}\left(1+\frac{\partial\left[\theta^*-\bar\theta(\theta^*)\right]}{\partial\theta^*}\frac{z}{\frac{1}{2}+\left[\theta^*-\bar\theta(\theta^*)\right]}\frac{d\theta^*}{dz}\right).$$
Substitute using the zero-profit condition for the trader with ability $$\theta^*,$$ and we obtain the expression in Equation (31). As we saw in Section 2.1, /$$\gamma$$ is the elasticity of the population when ability is symmetric. The second term amplifies or mollifies this effect depending on whether
(A7)
$$\frac{\partial\left[\theta^*-\bar\theta(\theta^*)\right]}{\partial\theta^*}\gt0,$$
because $$\frac{z^2}{cN\gamma}\gt0,$$ and the cutoff ability rises as profits fall, $$\frac{d\theta^*}{dz}\lt0.$$

We can use (28) and (29) to express the average ability in terms of the distribution of types in the entering cohorts:

(A8)
$$\bar\theta=\int^{\theta^*}_{\theta_L}\frac{1-\beta}{1-\beta F_0(\theta^*)}\theta f_0(\theta)d\theta+\int^{\theta^H}_{\theta*}\frac{1}{1-\beta F_0{(\theta^*)}}\theta f_0(\theta)d\theta$$
Now consider the derivative of $$\bar\theta$$ with respect to $$\theta^*.$$
(A9)
$$\begin{array}{l}\bar{\theta}'&=&\frac{\beta f_0(\theta^*)}{(1-\beta F_0(\theta^*))^2}\left[\int^{\theta^*}_{\theta_L}(1-\beta)\theta f_0(\theta)d\theta+\int^{\theta_H}_{\theta^*}\theta f_0(\theta)d\theta\right]\\&&+\frac{1}{1-\beta F_0(\theta^*)}[(1-\beta)\theta^*f_0(\theta^*)-\theta^*f_0(\theta^*)]\\&=&\frac{\beta f_0(\theta^*)}{1-\beta F_0(\theta^*)}(\bar\theta(\theta^*)-\theta^*).\end{array}$$
It then follows that
(A10)
$$\frac{\partial[\theta^*-\bar\theta(\theta^*)]}{\partial\theta^*}=\frac{1}{1-\beta F_0(\theta^*)}(1-\beta F_0(\theta^*)-\beta f_0(\theta^*)(\bar\theta-\theta^*)).$$
Opening the brackets, we can rewrite the above equation as
(A11)
$$\frac{\partial[\theta^*-\bar\theta(\theta^*)]}{\partial\theta^*}=1-(\bar\theta-\theta^*)\frac{\beta f_0(\theta^*)}{1-\beta F_0(\theta^*)}.$$

Proof of Lemma 4.

By Equations (20) and (27), the populations in the two steady states are related as

(A12)
$$N^*=N(1-\beta(1-f^H_0)),$$
where $$N$$ is the high-population level and $$N^*$$ is the low-population steady-state outcome. Neither depends on γ.

The profits for the high-ability agents in the high-population steady state, less their profits in the low-population steady state are:

(A13)
$$\frac{z}{(N)^\gamma}\left[\frac{1}{2}+(\theta_H-\bar\theta_0)\right]-\frac{z}{(N^*)^\gamma}\frac{1}{2}.$$
Using the relation between the two populations, this can be simplified to:
(A14)
$$\frac{z}{(N)^\gamma}\frac{1}{2}\left(\left[1-\frac{1}{(1-\beta(1-f_0^H))^\gamma}\right]+(\theta_H-\bar\theta_0)\right).$$
The sign of this expression clearly depends on the term in parentheses. The difference between $$\theta_H$$ and $$\bar\theta_0$$ is clearly positive, and does not depend on γ. Because both β and $$f_0^H$$ are fractions, the term in brackets goes to zero as $$\gamma\rightarrow 0,$$ leaving the term in parentheses positive. This term in brackets goes to $$-\infty$$ as $$\gamma\rightarrow\infty,$$ and will determine the sign of the term in parentheses.▪

#### Appendix B. Algorithm for Numerical Solution of System Dynamics

This section describes in detail the optimization problem solved by individual agents, the behavioral assumptions made to render the model tractable, and the steps taken in our paper to numerically compute equilibria and simulate system dynamics. First, we review the general model with a continuous distribution of types (ability levels). Then we describe how we implement the model with a discrete distribution of types. Finally, we describe the numerical algorithm used to solve for the value and policy functions associated with equilibrium outcomes.

##### B.1 Continuous distribution of types

An incumbent individual of type $$\theta_i$$ has the following problem to solve every period $$t:$$ either stay in the market and receive the immediate market payoff less operating costs plus the continuation value, or exit the market and receive reservation value 0. This decision is conditional on the type of the agent, current realization of aggregate productivity shock ($$z_t$$), and the two observable variables describing the competitive environment—the total number of agents active last period ($$(N_{t-1})$$) and the distribution of agents’ types in the previous period ($$(f_{t-1})$$).

Let $$\phi_i(\theta;z_t,N_{t-1},f_{t-1})$$ represent the forecast an agent of type $$\theta$$ makes about the policy pursued by agents of type θ. We consider only symmetric equilibria, where all agents of a given type follow the same strategy and share the same beliefs about the future choices of others. We then search for rational expectations equilibria at which these beliefs are, in turn, consistent with the realized distribution of types.

The agent’s value function can be written as:

(B1)
$$\mathcal{v}(\theta_i;z_t,N_{t-1},f_{t-1})=\max\limits_{p\in[0,1]}p\left(\pi(z_t,N_t)\left[\frac{1}{2}+(\theta_i-\bar \theta_t)\right]-c+\beta E_t[\mathcal{v}(\theta_i;z_{t+1},N_{t},f_{t})]\right),$$
where
(B2)
$$\bar\theta_t=\int^{\theta^H}_{\theta^L}\theta f_t(\theta)d\theta$$
is the average ability level, and where
(B3)
$$f_t(\theta)=\begin{cases}\frac{Mf_0(\theta)+\phi_i(\theta;z_t,N_{t-1},f_{t-1})\beta f_{t-1}(\theta)N_{t-1}}{N_t}&\theta\ne\theta_i,\\\frac{Mf_0(\theta)+p\beta f_{t-1}{(\theta)}N_{t-1}}{N_t}&\theta\ne\theta_i,\end{cases}$$
is the agent’s forecast of the distribution of types. The agent’s forecast of the population will be
(B4)
$$N_t=M+\int^{\theta_H}_{\theta_L}\mathbb{1}_{\theta\ne\theta_i}\phi_i(\theta;z_t,N_{t-1},f_{t-1})\beta f_{t-1}(\theta)N_{t-1}d\theta+ \int^{\theta_H}_{\theta_L}\mathbb{1}_{\theta\ne\theta_i}p\beta f_{t-1}{(\theta)}N_{t-1}\,d\theta.$$

Beliefs about actions of other traders influence the decision of the agent $$i$$ through their effects on $$N_t,$$$$f_t,$$ and $$\bar\theta_t,$$ and hence, on the magnitude of the payoff that this agent expects to receive if he stays in the market.

We assume that $$\int^{\theta_H}_{\theta_L}\mathbb{1}_{\theta=\theta_i}f_0(\theta)d\theta=0$$ or, in other words, that each type is of zero measure. From this it follows that $$\partial\bar\theta_t/\partial p=0$$ and $$\partial N_t/\partial p=0.$$ Neither the mass of agents active in the market nor the average ability level are affected by the individual decisions of agents. By varying $$p,$$ the agent only affects the multiplier on the brackets in the maximization problem, and not the value of the payoff inside the brackets. Then as long as his payoff to staying in the market (current payoff + continuation value) is nonnegative, $$p=1.$$ Otherwise, $$p=0.$$ When the distribution of types is discrete, we continue to assume the measure of any individual is zero, and that they view the effect of their own choice on the aggregate outcomes as negligible. As discussed in the next section, however, there may be interior choices of $$p,$$ or mixed strategies.

We are seeking a rational expectations equilibrium. That is, for each trader with ability level $$\theta_i,$$ we are looking to find $$\mathcal{v}(\theta_i;z_t,N_{t-1},f_{t-1})$$ and corresponding $$p(\theta_i;z_t,N_{t-1},f_{t-1})$$:

• that solve the trader’s problem;

• such that $$p(p(\theta_i;z_t,N_{t-1},f_{t-1})=\phi_j(\theta_i;z_t,N_{t-1},f_{t-1})$$ for all $$j.$$

In such an equilibrium, the state variables evolve according to the following laws of motion along the equilibrium path:

(B5)
$$N_t=M+\int^{\theta_H}_{\theta_L}p(\theta;z_t,N_{t-1},f_{t-1})\beta f_{t-1}(\theta)N_{t-1}\,d\theta,$$

(B6)
$$f_t(\theta)=\frac{Mf_0(\theta)+p(\theta;z_t,N_{t-1},f_{t-1})\beta f_{t-1}(\theta)N_{t-1}}{N_t}.$$

(B7)
$$Z_t=\mu e^{\varepsilon_t},\,\varepsilon_t=\rho\varepsilon_{t-1}+e_t$$
The exogenous productivity shock, $$e_t,$$ is iid normal with zero mean and variance σ, and μ plays the role of a scaling parameter.

As is evident from the analysis of the steady states in the deterministic case, there may be multiple equilibrium strategies at any point in time. Moreover, to solve the model, we must specify agents’ forecasts of the choices of others, which will depend on which equilibrium they think will prevail. In solving the model we select among the possible equilibria by favoring the high-population outcomes. That is, in looking forward, agents assume that others will coordinate on equilibria at which more people are employed in the industry. This choice of focus could be rationalized by arguing that decision makers in the financial industry gain nonpecuniary (and unmodeled) benefits from working in a growing industry, or bear costs associated with shrinking their organizations. The role this assumption plays in actually computing equilibrium outcomes will be apparent at the end of the final section of this note.

##### B.2 Two types of agents: high and low ability

Consider a version of the problem outlined above but where θ takes only two values $$\theta_H$$ and $$\theta_L.$$ The distribution of types over ability levels can then be summarized by a single number, denoted by $$f_t$$ the proportion of high-ability traders present in the market in period $$t.$$

Importantly, with discrete number of types, there is nonzero mass of agents of the same type. In principle, there might exist equilibria at which agents of the same type given the same set of state variables can make different choices. But we restrict our attention to symmetric equilibria—agents of the same ability act in exactly the same way. Because it might be the case that it is not profitable for all agents of the same type to stay in the market, we consider choices $$p(\theta_i;z_t,N_{t-1},f_{t-1})$$ between 0 and 1 as agents playing mixed strategies—to stay in the market with probability given by $$p.$$

(B8)
$$\mathcal{v}(\theta_i;z_t,N_{t-1},f_{t-1})=\max\limits_{p\in[0,1]}p\left(\pi(z_t,N_t)\left[\frac{1}{2}+(\theta_i-\bar\theta_t)\right]-c+\beta E_t[\mathcal{v}(\theta_i;z_{t+1},N_{t},f_{t})]\right).$$
where $$\theta_i\in\{\theta^L,\theta^H\},$$
(B9)
$$\bar\theta_t=\theta^Hf_t+\theta^L(1-f_t),$$

(B10)
$$f_t=\frac{Mf_0+\phi^H_i(z_t,N_{t-1},f_{t-1})\beta f_{t-1}N_{t-1}}{N_t}.$$

(B11)
$$N_t=M+\phi^H_i(z_t,N_{t-1},f_{t-1})\beta f_{t-1}N_{t-1}+\phi^L_i(z_t,N_{t-1},f_{t-1})\beta (1-f_{t-1})N_{t-1}$$

In this setting we see that the beliefs that agent $$i$$ has about the actions of other agents, of his type and of other types, are affecting the payoff through effect on the number of agents $$(N_t),$$ proportion of high-ability traders $$(f_t)$$ and average ability level $$(\bar\theta_t).$$ Each agent is infinitesimally small in mass, however, and ignores the effect his own choice of $$p$$ has on market characteristics (mass of active agents and average ability level).

We are looking for a rational expectations equilibrium, that is, a pair of $$\mathcal{v}(\theta_i=\theta^H; z_t,N_{t-1},f_{t-1})\equiv \mathcal{v}^H(z_t,N_{t-1},f_{t-1})$$ and $$\mathcal{v}(\theta_i=\theta^L; z_t,N_{t-1},f_{t-1})\equiv \mathcal{v}^L(z_t,N_{t-1},f_{t-1})$$ and the corresponding $$p^H(z_t,N_{t-1},f_{t-1})$$ and $$p^L(z_t,N_{t-1},f_{t-1}),$$ such that optimality conditions are satisfied for both high- and low-ability traders and beliefs are consistent with optimal choices.

To simulate the dynamics of the market for an arbitrary starting point $$t$$ = 0 and a path of $$z_t,$$ we solve numerically for $$\mathcal{v}^H(z_t,N_{t-1},f_{t-1})$$ and $$\mathcal{v}^L(z_t,N_{t-1},f_{t-1}).$$

##### B.3 Algorithm

The state space for $$z,$$$$N,$$ and $$f$$ is discretized. We make an initial guess for $$v^H$$ and $$v^L,$$ denote them by $$\mathcal{v}^H_0$$ and $$\mathcal{v}^L_0.$$ We approximate the value functions using a piecewise linear basis, so the initial guesses consist of two matrices. Then we update the guess on the two value functions.

Denote by $$u^H(p^H,p^L;z,N,f,\mathcal{v}^H_i,\mathcal{v}^L_i)$$ the profit of a high-type agent given decisions of low type $$(p^L)$$ and hight type $$(p^H),$$ as well as current guess on the continuation values. According to Lemma 1 in the paper, low-type agents will find it profitable to stay only if high-ability traders are earning nonzero profit.

For every state space node $$(z,N,f)$$:

1. If $$u^H(p^H=1,p^L=0;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i)\lt0,$$ that is, when only all high ability traders stay, they suffer losses, then $$p^L=0$$ and $$p^H$$ is found in such a way that $$u^H(p^H,p^L=0;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i)=0.$$ If $$u^H(p^H=1,p^L=0;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i)\gt0,$$ then there is enough profit to support all high ability traders, and hence we need to find out if it would be profitable for low ability traders to stay.

2. First we check “all stay” equilibrium, that is, when all low abiility traders decide to stay. If $$u^L(p^H=1,p^L=1;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i)\gt0,$$ then $$p^H=1,$$$$p^L=1.$$ If this is not the case, meaning $$u^L\lt0$$ with all low type agents staying, then we proceed to find smaller population equilibria.

3. We check if $$p^H=1,$$$$p_L=0$$ is an equilibrium. That is, if $$u^L(p^H=1,p^L=0;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i)\lt0.$$ Because of positive participation externality, $$u^L$$ is increasing in $$p^L$$ for the range of parameter values that we focus our analysis on.9 Hence, if $$u^L(p^H=1,p^L=0;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i)\gt0$$ we proceed to the next step.

4. It must be the case that $$p^L$$ is interior and $$p^H=1.$$ That is, we find $$p^L$$ such that $$u^L(p^H=1,p^L;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i)=0.$$

Next, we use computed $$p^H$$ and $$p^L$$ to update the value functions: $$\mathcal{v}^H_{i+1}(z,N,f)=u^H(p^H,p^L;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i)$$ and $$\mathcal{v}^L_{i+1}(z,N,f)=u^H(p^H,p^L;z,N,f;\mathcal{v}^H_i,\mathcal{v}^L_i).$$ Then we compute the distance from the guess that we have started with ($$\mathcal{v}^H_{i}$$ and $$\mathcal{v}^L_{i}$$) and if the distance is below a tolerance level, we declare $$\mathcal{v}^H_{i+1}(z,N,f)$$ and $$\mathcal{v}^L_{i+1}(z,N,f),$$ and the corresponding $$p^H(z,N,f)$$ and $$p^L(z,N,f)$$ to be a solution.

Note that each of possible 4 cases outlined above can happen to be optimal only if the previous did not fulfil the equilibrium requirement. That is, before checking if $$p^H=1,$$$$p^L=1$$ can be a solution (case 2), we first rule out $$p^H$$ interior, $$p^L=0$$ as an optimal solution. It is crucial to observe that because of positive participation externality, both $$p^H=1,$$$$p^L=1$$ (case 2) and $$p^H=1,$$$$p^L=0$$ (case 3) can satisfy optimality conditions. In that case, we favor larger population outcome, which is apparent in the ordering in which cases are being checked. That is, if both cases 2 and 3 satisfy equilibrium reguirements, we chose case 2.

The generalization of this approach to three types is straightforward, though a bit tedious.

2 See the report “Commodities Trading, March 2011,” at www.TheCityUK.com.
3 Although longer series are available for more aggregated industry sectors (i.e., financial services generally), this period allows us to consider employment at the more specific level of “Securities, Commodities and Investment,” more closely corresponding to the trading services that are the focus of our model.
4 “Wall Street Pay Hits a Wall,” Wall Street Journal, November 28, 2011
5 Clearly, if traders could collude or coordinate their actions, they could find ways to avoid redundant operating costs and increase aggregate surplus. As is typical in models of competition or oligopoly within industries, however, we assume they act noncooperatively.
6 The value function for new entrants, of course, is simply the expectation across the distribution of ability types, using the density $$f_0(\theta).$$
$$\mathcal{v}^0(z_t,N_{t-1},f_{t-1})=\int^{\theta_H}_{\theta_L}f_0(q)\mathcal{v}(q;z_t,N_{t-1},f_{t-1})dq,$$
which must be above the entry costs to be consistent with voluntary entry by the newly arriving generation of traders. With endogenous entry, agents that decide to continue in the market deter the entry of newcomers. Hence, an agent of a given type benefits from the presence of other agents of a similar type remaining in the market not only because of lower chances of trading with higher types, but also because of smaller numbers of entrants. Overall, endogenous entry leads to periods of greater contraction until natural attrition reduces the population sufficiently to trigger new entry.
7 At any conjectured $$p^*\lt1,$$ any incumbent would earn higher expected profits by choosing $$p^i_t\gt p^*$$ for the current period, and could always exit to avoid expected losses in subsequent periods. Hence, $$p^*=1.$$
8 The values of μ and $$c,$$ the per-period operating cost, jointly determine the average profitability. For instance, the expected periodic payoff (with symmetric ability) is $$\frac{1}{2}\frac{\mu}{N^\gamma_t}-c.$$ In other words, for one \$1 of revenue, the expected cost is $$\frac{2cN^\gamma_t}{\mu}.$$
9 We have a formal proof that profit of low-type agent is increasing in the proportion of low-type agents staying in the market in the setting with nonstochastic profit and endogenous entry.

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