Blockbusting: Brokers and the dynamics of segregation€¦ · models of urban segregation (Schelling [33], Benabou [6,7], Becker and Murphy [5], Frankel and Pauzner [19]) do not explain

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Journal of Economic Theory 157 (2015) 811–841

www.elsevier.com/locate/jet

Blockbusting: Brokers and the dynamics

of segregation ✩

Amine Ouazad 1

Received 5 July 2013; final version received 7 February 2015; accepted 13 February 2015

Available online 23 February 2015

Abstract

The paper presents a dynamic model of neighborhood segregation where fee motivated real estate bro-kers match sellers optimally either to minority or to white buyers. In an initially all-white neighborhood, real estate brokers thus either keep the neighborhood in a steady-state white equilibrium or trigger racial transition by matching sellers to minority buyers, a process called blockbusting. Racial transition leads to a higher rate of property turnover in the neighborhood once the fraction of minorities has reached a tipping point—but racial transition also leads to lower prices, and this is the trade-off faced by a broker. The model shows that with multiple brokers, blockbusting profit per broker is lower as brokers free ride on each other’s groundbreaking efforts. The model predicts that racial transition will happen in the neighborhood when (i) the number of brokers is limited, (ii) racial preferences lie in an intermediate range, (iii) the arrival rate of offers is intermediate. Otherwise, real estate brokers steer white households toward white buyers.© 2015 Elsevier Inc. All rights reserved.

JEL classification: C72; C73; D62; R00

Keywords: Dynamic games; Segregation; Multiple equilibria

✩ I would like to thank two anonymous referees as well as Roland Bénabou, Leah Platt Boustan, Ambika Gandhi, Denis Gromb, Maria Guadalupe, David Hemous, Francis Kramarz, Guy Laroque, François Maniquet, Vladimir Mares, Eric Maurin, Scott Page, Thomas Piketty, Romain Rancière, Albert Saiz, Xavier Vives, Tim Van Zandt, and Yves Zénou for suggestions and comments on preliminary versions of the paper. I would also like to thank the audiences of the Urban Economics Conference, CREST, the London School of Economics, INSEAD, and CORE at the Université Catholique de Louvain. The usual disclaimers apply.

E-mail address: [email protected] Assistant Professor of Economics, INSEAD.

http://dx.doi.org/10.1016/j.jet.2015.02.0060022-0531/© 2015 Elsevier Inc. All rights reserved.

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812 A. Ouazad / Journal of Economic Theory 157 (2015) 811–841

Typically the practice, known as blockbusting, involves an agent claiming that property values will drop because members of minority groups are moving into the neighborhood. The agent tries to persuade the homeowner to let him sell the house before the values drop, and the agent then gains a commission.

[U.S. is Investigating Charges of Blockbusting, New York Times, October 1995.]

1. Introduction

Despite increasing levels of ethnic and racial diversity, racial segregation is a defining feature of American cities. According to the 2010 Census, the average urban,2 African American house-hold lives in a neighborhood that is only 35% white (Logan and Stults [27]).3 Empirical evidence suggests that racial segregation has adverse welfare consequences (Cutler and Glaeser [16], Alesina and Ferrara [3], Card and Rothstein [15], Boustan and Margo [13]). It is interesting that, while overall racial segregation across neighborhoods remains high, the racial composition of some neighborhoods changes dramatically over short periods of time.

Social interaction models explain the mechanisms of neighborhood tipping (Schelling [33]), whereby the entry of a small number of minority residents in a neighborhood is followed by large outflows of white households, departures that are often referred to as white flight (Grubb [22], Boustan [11,12]). Card, Mas and Rothstein [14] present U.S. evidence of neighborhood tipping in recent decades, where the fraction of minority residents that triggers large departures of white households ranges from 5% to 20%.4

Historical evidence (Helper [23], Orser [31], Gotham [21]) as well as law articles (Glass-berg [20], Mehlhorn [29]) suggests that brokers play a decisive role in neighborhood tipping. The U.S.-based National Association of Real Estate Boards found the issue sufficiently concerning that, until 1956, Article 34 of Part III of its Code of Ethics specified that “a realtor should never be instrumental in introducing into a neighborhood a character of property or occupancy, mem-bers of any race or nationality, or any individuals whose presence would clearly be detrimental to property values in that neighborhood.”5 A series of congressional hearings during develop-ment of the 1968 Civil Rights Act considered the role of real estate brokers in neighborhood change. These hearings led to section 804[e] of Title VIII of that legislation,6 which prohibits blockbusting7:

[e] For profit, to induce or attempt to induce any person to sell or rent any dwelling by rep-resentations regarding the entry or prospective entry into the neighborhood of a person or persons of a particular race, color . . . .

2 We consider a household to be urban when it resides in a metropolitan statistical area (MSA).3 Bayer, McMillan and Rueben [4] suggest that segregation by socioeconomic characteristics does not fully explain

racial segregation.4 Swedish evidence of neighborhood tipping in response to immigrant inflows is described by Aldén, Hammarstedt

and Neuman [1]. The U.S. evidence on tipping is disputed by Easterly [18].5 “Race or nationality” was removed in 1956, but the rest of the article remains in the Code of Ethics.6 Title VIII of the 1968 Civil Rights Act is also called the 1968 Fair Housing Act.7 Newspapers also reported a large number of alleged cases of blockbusting. At the time this article was written, the

New York Times had published 844 real estate articles on blockbusting, e.g. “New Neighbors Pushing at the Edge; Brooklyn Hasidim Seek to Expand Into a Black and Hispanic Area,” New York Times, July 19, 1999; “Town Tries to Keep Its Balance in Wake of White Flight,” March 11, 1996.

A. Ouazad / Journal of Economic Theory 157 (2015) 811–841 813

And although the word blockbusting was arguably coined in U.S. literature, there is evidence in other countries including Canada (Teixeira and Murdie [35]), France (Bonnet, Safi and Lalé [10]), and Greece (Drydakis [17]) that real estate brokers’ steering behavior may be able to shape urban racial segregation.

However, to our knowledge, no theoretical framework describes real estate brokers’ incen-tives and optimal strategies in the process of neighborhood segregation, and in particular in the process of neighborhood tipping. There is substantial evidence that brokers steer—i.e., match—minority and white buyers and sellers to particular houses, particular households, and/or partic-ular neighborhoods (Turner and Mikelsons [36], Ondrich, Ross and Yinger [30]). Yet dynamic models of urban segregation (Schelling [33], Benabou [6,7], Becker and Murphy [5], Frankel and Pauzner [19]) do not explain whether brokers have an incentive to match sellers to same-race buyers or to buyers of another race; and when brokers have incentives and are able to substan-tially change a neighborhood’s racial composition.

This paper focuses on the ability and the incentives of a real estate broker to engage in block-busting. It presents a dynamic model of a neighborhood where, in each period, a fee-motivated real estate broker matches sellers to either white buyers, or to black buyers, i.e. blockbusts.8

Starting with an initially all-white neighborhood,9 the broker chooses to steer sellers to black or white buyers based on comparing the present discounted value of his brokerage fee revenue in an all-white neighborhood with that results if the same neighborhood experiences racial transition.10

We assume that white households prefer living with white neighbors to live with minority neighbors. Households have forward-looking expectations of the neighborhood’s racial compo-sition,11 and may experience negative shocks, e.g. a job loss on the base utility they derive from the neighborhood. With such shocks the neighborhood has a natural turnover rate that gener-ates brokerage fees even when its racial composition does not change. When the broker matches white sellers to minority buyers, there is an increase in the fraction of minority households in the neighborhood. At first, only Whites experiencing a negative shock sell to minority buyers, turnover is low. We show that there is a unique tipping point: a fraction of minorities in the neighborhood beyond which all white households—regardless of whether they are experiencing the shock or not—are ready to sell. Then turnover is high, although prices (and thus brokerage fees per transaction) are lower than in the all-white neighborhood.

The broker thus faces a trade-off. On the one hand, maintaining an all-white neighborhood ensures high prices and high brokerage fees but results in low turnover. On the other hand, triggering a racial transition generates sequential patterns of transactions and fees. Prices and turnover are initially low: high-valuation white households do not sell (low turnover) and white households expect a neighborhood change (low prices). But when the neighborhood reaches its tipping point, turnover increases and the broker realizes higher revenues. After the transition, however, turnover and prices decline to level below those for an all-white neighborhood. Given

8 The model’s mechanisms do not rely on the particular labeling “black” and “white” of these specific groups. Evidence suggests that immigrant inflows may also cause neighborhood tipping (Saiz and Wachter [32]). Section 804 of the 1968 Civil Rights Act mentions “race, color, religion, sex, familial status, or national origin.”

9 In law, Bisceglia [8] defines blockbusting as “the first sale in a previously all white area to a black—so that the racial homogeneity of the block is ‘busted’.”10 Thus the model endogenizes the steering behavior of real estate brokers. Aleinikoff [2] states: “The quickest and surest sales can be made by satisfying buyer preferences [...] members of the buyer’s own race.”11 In contrast, Becker and Murphy [5] assume myopic expectations. Frankel and Pauzner [19] feature forward-looking expectations: when white households expect neighborhood change, tipping occurs earlier than in a model with myopic expectations; and the number of equilibria is reduced. But the paper does not address steering by brokers.


these dynamics, the broker’s incentive to trigger “white flight” depends on whether or not the medium-term benefits of a large number of transactions (more than) offset the lower transaction fees throughout the transition and in the long run.

The broker’s fee incentives provide a simple necessary and sufficient condition for racial transition: the broker’s blockbusting is a Markov perfect equilibrium strategy if and only if the present discounted value of brokerage fees is higher in racial transition than in a steady-state white neighborhood. Moreover, if the arrival rate of offers is low, the neighborhood’s equilibrium path and long-run racial composition are uniquely determined by the broker’s fee incentives.

We then turn to the effect of key parameters on the occurrence of blockbusting. First, impatient brokers may be less inclined to practice blockbusting because of the lower short-term brokerage fees (before the tipping point is reached). Moderately patient brokers are willing to earn lower revenues at first as long as higher revenues can be reasonably anticipated in the medium term (i.e., for some time at and after the tipping point). Finally, brokers who are extremely patient do not benefit from blockbusting owing to the lower long-term brokerage fees. Hence, blockbusting occurs only for a bounded range of broker discount rates. Second, we analyze the effect of white households’ racial preferences. In the case of strong racial preferences, the tipping point would be reached rapidly but transaction prices would fall rapidly as well. When racial preferences are weak, the tipping point is not reached until after a long period of time characterized by low turnover and low transaction fees. Thus blockbusting generates more revenue only for an intermediate range of racial preferences.

Third, blockbusting incentives are higher when minorities’ value of housing outside the neigh-borhood is lower. This result reflects the impact of minorities’ outside options on transaction prices and on the tipping point. When the options of minority households improve, transaction prices (and brokerage fees) decrease. At the same time, the tipping point occurs later because minority households have a lower relative valuation of the target neighborhood. Both effects re-duce the broker’s revenue. Hence, blockbusting occurs only when the outside option for minority households is sufficiently low. These results shed light on historical accounts that brokers’ block-busting activity generated substantial brokerage revenue in metropolitan areas where the quality of housing supplied to minority households was low.12

A higher rate of arrival of offers has an ambiguous effect on blockbusting revenues: although a higher rate hastens the tipping point, it also implies a more rapid entry of minorities; that, in turn, reduces prices because white households’ valuation of the neighborhood incorporates forward-looking expectations of neighborhood change. For a bounded range of offer arrival rates, the broker’s blockbusting revenue is higher than his revenue from a steady-state white neighbor-hood. For high rates of offers, blockbusting revenue is actually lower than revenue generated from a steady-state white neighborhood.

Thus racial transition will only occur when (i) the value of housing outside the neighborhood for minority households is low, and (ii) white households’ racial preferences, (iii) the broker’s time discount factor, and (iv) the arrival rate of offers lie in an intermediate range. In particular, the model suggests that steering white households toward black sellers does not generate more revenue if any of racial preferences, offer arrival rates, or broker impatience are excessively high.

After the one-broker case, we then consider a two-broker model—our second possible mar-ket mechanism. With two brokers, in equilibrium, one broker (say, broker 1) matches sellers to

12 Mehlhorn [29] and Orser [31] describe successful blockbusting in white neighborhoods that were close to black neighborhoods with slum or low-quality housing.


minority buyers all along while the other broker (say, broker 2) free-rides on the groundbreaking efforts of broker 1 to change the neighborhood’s racial composition. Indeed, blockbusting ini-tially involves lower commission fees. In particular, broker 2 initially matches sellers to white buyers and only later—when the commissions from matching to minority buyers exceed those from matching to white buyers—sells also to minorities.

These results—when combined with those on the arrival rate of offers—imply that, with a large number of brokers, blockbusting revenue is less than the revenue from a steady-state white neighborhood.

Finally, we consider a neighborhood where trades are not intermediated, our third possible market mechanism. Households sell their house to the highest-valuation buyer. In contrast, the broker may match sellers to black buyers even when such black buyers are not the highest-valuation buyers, and the broker can trigger racial transition when, with the same parameters, and with direct trades, racial transition would not occur in equilibrium.

The paper proceeds as follows. Section 2 presents the model and provides existence and uniqueness results. Section 3 gives a criterion for the existence of a blockbusting equilibrium. Section 4 analyzes the effects of key parameters on the broker’s incentives to blockbust: the effects of racial preferences, the broker’s discount rate, the minorities’ outside option, and the arrival rate of offers. Section 5 presents the second alternative market mechanism for trades: two brokers. Section 6 introduces the third market mechanism, direct, non-intermediated trades. Sec-tion 7 shows that blockbusting leads to a trade-off between incumbent Whites’ welfare losses and black households’ welfare gains. Section 8 concludes. The online appendix presents the model with a continuum of brokers (the fourth market mechanism), and the impact of endogenous com-mission fees.

2. The model

We model, in discrete time t = 1, 2, 3, . . . , the evolution of a neighborhood that has a con-tinuum i ∈ [0, 1] of houses. Each house is occupied by one household and each household lives in one house. Households are either black, r = b, or white, r = w.13 The race of the house-hold living in house i in period t is noted r(i, t). The fraction of Blacks in the neighborhood is xt = ∫ 1

0 1(r(i, t) = b)di.A household derives utility vlr > 0 per unit of time when living outside the neighborhood.

A household of race r ∈ {w, b} derives utility vr − ρrxt in period t when living in the neigh-borhood, with vr > 0. Thus, households of race r incur a disutility from black neighbors when ρr > 0, and have a preference for black neighbors when ρr < 0.

In addition, the household living in house i ∈ [0, 1] is affected by an i.i.d. shock ε with proba-

bility λ ∈[0, 1

2

]in each period t ; when a household of race r is affected by the shock, he derives

utility vr − ρrxt − ε in the period instead of vr − ρrxt .14 When household i is affected by the shock in period t , the household is said to be in the distressed state (s(i, t) = D), and households who are not affected by the shock are in the relaxed state (s(i, t) = R).

13 For simplicity the two groups are labeled “black” and “white.” Instances of blockbusting with other minority groups have been described (Seligman [34]). The model’s intuition does not depend on the particular labeling of racial groups, but only on households’ preference for same-group neighbors.14 The shock is, for instance, due to job loss, or family events such as divorce or changes in family size.


Households’ time discount factor is β ∈ (0, 1). The value of living outside the neighborhood for a household of race r is noted Vlr = vlr/ (1 − β).15

2.1. Market mechanism

We consider here, and up to Section 4, a simple market mechanism: all trades between buyers and sellers are intermediated by a single broker. Alternative market mechanisms (two brokers, continuum of brokers, direct trades) are introduced in Sections 5 and 6 and in the online appendix.

There is one broker who intermediates all trades. With i.i.d. probability δ ∈ (0, 1) in each period, the household living in house i ∈ [0, 1] is matched to a buyer by the broker. There is an infinite pool of potential buyers. In each period t , the broker chooses whether to match sellers to buyers of race b or of race w. We write μ(i, t) = ∅ if household i has not been matched to a buyer, so that P(μ(i, t) = ∅) = 1 − δ. We write μ(i, t) = b when household i has been matched to a black buyer by the broker and μ(i, t) = w when household i has been matched to a white buyer. A transaction happens whenever the broker has matched a buyer and a seller and the seller chooses to sell to the buyer. If household i sells in period t to its matched buyer, a(i, t) = sell, otherwise a(i, t) = stay.

The broker does not observe the state (s(i, t) ∈ {D, R}) of each individual household but the probability λ of the shock ε is common knowledge. The broker observes the fraction of Blacks xt in the neighborhood at time t .

At time t , a history of the neighborhood ht is an initial racial composition r(i, t = 0), a sequence of matches μ(i, k), actions a(i, k), and shocks s(i, k) for all previous periods k = 1, 2, . . . , t and for all households i ∈ [0, 1].

2.2. Strategies

The set of broker and household strategies is potentially large, as a strategy for the broker or a household is a complete contingency plan that specifies actions at each period t and for each hypothetical history ht of the game. Following Maskin and Tirole [28], we thus restrain the set of possible household and broker strategies to the set of Markovian strategies, i.e. strategies depending only on payoff relevant state variables.

The broker’s payoff-relevant state variable is the fraction of Blacks in the neighborhood.16

A strategy for the broker is a race m(x) ∈ {w, b} of the matched buyers for the households of a neighborhood where the fraction of blacks is x, for each x ∈ [0, 1]. While the race of the match is chosen by the broker, the probability of matching is exogenous and i.i.d.

The strategy of the household living in house i in period t assigns an action σ it (x, s, μ) ∈

{sell, stay} to a household in state s ∈ {D, R} matched (μ �= ∅) to a buyer of race μ ∈ {w, b}, in a neighborhood with a fraction x ∈ [0, 1] of Blacks. We focus on symmetric Markovian strategies, i.e. strategies that depend on the race of the household in house i at time t , so that, σ i

t (x, s, μ) =σ r(i,t)(x, s, μ) where r(i, t) ∈ {b, w} is the race of household i in period t .

15 A summary of notations is provided in Table 1.16 In the language of Maskin and Tirole [28], the broker’s preferences are identical over two histories ht and h′

t with the same fraction xt of Blacks at time t , and hence a partition of histories at t by their fraction of black households xt at t is action-space-invariant.


Definition 1 (Strategy profile). The set S of symmetric Markovian strategy profiles is the set of triplets of broker and household strategies (m, σw, σb) where m : x ∈ [0, 1] → {w, b}, σ r :(x, s, μ) ∈ [0, 1] × {R, D} × {w, b} → {sell, stay} for r ∈ {w, b}.

2.3. Profit and valuation functions

Given a strategy profile (m, σw, σb), the neighborhood’s racial dynamics, households’ valua-tion of the neighborhood, and the broker’s profit are defined as follows.

Racial dynamics

The neighborhood’s racial dynamics are driven by the broker’s choice of buyers’ race m(x)

at any black fraction x and by households’ decision to sell σw(x, s, μ) and σb(x, s, μ) for each fraction of Blacks x, in each state s and for each matched race μ = m(x). The next period’s black fraction x′(x) is:

x′(x) = x + δ{1m(x)(b) · (1 − x) · [λ · 1σw(x,D,b) + (1 − λ) · 1σw(x,R,b)

]− (1 − 1m(x)(b)) · x · [λ · 1σb(x,D,w) + (1 − λ) · 1σb(x,R,w)

]}(1)

where 1m(x)(b) = 1 if and only if m(x) = b, and 0 otherwise; and 1σ = 1 if and only if σ = sell, 0 otherwise. The change x′ −x in the fraction of Blacks in the neighborhood is due to two terms. The first term (positive) in δ is the number of Whites who sell to black buyers, thus increasing the fraction of black households in the neighborhood. The second term (negative) in δ is the number of Blacks who sell to white buyers. When the broker matches sellers to white buyers, 1m(x) = 0(resp., to black buyers, 1m(x) = 1) the fraction of Blacks weakly decreases, x′ ≤ x (resp., weakly increases, x′ ≥ x).

Households’ valuation

The value of living in the neighborhood for household i of race r depends on (i) the flow utility vr of the neighborhood, (ii) the current racial composition of the neighborhood x, (iii) the shock ε if the household is distressed (s = D) and on (iv) household i’s expected valuation of the neighborhood Es′,μ′

[V r(x′, s′,μ′))

]in the next period. The household may also have an

opportunity to sell to a buyer of race μ ∈ {w, b} if μ �= ∅:

V r(x, s,μ) =⎧⎨⎩

vr − ρrx − 1s(D)ε + βE(V r(x′, s′,μ′)

)if μ = ∅(

1 − 1σ r(i,t)(x,s,μ)

) · [vr − ρrx − 1s(D)ε + βEs′,μ′(V r(x′, s′,μ′)

)]+ 1σ r(i,t)(x,s,μ) · [Vlr + (1 − α) · pr(x, s,μ)

]if μ ∈ {w,b}

(2)

with x′ as above, μ′ the matching outcome in the next period, and s′ the household’s state, relaxed or distressed, in the next period. μ′ = ∅ with probability 1 − δ and μ′ = m(x′) with probability δ. 1σ = 1 if and only if σ = sell, and 1σ = 0 otherwise. pr is the transaction price, and α the commission rate; both are defined below.

Household i’s expected valuation of the neighborhood in the next period depends on the matching probability δ and the probability of the shock λ:

Es′,μ′(V r(x′, s′,μ′)

)= δ · {λ · V r(x′,D,μ′) + (1 − λ) · V r(x′,R,μ′)}

+ (1 − δ) · {λ · V r(x′,D,∅) + (1 − λ) · V r(x′,R,∅)}

(3)

with x′ as above given the broker’s and households’ strategies.


If household i sells (μ ∈ {w, b} and σ r(i,t)(x, s, μ) = sell), the seller of race r(i, t) and the matched buyer of race μ set a price pr(x, s, μ) that splits the transaction surplus equally. When a buyer moves into the neighborhood in period t , he is relaxed for the initial period t , and incurs the shock ε with probability λ in each subsequent period.17 Hence the transaction price that splits the transaction surplus is:

pr(x, s,μ) = 1

2[(V r(x, s,∅) − Vlr ) + (V μ(x,R,∅) − Vlμ)], (4)

for x ∈ [0, 1], s ∈ {D, R}, and μ ∈ {w, b}. The transaction is beneficial for the seller whenever his valuation of leaving the neighborhood Vlr + (1 − α)pr(x, s, μ) (including the commission fee) is greater than the valuation V r(x, s, ∅) of staying in the neighborhood. The definition of prices (4) implies that a transaction that is beneficial for the seller is also beneficial for the buyer: the buyer’s valuation of moving into the neighborhood V μ(x, R, ∅) − (1 + α)pr(x, s, μ) is then greater than the valuation Vlμ of living outside the neighborhood.

Broker profit

The broker, in a neighborhood with a fraction x ∈ [0, 1] of Blacks, who matches a seller of race r in state s ∈ {D, R} with a buyer of race m(x) ∈ {b, w}, earns a commission fee whenever a transaction occurs. Such commission fee is a fraction 2α of the transaction price pr(x, s, μ), with the buyer and the seller each paying αpr(x, s, μ).18 Thus the broker’s one-period profit when matching sellers to buyers of race μ = m(x) is:

π(x,μ)

= 2αδ{x ·

[λ · 1σb(x,D,μ) · pb(x,D,μ) + (1 − λ) · 1σb(x,R,μ) · pb(x,R,μ)

]+ (1 − x) · [λ · 1σw(x,D,μ) · pw(x,D,μ) + (1 − λ) · 1σw(x,R,μ) · pw(x,R,μ)

]}(5)

where 1σ = 1 whenever σ = sell, and 0 otherwise. The broker discounts future commission fees at a rate 1

1+r. Thus, the broker’s forward-looking present discounted value of profits in a

neighborhood with a fraction x ∈ [0, 1] of Blacks and with strategy m(·), given households’ strategies σw and σb, is:

(x) = π(x,m(x)) + 1

1 + r(x′(x)) (6)

Appendix A shows (Lemma 1) that Eqs. (1)–(6) together define a mapping � : (m, σw, σb) →(, V w, V b) that assigns a unique valuation profile (, V w, V b) to any strategy profile (m, σw, σb).

Definition 2 (Valuation profile). The set V of valuation profiles (, V w, V b) is the set of triplets of measurable functions: a profit function : [0, 1] → [0, ∞), and households’ valuation func-tions V r : [0, 1] × {R, D} × {b, w} → [0, ∞) for r ∈ {b, w}.

17 The assumption is not necessary but simplifies algebra. A model where the entrant is distressed with prob-

ability λ yields similar dynamics. In such a case, pr (x, s, μ) = 12 [(V r (x, s, ∅) − Vlr ) + ((1 − λ)V μ(x, R, ∅) +

λV μ(x, D, ∅) − Vlμ)].18 We model the brokerage fee as a fixed percentage of the transaction price. This accords with Hsieh and Moretti [24], who use the Consumer Expenditure Survey to show that the median commission fee paid was 6.1% of the transaction price and exhibited little variance. The online appendix’s Section 3 endogenizes the commission rate α.


2.4. Neighborhood equilibrium

In equilibrium, the broker’s strategy maximizes broker profit given households’ strategies, for each fraction x ∈ [0, 1] of black households in the neighborhood. Households’ strategies maximize household utility, for each fraction x ∈ [0, 1], for each state (distressed or relaxed), and for each matched seller, given the broker’s and other households’ strategies.

Definition 3 (Neighborhood equilibrium). Note � :V → S the mapping that, given any valuation profile (, V w, V b) ∈ V, defines a profile (m, σw, σb) ∈ S of optimal strategies:

• Household’s strategies σ r , r ∈ {w, b}, are optimal given the valuation functions V r , i.e. ∀(x, s, μ) ∈ [0, 1] × {D, R} × {∅, b, w}, σ r(i,t)(x, s, μ) = sell if and only if:

Vlr + (1 − α) · pr(x, s,μ) ≥ V r(x, s,∅) (7)

i.e. if and only if the value of leaving the neighborhood is higher than the value of staying in the neighborhood; σ r(i,t)(x, s, μ) = stay otherwise. The price pr(x, s, μ) = 1

2 [V r(x, s, ∅) −Vlr + V μ(x, s, ∅) − Vlμ], that splits the transaction surplus equally is a function of V w

and V b .• The broker’s strategy maximizes the present discounted value of his profit, i.e. at any given

x ∈ [0, 1], m(x) = w if and only if:

π(x,w) + 1

1 + r(x′(x,w)

)≥ π(x, b) + 1

1 + r(x′(x, b)

)(8)

where racial dynamics are x′(x, b) = x + δ(1 − x) · [λ · 1σw(x,D,b) + (1 − λ) · 1σw(x,R,b)

]and x′(x, w) = x − δx · [λ · 1σb(x,D,w) + (1 − λ) · 1σb(x,R,w)

], with σw and σb defined by

their best response Eq. (7); and π(x, m) is defined according to Eq. (5) as a sole function of V r given and σ r defined as in (7). Then x′(x) ≡ x′(x, m(x)).

Then (m∗, σw∗, σb∗) is an equilibrium strategy profile (resp., (∗, V w∗, V b∗) an equilibrium valuation profile) if and only if it is a fixed point of = � ◦ � : S → S (resp., � = � ◦ � :V → V).

Proposition 1 below shows equilibrium existence. Uniqueness occurs for low offer arrival rates, in a way reminiscent of Krugman [26]: when the rate of arrival of offers is high, the neighborhood’s racial composition can experience large and sudden changes, and broker and household expectations determine the selection of the equilibrium.19 When the rate of arrival of offers is low, households’ preferences, the broker’s commission rates, the probability and the magnitude of the shock ε, and the initial racial composition of the neighborhood uniquely deter-mine the neighborhood’s racial dynamics.

Proposition 1. There exists an equilibrium valuation profile (∗, V w∗, V b∗) ∈ V, and a cor-responding equilibrium strategy profile (m∗, σw∗, σb∗) ∈ S. Moreover, if the offer arrival rate δ ∈ [0, 1] is strictly lower than 1

2α· r

1+r, the equilibrium valuation profile (∗, V w∗, V b∗) is

19 Online appendix Section 7 presents an example of equilibrium multiplicity when the condition of Proposition 1 below is not satisfied. In the example, household expectations determine the neighborhood’s equilibrium path.


unique in the space V of measurable valuation profiles; and the equilibrium strategy profile is unique in the set S of symmetric Markovian strategy profiles.

Proof. See Appendix A. Existence of an equilibrium valuation profile (∗, V w∗, V b∗) follows from the application of Brouwer’s fixed point theorem applied to �. Uniqueness follows from Banach’s fixed point theorem: we endow the set V of valuation profiles with the ‖‖∞ norm: ‖ (, V w, V b) ‖∞ is the max

{‖ ‖∞,‖ V w ‖∞,‖ V b ‖∞}; then the functional mapping � is a

contraction mapping on the normed space (V,‖‖∞) whenever 2αδ + 11+r

< 1, i.e. δ < 12α

· r1+r

.

The term 2αδ + 11+r

comes from applying Blackwell’s [9] theorem to Bellman equation (6).

Write ( + c, V w + c, V b + c) = (, ˜V w, V b) and π the one-period profit function defined fol-lowing Eq. (5) using the valuation profile (, ˜V w, V b). Note � (resp., �) the profit func-tion in the triplet �(, V w, V b) (resp., �(, ˜V w, V b)). Then: �(x) = π (x) + 1

1+r(x′) ≤

π(x) +2αδ ·c+ 11+r

(x′) + c1+r

which is �(x) +k ·c with k = 2αδ+ 11+r

< 1 if δ < 12α

· r1+r

. Appendix A shows that this is a sufficient condition that guarantees equilibrium uniqueness. �

Throughout the paper we maintain the assumption δ < 12α

· r1+r

and denote by (m∗, σw∗, σb∗)the unique equilibrium strategy profile.

3. Dynamics of the initially all-white neighborhood

This section provides a necessary and sufficient condition for blockbusting to occur on the equilibrium path, formally m∗(x) = b, ∀x ∈ [0, 1]; and an all-white neighborhood becomes all-black, xt → 1 as t → ∞.

We derive household strategies σw and σb, valuation and profit functions (, V w, V b) when the broker maintains the neighborhood all-white, m(0) = w, and then when the broker block-busts, m(x) = b, ∀x ∈ [0, 1]. Without loss of generality, we assume ρw and ρb are such that Whites’ valuation V w(x, s, μ) is a weakly decreasing function of x ∈ [0, 1] for any s ∈ {D, R}and μ ∈ {w, b, ∅}.20

3.1. Steady-state white neighborhood

When the broker matches sellers to white buyers, m(0) = w, an all-white neighborhood stays all white. Whites’ valuation of the neighborhood does not vary over time. In such an equilibrium, distressed white households sell to white buyers. To see this, we note that, as mentioned in Section 2, white households who buy in the neighborhood are initially relaxed, and can only get matched in the next period, and thus value the neighborhood at V w(0, R, ∅); and distressed white sellers have a valuation V w(0, D, ∅) = V w(0, R, ∅) − ε of the neighborhood. Hence transactions between distressed white sellers and white buyers are mutually beneficial. Also, relaxed white households do not sell to white buyers: their valuation of the neighborhood is equal to buyers’ valuation, and both buyer and seller would pay commission fees.

20 Such monotonicity of V w(x, s, μ) is satisfied whenever |ρw | > (1−α)βδ2(1−β(1−αδ))

|ρb|. If such condition is not satisfied, then V w(x, s, μ) may be increasing in x ∈ [0, 1] as white households can sell to black buyers with increasing valuations V b(x, s, ∅) of the neighborhood. Similar blockbusting results apply since the broker’s incentive to blockbust increases in x: the broker does not face a trade-off between lower commission fees and medium-run turnover.


Relaxed white households’ neighborhood valuation is given by the steady-state Bellman equa-tion:

V w(0,R,∅) = vw + β{λ[δ(Vlw + (1 − α)pw(0,D,w)

)+ (1 − δ)V w(0,D,∅)]

+ (1 − λ)V w(0,R,∅)}

(9)

vw is Whites’ one period utility, β their time discount factor, and V w(0, D, ∅) = V w(0, R, ∅) −ε. The price pw(0, D, w) splits the transaction surplus as described in Eq. (4): pw(0, D, w) =12

[(V w(0,D,∅) − Vlw) + (V w(0,R,∅) − Vlw)

].

In an all-white neighborhood the broker receives a constant stream of per period revenue π , and thus his present value of future revenues is:

ss-white =(

1 + 1

r

)·⎛⎜⎝ λδ︸︷︷︸

turnover

·2αpw(0,D,μ = w)︸︷︷︸commission fee

⎞⎟⎠ (10)

3.2. Blockbusting dynamics

This section derives the broker’s profit function and households’ valuations V w and V b

assuming m(x) = b for all x ∈ [0, 1], i.e. blockbusting. Section 3.3 shows when such strategy is played in equilibrium.

Households’ strategies and valuations

With m(x) = b for all x ∈ [0, 1], the fraction of black households weakly increases over time, i.e. x′(x) ≥ x for all x ∈ [0, 1], and white households’ valuation weakly decreases in x.

The broker’s strategy m(·) leads to the entry of Blacks in an initially all-white neighborhood if distressed white households and black buyers find it mutually beneficial to trade in the all-white neighborhood (x = 0)—otherwise no transaction occurs and the present discounted value of blockbusting revenues blockbusting = 0. A necessary condition for blockbusting is thus:

Vlw + (1 − α)pw(0,D,w) ≥ V w(0,D,∅) (11)

Relaxed white households start selling when the fraction of Blacks has reached a threshold x, where Whites are indifferent between selling and staying:

Vlw + (1 − α)pw(x,R, b) = V w(x,R,∅) (12)

equivalently V w(x, R, ∅) reaches the threshold 1−α1+α

[V b(x, R, ∅) − Vlb] + Vlw .The fraction of Blacks x is a tipping point, following Schelling [33]: as relaxed Whites start

selling at x = x, black inflows �x = x′ − x jump discontinuously from λδ(1 − x) to δ(1 − x)

per period:

x′ = x +{

λδ(1 − x) for x < x

δ(1 − x) for x ≥ x

Household strategies are thus: σw(x, D, b) = sell for all x ∈ [0, 1], and σw(x, R, b) = sell if and only if x ≥ x. Households’ valuations are derived from their Bellman equation (2): ∀x ∈ [x, 1],

V w(x,R,∅) = vw − ρwx + β{δ[Vlw + (1 − α)Es′pw(x′, s′, b)

]+ (1 − δ)

[V w(x′,R,∅) − λε

]}(13)


where the fraction of Blacks in the next period is x′ = (1 − δ) · x + δ and the state s′ in the next period is D with probability λ, R with probability 1 − λ. The transaction price pw(x′, s′, b) is a linear function of V w(x′, R, ∅), thus V w(x, R, ∅) is linear V 0w − V 1wx with V 0w , V 1w > 0.21

The broker’s blockbusting revenue

We then turn to the broker’s profit functions when blockbusting the initially all-white neigh-borhood, m(x) = b, ∀x ∈ [0, 1]. The broker, whose time discount factor is r , collects commission fees π(xt ) in each period t . The present value of future revenues starting from a initial fraction of x0 black households is:

blockbusting(x0) =∞∑t=0

1

(1 + r)tπ (xt )

=t (x0)−1∑

t=0

1

(1 + r)t2δ(1 − xt )αλpw(xt ,D,b)︸︷︷︸Before tipping

+∞∑

t=t (x0)

1

(1 + r)t2δ(1 − xt )α[λpw(xt ,D,b) + (1 − λ)pw(xt ,R,b)]

︸︷︷︸After tipping

+∞∑t=0

1

(1 + r)t2δλxtαpb(xt ,D,b)︸︷︷︸

Black sellers’ brokerage fees

, with xt+1 = x′(xt ) (14)

The time t (x0) to reach the tipping point is the first time period t ∈ {1, 2, . . .} such that xt ≥ x. For simplicity we note blockbusting ≡ blockbusting(0) and t ≡ t (0).

Before tipping, i.e. for t < t , distressed Whites sell to black buyers—relaxed Whites do not. Similarly, black households that moved in the neighborhood in periods t > 0 sell when distressed and matched to black buyers. Hence turnover is low, at δλxt + δλ(1 −xt ) = δλ. Trans-action prices pw(x, D, b) are also lower than transaction prices when white households sell to Whites. After the tipping point, for t ≥ t , both distressed and relaxed Whites sell to black buyers. Hence turnover is high (δ(1 − x) + δλx, which includes distressed black sales to black buy-ers δλx). Average transaction prices jump as well as both relaxed (with a higher valuation of the neighborhood) and distressed Whites sell. In any period t , black households who entered the neighborhood previously, for t > 0, also are distressed with probability λ > 0 in subsequent periods and thus sell to matched black buyers. Such trades between black households generate a turnover δλx and transaction prices pb(x, D, b). Noting turnover(x) the turnover of the neigh-borhood when it has reached a fraction x of black households, turnover(x) = δλ when x < x and δ(1 − x) + δλx when x ≥ x. And commissions(x) the average commission fees per transaction:

commissions(x) ={

2αδλ(1−x)·pw(x,D,b)+δλx·pb(x,D,b)

δλwhen x < x

2αδλ(1−x)·pw(x,D,b)+δ(1−λ)(1−x)·pw(x,R,b)+δλx·pb(x,D,b)

δ(1−x)+δλxwhen x ≥ x

21 Before tipping, x ≤ x, Appendix A shows that household valuations are the sum of a decreasing linear function and a decreasing convex function of x.


Dynamics of an initially all-white neighborhood: x0 = 0, when matching to Blacks, m(x) = b, ∀x ∈ [0, 1], and when matching to Whites, m(0) = w. π(xt ) = turnover(xt ) · commissions(xt ). Closed forms of the broker’s one-period profit π (xt ) at time t provided in Appendix A.

P.d.v. of revenue =∑∞t=0

1(1+r)t

π(xt ) of an initially all-white neighborhood when matching to Blacks, m(x) = b, ∀x ∈ [0, 1], and when matching to Whites, m(0) = w.

Fig. 1. Per period revenue π(xt ) and p.d.v. of revenue (0)—steady-state white and blockbusting strategies.

Fig. 1(a) shows the sequence of profits π(xt ) = turnover(xt ) · commissions(xt ) at each period twhen the broker chooses the blockbusting strategy m(x) = b for all x ∈ [0, 1] (solid line, match-ing to black buyers) and when the broker chooses the steady-state white strategy m(0) = w

(dashed line). Fig. 1 thus describes the broker’s trade-off between low average commission fees with a higher turnover rate between t and t (blockbusting) and high average commission fees with low and constant turnover (steady-state white neighborhood).

3.3. Equilibrium blockbusting

This section provides a sufficient and necessary condition for equilibrium blockbusting, i.e. m∗(x) = b for all x ∈ [0, 1] if and only blockbusting ≥ ss-white.

First, if m∗(x) = b, ∀x ∈ [0, 1] then the blockbusting profit blockbusting at x = 0 is higher than the steady-state white equilibrium revenues since the blockbusting strategy’s p.d.v. of profits is higher than the profit of any alternative strategy, including any strategy such that m(0) = w. Therefore blockbusting ≥ ss-white is a necessary condition.


The converse requires showing that blockbusting ≥ ss-white implies that the broker does not have an incentive to deviate at any x ∈ [0, 1], i.e.:

∀x ∈ [0,1] π(x, b) + 1

1 + rblockbusting(x

′(x, b)) ≥ π(x,w)

+ 1

1 + rblockbusting(x

′(x,w)), (15)

where x′(x, w) is the fraction of Blacks in the next period when the broker deviates from his equilibrium strategy for one period.

If the blockbusting profit blockbusting at x = 0 is greater than the steady-state white revenue ss-white, then the one-deviation condition is satisfied at x = 0. Indeed, at x = 0 the left hand side of the no-deviation condition (15) is the blockbusting revenue blockbusting. The right-hand side of this expression is the profit of delaying blockbusting by one period. As x ′(0, w) = 0, condition (15) at x = 0 is thus equivalent to blockbusting ≥ (1 + 1

r) · π(0, w; σw∗, σb∗). The

steady-state white profit ss-white is greater than (1 + 1r) ·π(0, w) as the one period profit π(0, w)

of matching to Whites is based on white households’ expectations of neighborhood change. If blockbusting ≥ ss-white, therefore, condition (15) holds at x = 0.

Section A.4 in Appendix A shows that blockbusting(0) ≥ ss-white implies condition (15) at any x > 0 and thus the subgame perfection of the blockbusting strategy. Subgame perfection is a key condition of the equilibrium Definition 3 as households set trading prices pr(x, s, μ) based on the broker’s and their neighbors’ strategies in future periods.

The proof’s intuition is as follows: as black households x increase up to the tipping point x, incentives to deviate from the equilibrium strategy decline as white households’ valuation of the neighborhood declines, while incentives to blockbust increase, as blockbusting(x) is increasing and convex in x. After tipping, the proof in Section A.4 in Appendix A shows that convexity properties of the profit functions π(·) and blockbusting(·) imply the no-deviation condition for all xs in [x, 1]: the broker does not have an incentive to reverse the process of racial transition. Finally,

Proposition 2 (Blockbusting equilibrium). In equilibrium, a broker adopts the blockbusting strat-egy (m∗(x) = b for all x ∈ [0, 1]) if and only if the blockbusting revenue blockbusting is greater than the revenue ss-white of a steady-state white neighborhood. Households’ strategies are as follows:

• Distressed Whites sell when matched to a black buyer, i.e. σw∗(x, D, b) = sell for any x ∈[0, 1].

• Relaxed Whites sell to a black buyer when the fraction of Blacks has reached a tipping point x ≥ x, i.e. σw∗(x, R, b) = sell if and only if x ≥ x.

• Distressed Blacks sell to black buyers whenever matched, σb∗(x, D, b) = sell. Relaxed Blacks do not, σb∗(x, R, b) = stay.

• Relaxed Blacks sell to white buyers whenever x ≤ x, and distressed Blacks sell to Whites whenever x ≤ x.

From the proposition follow the cases of gentrification (m∗(x) = w for all x ∈ [0, 1]) and a characterization of the equilibrium strategy m∗(x) (online appendix Sections 5 and 6).


4. Implications of the model for blockbusting

This section analyzes how the broker’s strategy depends on the values of the broker’s discount factor, the arrival rate of offers, white households’ racial preferences, and black households’ outside option.

4.1. The broker’s discount factor and blockbusting profits

The broker’s discount rate r determines the relative weight given to revenues at each stage of the neighborhood’s racial transition from all white to all black. Fig. 1(a) described these three stages. From t = 0 to the tipping point t = t , the broker’s revenue π(xt ) is less when steering sellers to black than to white buyers. For t ∈ [t , t], the real estate broker can make more revenue by steering sellers to black buyers. But for t ≥ t , steering to Blacks once again yields lower profit.

A high time discount rate r (i.e., an impatient broker) gives greater weight to early revenues π(x(t)), t ≤ t , than to medium-term revenues, t ∈ [t , t], and long-term revenues, t ≥ t . Con-versely, a low time discount rate r (i.e., a patient broker) gives greater weight to long-term revenues than to medium- and short-term revenues.

Proposition 3 (Blockbusting and the broker’s discount factor). If the tipping point is strictly positive and finite x > 0, then there are two values r, r ∈ [0, ∞) of the time discount factor such that, when r /∈ [r; r], the broker does not blockbust an all-white neighborhood in equilibrium, i.e. the broker matches sellers to white buyers in an all-white neighborhood m∗(x = 0) = w. If the tipping point is x = 0 (both relaxed and distressed Whites sell at t = 0), then there exists a finite r such that the broker blockbusts an all-white neighborhood for r > r .

The results of this proposition are illustrated in Fig. 1(b) (bottom panel), which presents the scenario where the tipping point x is nonzero. We fix all parameters except r , which varies from zero to infinity.

For values of r below r (a patient broker), the p.d.v. of revenue from steering to Blacks in an all-white neighborhood is lower than the steady-state white profit. The reason is that, for low values of r , this broker’s present discounted value gives more weight to long-term revenues, which are lower if the broker steers sellers to black buyers. For r > r (an impatient broker), the revenue from steering to Whites is greater than that from steering to Blacks. This is because the p.d.v. of an impatient broker’s revenue gives more weight to the period t ∈ [0, t] (i.e., before the tipping point), when turnover and prices are both low.

4.2. Arrival rate of offers and blockbusting profits

The rate δ of arrival of offers has an ambiguous effect on the broker’s blockbusting revenue. In the first place, a higher arrival rate decreases the time t required to reach the tipping point, as more black households enter the neighborhood at each time t ; this increases the broker’s blockbusting revenue.

Second, a higher rate of arrival of offers lowers the tipping point x. Households have forward-looking expectations of neighborhood change. A higher δ implies a faster racial transition, and thus lower white valuations. Thus, the tipping point x is lower when the arrival rate of offers is higher. This effect, too, increases the broker’s blockbusting revenue. The tipping point effectis illustrated in Fig. 2(a), which shows the time to reach the tipping point t for three different


The plots are generated by the closed-form solutions derived in Appendix A.

Fig. 2. Offer arrival rate δ, racial preference ρw and the revenues from steering to black buyers.

increasing values δ3 > δ2 > δ1, while holding all other parameters constant. We use δ to denote the smallest value of δ for which the tipping point is zero.

However, a third effect of higher δ is one that reduces the broker’s blockbusting revenue. Namely, households’ forward-looking expectations of neighborhood change imply that the value of living in the neighborhood is lower when black households enter the neighborhood more rapidly. Such price effect implies lower transaction prices for higher offer arrival rates δ.

These three effects are illustrated in Fig. 2(a), which shows blockbusting one-period revenue π(xt ) at time t for three different values δ1 < δ2 < δ3. If the offer arrival rate increases from δ = δ1 to δ = δ2, then the time until the tipping point is reduced from t1 to t2 and the neighbor-hood turnover increases. However, as t → ∞, the revenue for δ = δ2 becomes lower than the revenue for δ = δ1 because the fraction of Whites in the neighborhood is smaller in the case of a high arrival rate and so transaction prices are lower. At δ = δ3 the tipping point is zero and so increasing δ can no longer affect the tipping point. However, increasing δ will continue to induce falling transaction prices.

Fig. 2(b) plots the present discounted value of the blockbusting revenue at t = 0 in an initially all-white neighborhood (solid line). The same figure also compares such p.d.v. of blockbusting revenue to the steady-state white revenue ss-white. In Section 3.1 we showed that the p.d.v. of revenues in an all-white neighborhood increases in δ. For values of δ close to 0, the p.d.v. of the revenue from blockbusting is less than the p.d.v. of revenue from maintaining an all-white neigh-borhood. The reason is that the time to reach the tipping point approaches infinity as the offer arrival rate approaches zero. For values of δ close to 1, the p.d.v. of the blockbusting revenue can


also be less, because of the price effect. For values of δ ∈ [δmin, δmax] ⊆ [0,1], the revenue from blockbusting revenue is higher than from the all-white neighborhood. The difference between these two revenue levels decreases from δ onward, the point at which δ ceases to have any effect on the tipping point.

4.3. Racial preferences and blockbusting profits

We turn to the impact of Whites’ racial preference ρw on blockbusting profit. White house-holds’ racial preference ρw has two effects on the broker’s blockbusting profit. First, a higher ρw

(greater aversion) tends to lower the tipping point, which in turn raises broker revenue; a tipping point effect. Second, a higher ρw reduces transaction prices (and hence commission fees); a price effect, which lowers the broker’s profit.

The tipping point effect is illustrated in Fig. 2(c), which plots the revenue at time t for three different values of ρ. For values of ρw below ρ, the tipping point is x = 1 and only distressed Whites sell to black households at any time t . In this case, the neighborhood’s turnover and the commission fees are lower than in the steady-state all-white neighborhood; therefore, the broker’s blockbusting revenue is lower than the steady-state white revenue. For higher values of ρw, the tipping point becomes x < 1. Fig. 2(c) shows a case where, at ρw = ρ1, the present discounted value (p.d.v.) of blockbusting revenue is higher than the steady-state white revenue. An increase from ρw = ρ1 to ρw = ρ2 lowers the tipping point x > 0, which increases the broker’s p.d.v. of revenue from steering sellers to black buyers. However, the attendant decline in prices will decrease that p.d.v. The new curve of revenue at time t is depicted by the dashed line. The effect of ρw on total blockbusting revenue is ambiguous because it depends on which of these two effects dominates.

For values of ρw above ρ, the tipping point is x = 0. In that case, the only effect is the price effect of ρw . This is illustrated in Fig. 2(c) as ρw goes from ρw = ρ2 to ρw = ρ3. The value ρw = ρ3 is greater than ρ. At ρ3, the only effect of an increase in ρw is a decline in average prices (and hence in average commission fees), so at this point, too, there is only a price effect. In short, the blockbusting p.d.v. declines with increasing ρw .

These findings are summarized in Fig. 2(d), which plots the present discounted value of the broker’s revenue in an all-white neighborhood, at time t = 0, as a function of white households’ racial preference ρw . If the tipping point is x = 0 (for ρw > ρ), then the broker’s revenue is a decreasing function of racial preference ρw . For ρw > ρ therefore, the price effect dominates. We show that revenue is a linear decreasing function of ρw for ρw ≥ ρ. Fig. 2(d) thus illustrates the results of

Proposition 4 (Blockbusting and white households’ racial preference). An all-white neighbor-hood experiences blockbusting for a bounded set R of white households’ racial preference ρw. Let ρmin and ρmax denote, respectively, the least and most racial aversion for which blockbusting is the broker’s strategy in an all-white neighborhood. Then ρmax < ∞. If, in addition, distressed white households value an all-white neighborhood more than black buyers, then ρmin > 0. Block-busting revenue is linear and decreasing for values ρw ≥ ρ, where ρ is the weakest level of racial preferences for which the tipping point x is zero.

For ρw < ρ, the effect of an increase in ρw on broker revenue is ambiguous because the tipping point effect and the price effect are opposed: the former increases revenue whereas the latter reduces it. Numerical simulations suggest that the tipping point effect dominates the price effect for ρw ≤ ρ (Fig. 2(d)).


4.4. Black households’ outside option and blockbusting profits

Black households’ outside option vlb reflects both the quality and the price of housing out-side the neighborhood. Empirical evidence (Ondrich et al. [30]) suggests that minorities are not presented with the same housing opportunities as white households. Thus, the value of Blacks’ outside option vlb = Vlb/β in the model may be critical to the existence of a blockbusting equi-librium. When this value vlb increases, the tipping point x and the time needed to reach that point also increase, dx/dvlb > 0 and dt/dvlb > 0. Indeed, with higher vlb, relaxed white households’ value of living in the neighborhood needs to decline by a larger amount in order to reach the time t at which selling to Blacks yields more revenue than does selling to Whites. When vlb in-creases, transaction prices also decline, dpw(x, s, b)/dvlb < 0 for s ∈ {D, R}. Hence, the p.d.v. of blockbusting is decreasing in vlb, while ss-white is unaffected by vlb. Formally,

Proposition 5 (Black households’ outside option and blockbusting revenue). There is a value of black households’ outside option vlb such that, for vlb < vlb , blockbusting is the broker’s equilibrium strategy: the blockbusting revenue blockbusting is a decreasing function of Blacks’ valuation of housing outside the neighborhood.

5. Market mechanism #2—multiple brokers

This section defines and proves equilibrium existence with J ≥ 1 brokers. We show existence a free-riding blockbusting equilibrium with J = 2 brokers.

J brokers

The city has J brokers through which all trades take place. We look at the impact of multiple brokers separately from the impact of the arrival rate of offers by keeping the total arrival rate of offers constant, at δ. A household is matched with probability δ, and conditional on being matched, is matched to broker j with probability 1

J. Each broker j ’s Markovian strategy is

noted mj(x) ∈ {w, b}. Household strategies are, as in the one-broker case, noted σ r(x, s, μ). All brokers charge an identical fixed commission rate α.22

Definition 4 (Equilibrium definition with J brokers). An equilibrium strategy profile((m∗

j

)j=1,...,J

, σw∗, σ b∗)

with J ≥ 1 brokers is such that:

• Racial dynamics are driven by both brokers’ actions and households’ strategies:

x′(x) = x + δ

J

⎧⎨⎩⎛⎝ J∑

j=1

1m∗j (x)(b)

⎞⎠ · (1 − x) · [λ · 1σw∗(x,D,b) + (1 − λ) · 1σw∗(x,R,b)

]

−⎛⎝ J∑

j=1

1m∗j (x)(w)

⎞⎠ · x · [λ · 1σb∗(x,D,w) + (1 − λ) · 1σb∗(x,R,w)

]⎫⎬⎭ (16)

22 The commission rate α is endogenized in online appendix Section 2. Allowing for endogenous commission rates αstrengthens the conclusion, presented later in the section, that multiple brokers lowers blockbusting profit and makes it thus less likely.


• ∀(x, s, μ) ∈ [0, 1] × {R, D} × {w, b}:σ r(i,t)∗(x, s,μ) = sell if and only if Vlr + (1 − α) · pr∗(x, s,μ) ≥ V r∗(x, s,μ) (17)

and V r∗(x, s, μ) = Vlr + (1 − α) · pr∗(x, s, μ) if σ r∗(x, s, μ) = sell and V r∗(x, s, μ) =vw − ρwx − 1D(s)ε + βEs′,μ′V r∗(x′, s′, μ′) if σ r∗(x, s, μ) = stay.

• For each x ∈ [0, 1], broker j ’s action m∗j (x) ∈ {w, b} maximizes the present discounted value

of his profits, i.e. ∀x ∈ [0, 1]:

m∗j (x) = arg maxm(x)∈{w,b}

{πj (x,m∗

j (x)) + 1

1 + r∗

(x′(

x,(m∗

j (x))

j=1,...J

))}(18)

given his strategy m∗j for all other values of x in [0, 1], given other brokers’ strategies m∗−j

and households’ strategies σw∗ and σb∗. The one-period profit function πj(x, m∗(x)) is:

2αδ

J

{x ·

[λ · 1σb∗(x,D,m) · pb∗(x,D,m) + (1 − λ) · 1σb∗(x,R,m) · pb∗(x,R,m)

]+ (1 − x) · [λ · 1σw∗(x,D,m) · pw∗(x,D,m)

+ (1 − λ) · 1σw∗(x,R,m) · pw∗(x,R,m)]}

(19)

for all x ∈ [0, 1]. Broker j ’s profit function ∗j (x) is Eq. (18)’s maximand at each x ∈ [0, 1].

Equilibrium existence is guaranteed under the same conditions as in the case of J = 1 broker (Proposition 1). The equilibrium is in general not unique. Each broker j ’s payoff (right-hand side of Eq. (18)) depends on the other brokers’ actions m∗−j (x).

Conditions for the existence of an interesting free-riding equilibrium are proven in the online appendix. In such an equilibrium broker j = 1 matches to black buyers for any x ∈ [0, 1], and broker j ∈ {2, 3, . . . , J } matches to black buyers for x ≥ xj and to Whites otherwise, where brokers j = 1, . . . , J are ordered, xj+1 ≥ xj .

Free-riding equilibrium with J = 2 brokers

With such a strategy profile, 1,blockbusting is broker 1’s profit when m1(x) = b, ∀x ∈ [0, 1] as a function of the model’s exogenous parameters; and 1,ss-white is broker 1’s profit in a steady-state white neighborhood.

Fig. 3(i) depicts the trade-off that the free-riding broker 2 faces. Before the tipping point, white buyers have a higher valuation of the neighborhood than black buyers, thus broker 2 makes a higher one-period profit when matching sellers to Whites than to Blacks. But by doing so, broker 2 also increases the time to tipping t . Broker 2 matches sellers to Whites as long as:

π∗2 (x,w) + 1

1 + r∗

2,blockb.

(x′ (x, b,w)

)≥ π∗2 (x, b) + 1

1 + r∗

2,blockb.

(x′ (x, b, b)

)(20)

The left-hand side of this inequality is the p.d.v. of broker 2’s profits when matching sellers to white buyers. In such a case, the fraction of blacks increases by x ′ − x = x′(x, b, w) − x =δ2λ(1 − x) in each period, and only distressed Whites sell to white buyers. The right-hand side of the inequality is the p.d.v. of profits when matching sellers (black and white) to black buyers. In such a case, both distressed Blacks and Whites of the neighborhood sell, and the fraction of Blacks increases by x′ − x = δλ(1 − x).


The figure shows the p.d.v of broker 2’s commission fees when matching sellers to white buyers, and when matching sellers to black buyers. The online appendix shows that both profits are linear decreasing functions of x. Broker 2 matches sellers to black buyers for x ≥ x2.

The figure shows black inflows �x per period at each stage of the neighborhood’s racial transition. For x ≤ x2, only the first broker matches sellers to Black buyers. The inflow is �x = δ

2 λ(1 − x). For x ∈ [x2, x], both brokers match sellers to Black buyers, and the inflow is δλ(1 − x). The tipping point is reached at x = x, and all Whites sell, the inflow of blacks per period is �x = δ(1 − x). Plots generated using a numerical simulation of the model.

Fig. 3. Model with 2 brokers: broker 2’s free riding behavior.

x2 is the value of x at which inequality (20) is an equality (Fig. 3(i)). Note that after the tipping point x, the buyer with the highest willingness to pay is black; therefore broker 2 starts matching to white buyers at the tipping point at the latest: x2 ≤ x. Naturally, if the tipping point is zero, x = 0, then x2 = 0 and there is no free-riding.

Fig. 3(ii) shows the racial dynamics of the neighborhood for x ≤ x2, where x′−x = δ2λ(1 −x),

for x ∈ [x2, x], where black inflows double, to x′ − x = δλ(1 − x), and for x ≥ x, where black inflows jump to x′ − x = δ(1 − x).

Overall, in a free-riding equilibrium, broker j = 1 realizes a lower present discounted profit than when he is the only broker operating in the neighborhood, because he matches only δ

2households. But he also realizes a lower present discounted value than half of the blockbusting profit with one broker, 1,blockbusting ≤ 1

2blockbusting (defined in Section 3.2). Indeed: (i) as in the one-broker case, broker 1 is the only broker that matches sellers to black buyers initially, for x ∈ [0, x2], but (ii) at the latest when the tipping point has been reached, broker 2 matches sellers to Blacks, which doubles the arrival rate of black offers and lowers transaction prices for broker 1. However 1,ss-white = 1ss-white. Hence, combining 1,blockbusting ≤ 1blockbusting
2 2


with 1,ss-white = 12ss-white, the set of exogenous parameters where the free-riding equilibrium

exists with J = 2 brokers is smaller than the subset of exogenous parameters where blockbusting is the equilibrium with J = 1 broker.

6. Market mechanism #3—direct trades

We analyze neighborhood dynamics when households can sell directly to the buyer with the highest valuation. We then compare such a model with direct trades to the main model of the paper (Sections 1–4) with the same parameters, where all trades are intermediated by one broker.

There is a range of parameters where, when all trades are intermediated, a blockbusting equi-librium exists whereas when all trades are direct, with the same parameters, the neighborhood stays all white. In other words, the broker’s ability to select the race of buyers leads to the possi-bility of racial transition in the neighborhood.

The model with direct trades is as follows.23 In each period t , each household i ∈ [0, 1] of the neighborhood gets the opportunity to sell to directly to a buyer with probability δdirect ∈ (0, 1). We write μ = 1 in such a case, and μ = ∅ otherwise. A matched μ = 1 household of race r ∈ {w, b} in state s ∈ {D, R} can either sell to a white buyer (σ r(x, s) = w), sell to a black buyer (σ r(x, s) = b), or not sell (σ r(x, s) = stay).

Buyer and seller incur transaction costs with the same formula αpr(x, s, μ) as the brokerage commission fees of the main model, so that the only difference between the present model of purely direct trades and the intermediated model is households’ ability to sell to the highest bidder. The online appendix Section 2.2 formally defines an equilibrium strategy profile and proves existence and uniqueness.

Considering an initially all-white neighborhood, xt=0 = 0, racial transition happens on the equilibrium path if and only if black buyers’ have the highest relative valuation of the neigh-borhood, i.e. V w(0, R, ∅) − Vlw ≤ V b(0, R, ∅) − Vlb . Distressed and relaxed Whites then sell to a black buyer when matched with probability δdirect. Then, xt → 1 as t → ∞ as V w(x, R, ∅)

declines in x and V b(x, R, ∅) weakly increases in x.For racial transition to happen, a transaction between a distressed white seller and a black

buyer needs to be not only mutually beneficial, but also the most beneficial. This stands in con-trast with blockbusting equilibria with intermediated trades, as black buyers need not be the highest-valuation buyers.

Transaction with black buyers are the most beneficial when racial preferences ρ are above a threshold ρdirect. Such threshold ρdirect is thus higher than the threshold ρ such that trades between a black buyer and a distressed seller are mutually beneficial (Condition (11)). Racial transition starting from an all-white neighborhood is thus the equilibrium in the model with direct trades for ρw ≥ ρdirect. We note Rdirect = [ρdirect, ∞) the set of racial preference parameters ρw

where racial transition is an equilibrium, holding all other exogenous parameters constant.Proposition 4 has shown that, in the model with one broker, blockbusting occurs when

ρw ∈ R+ belongs to the set R, where either R = ∅ or R = [ρmin, ρmax], with ρmax < +∞.

The value ρdirect in the direct trades model is the lowest ρw such that black buyers outbid white buyers. It corresponds in the one broker model to the lowest ρw such that the tipping point xis 0, i.e. ρdirect = ρ ∈ [ρmin, ρmax] (Fig. 2(d)). Finally then, either blockbusting(0) < ss-white

23 A model with both direct and intermediated trades is also available but not presented here for clarity. Such model shows that blockbusting profits are a decreasing function of the rate of arrival of direct offers.


for all ρw , and Rdirect = R = ∅ (no value of ρw leads to a blockbusting equilibrium) or ρdirect

belongs to the interval [ρmin, ρmax], and Rdirect ∩R = [ρdirect, ρmax]. Finally,

Proposition 6 (Racial transition with direct trades vs. blockbusting). Holding all other ex-ogenous parameters than ρw constant, note the set Rdirect ⊂ [0, ∞) of racial preferences ρw

where, with only direct trades, an equilibrium of racial transition exists, starting from an all-white neighborhood. Such a set Rdirect is an interval [ρdirect, ∞). Using the notations of Proposition 4, the set R of racial preferences where the broker blockbusts is either R = ∅or such that ρdirect ∈ R ⊂ [ρmin, ρmax]. Hence, white households’ incentive to sell to black and the broker’s incentives to blockbust are aligned on the segment ρw ∈ [ρdirect, ρmax], i.e. σw∗

direct(0, R) = b = m∗(0).

7. Blockbusting and household welfare

The legal literature presents a variety of opinions and anecdotal evidence regarding blockbust-ing’s welfare impacts. Mehlhorn [29] describes the public’s perception of blockbusters: “During this period, blockbusters were national pariahs; whites hated them for dismantling their cozy neighborhoods,’ progressives hated them for harming blacks,” broadly suggesting that block-busting leads to welfare losses for both white sellers and black buyers. In contrast, Jung’s [25]analysis of an anti-blockbusting ordinance suggests that blockbusting leads to welfare gains for black buyers, as such ordinance “freez[es] in past discrimination” and “den[ies] blacks a fair opportunity to find suitable housing.”

This paper’s model predicts that blockbusting leads to welfare gains for blacks as blockbusting “opens up” housing in the neighborhood.

Specifically, consider a set of exogenous parameters where, with one market mechanism (say, one broker intermediating all trades), the neighborhood would experience racial transition; but would remain steady-state all-white with another market mechanism (either all trades intermedi-ated by J brokers, market mechanism #2, or direct trades, market mechanism #3).

In a steady-state white neighborhood, black buyers do not experience any welfare gain. In a blockbusting equilibrium, black households do experience welfare gains. For each transaction between a white seller in state s and a black buyer, the buyer’s welfare gain is the value of moving into the neighborhood V b(x, R, ∅) − (1 + α)pw(x, s, b) minus the value of the outside option Vlb . Hence, black households’ total welfare gains �Wb in blockbusting are:

�Wb = Wbblockbusting −

0︷︸︸︷Wb

ss-white

=∞∑t=0

βt ·{δ(1 − xt ) ·

[λ(V b(xt ,R,∅) − (1 + α)pw(xt ,D,b) − Vlb

)+ 1(xt > x) · (1 − λ)

(V b(xt ,R,∅) − (1 + α)pw(xt ,R,b) − Vlb

)]+ δxt · λ

[ε − 2αpb(xt ,R, b)

]}, with xt+1 = x′(xt , b) (21)

The first two terms (V b(x,R,∅) − (1 + α)pw(x, s, b) − Vlb

)and

(V b(xt , R, ∅) −

(1 +α)pw(xt , R, b) −Vlb

), with s ∈ {D, R} are the gains from trade of black households’ buying

from distressed (resp., relaxed) white sellers. The third term [ε − 2αpb(xt ,R, b)

]is distressed


black to relaxed black gains from trade. In any case, �Wb ≥ 0, i.e. black households benefit from blockbusting for any set of exogenous parameters for which blockbusting is an equilibrium.

White households’ welfare gains are similarly due to sales to black buyers. The difference between white households’ total gains from trade in blockbusting and the same gains from trade in a steady-state white neighborhood are:

�Ww = Wwblockbusting − Ww

ss-white

=[ ∞∑

t=0

βt · δ(1 − xt ) · {λ · (Vlw + (1 − α)pw(xt ,D,b) − V w(xt ,D,∅))

+ 1(xt > x) · (1 − λ) · (Vlw + (1 − α)pw(xt ,R,b) − V w(xt ,R,∅))}]

− 1

1 − β· δ · λ(ε − 2αpw(0,R,w)), with xt+1 = x′(xt , b) (22)

Proposition 7 (Welfare impact of blockbusting). �Wb ≥ 0. Further, (i) as white households’ racial preference ρw increases, black buyers’ welfare gains �Wb ≥ 0 in racial transition in-crease; white households’ welfare losses −�Ww increase in magnitude. Formally, there is a segment [ρ

w, ∞) such that ρw ∈ [ρ

w, ∞) implies �Ww ≤ 0. (ii) As black households’ outside

option value Vlb decreases, black buyers’ welfare gains �Wb increase, and white sellers’ wel-fare loss �Ww declines in magnitude compared to their respective welfare in the steady-state white neighborhood.

Proof. See Appendix A. �When ρw > ρ

wtherefore, blockbusting gains from trade are lower than steady-state white

gains from trade, �Ww ≤ 0, i.e. this paper’s model presents a trade-off between the welfare of white sellers (lower in a blockbusting equilibrium) and the welfare gains of black buyers (higher in a blockbusting equilibrium).24

Black households’ outside option value Vlb is exogenous in the model but would be pinned down in a model featuring a city with multiple neighborhoods. Generally speaking, in a broader model featuring multiple neighborhoods, discrimination in other neighborhoods would lower Vlb , increase the broker’s blockbusting profit in this specific neighborhood, increase buy-ers’ blockbusting welfare gains in this specific neighborhood; and lower the welfare losses of white sellers in blockbusting vs the steady-state white neighborhood.

8. Conclusion

This paper focuses on the ability and incentives of a real estate broker to change the equi-librium racial composition of a neighborhood. We show that an all-white neighborhood turns all-black when the broker blockbusts the neighborhood by steering white sellers to black buyers. Once the neighborhood contains a threshold fraction of black households, all white households

24 Black buyers’ welfare gains are weakly positive even if the broker is able to either charge a higher commission rate than the fixed rate α or price discriminate as in the extension with dynamic commission fess α(x, s, r, μ) presented in the online appendix.


(whether relaxed or distressed) sell and turnover is high. The broker has a financial incentive to change a neighborhood’s racial composition if doing so yields a greater present discounted value of revenues than does maintaining a stable all-white neighborhood.

The mechanisms described here suggest that we adopt a nuanced view of the broker’s incen-tives to steer sellers to black buyers. Neighborhoods where white households strongly dislike black neighbors are, in fact, not the best candidates. Fears of racial change (i.e., racial pref-erences) are key to the blockbusting process. And with respect to such racial preferences, the United States District Court for the northern district of Illinois states that “brokers encourage owners to list their homes for sale by exploiting fears of racial change”.25 However, strong racial preferences lead to lower valuations of the neighborhood by white households and hence to lower transaction prices and commissions. These effects may discourage blockbusting. Thus, the broker engages in blockbusting only within a (limited) range of racial preference values (ρw)

for which steering to black buyers leads to greater present discounted revenue.The model also demonstrates why an impatient broker is less likely to be a blockbuster—

namely, because of the initially lower fees generated thereby. Extremely high rates of arrival of offers are also unlikely to encourage blockbusting because then both short- and long-term trans-action prices are lower. Extending the model to the case of two brokers indicates that the second broker free-rides on the first broker’s initial blockbusting efforts. Finally, blockbusting occurs only with a limited number of brokers. When there are too many brokers, blockbusting incentives decline because then the offer arrival rate is high and so prices are low. Increasing competition in the real estate brokerage industry should therefore diminish incentives to blockbust.

More generally, this paper models brokers whose actions affect market equilibrium in a setting where social interactions could imply multiple equilibria. The incentives of a third party, here the broker, provide an equilibrium selection mechanism. Hence the model could have broader applicability and relevance beyond the issue of blockbusting and white flight.

Appendix A. Proofs and closed forms

A.1. Equilibrium existence and uniqueness

Lemma 1 (Existence of the mapping � : S → V). There exists a mapping that assigns a unique valuation profile (, V w, V b) to any strategy profile (m, σw, σb), defined by Eqs. (1)–(6).

Proof. Racial dynamics x′ : [0, 1] → [0, 1] are unambiguously given by Eq. (1). The existence of V w and V b given (m, σw, σb) and given x′ needs to be proven. For r ∈ {w, b}, define the functional mapping fV r from the set of functions [0, 1] × {R, D} × {w, b, ∅} → (−∞, +∞), noted V , into itself:

fV r

(V r

)(x, s,μ) =

⎧⎪⎪⎨⎪⎪⎩(1 − 1σ r(i,t)(x,s,μ)

)· [vr − ρrx − 1s(D)ε + βEs′,μ′

(V r(x′, s′,μ′)

)]+ 1σ r(i,t)(x,s,μ) · [Vlr + (1 − α) · pr(x, s,μ)

]if μ ∈ {w,b}

vr − ρrx − 1s(D)ε + βE(V r(x′, s′,μ′)

)if μ = ∅

(A.1)

with x′ given by (1), Es′,μ′(V r(x′, s′,μ′)

)from Eq. (3), and pr(x, s, μ) from Eq. (4). fV r is

continuous on the Banach space V . fvr is a contraction mapping with modulus max{β, 1 − α}

25 Pearson v. Edgar, 965 F. Supp. 1104, 1108–09 (N.D. Ill. 1997), cited in Mehlhorn [29].


Table 1Summary of notations.

Description

Parametersα ∈ [0,1] Broker’s commission rateδ ∈ [0,1] Arrival rate of offers

λ ∈[0, 1

2

]Probability of distressed shock

r ∈ {w,b} Racevr − ρrx One-period neighborhood utility (relaxed, race r)vr − ρrx − ε One-period neighborhood utility (distressed, race r)vlr One-period utility of the outside optionVlr P.d.v. of utility of the outside optionβ ∈ [0,1] Household time discount factor

11+r

∈ [0,1] Broker’s time discount factor

State variablesx ∈ [0,1] Fraction of Blacks in the neighborhoods ∈ {R,D} Household state: Relaxed or Distressedμ ∈ {w,b,∅} Race of the matched buyer (μ = ∅, no matched buyer)

Strategiesm(x) ∈ {w,b} Broker’s strategy: race of the matched buyerσ r (x, s,μ) ∈ {sell, stay} Race r households’ strategy

Value functionsV r (x, s,μ) P.d.v. of neighborhood utility for a household of race r in state s matched to a buyer μ

pr (x, s,μ) Transaction price between (i) a seller of race r in state s and (ii) a buyer of race μ

(x) Broker’s profit functionπ(x) Broker’s one-period commission fees

Endogenous parametersx ∈ [0,1] Tipping point in blockbusting (Proposition 2)t Time to reach the tipping point in blockbusting (Proposition 2)

for the norm ‖‖∞ on the space V . By Banach’s fixed point theorem fV r has a unique fixed point in V , which is the valuation function V r that satisfies households’ Bellman equation (2) given m, σw , σb, x′.

Given m, σw , σb , x′, V w , and V b , the profit function similarly exists; the mapping f from the set P of functions [0, 1] → (−∞, +∞) into itself, f()(x) = π(x, m(x)) + 1

1+r(x′(x))

with x′ given by (1) and π a function of m, σw , σb , V w , and V b given. f is a continuous contraction mapping with modulus 1/(1 + r) on the Banach space P ; and thus f admits a unique fixed point that satisfies the broker’s Bellman equation (6). �Proposition 1. (First part, equilibrium existence.)

Proof. The functional � = � ◦ � is a mapping of V into itself. The normed vector space (V,‖‖∞) is complete as a Cartesian product of complete spaces (the set of measurable func-tions over [0, 1] is complete for the norm ‖‖∞), and thus a Banach space. The mapping � is continuous. Finally consider the set P (resp., the set V) of profit functions (resp., of valuation

functions) taking values between (

1 + 1r

)· 2αδ · min{vw − ρw − ε, vb − ε} and

(1 + 1

r

)· 2αδ ·

max{vw, vb − ρb} (resp., between 1 min{vw − ρw − ε, vb − ε} and 1 max{vw, vb − ρb}).
1−β 1−β


� maps the convex and compact subset P × V × V into itself. By Brouwer’s theorem, � admits a fixed point (∗, V w∗, V b∗) in P × V × V . The corresponding strategy profile �(∗, V w∗, V b∗) = (m∗, σw∗, σb∗) is a fixed point of = � ◦ � as � ◦ �(∗, V w∗, V b∗) =(∗,V w∗,V b∗) implies � ◦� ◦�(∗, V w∗, V b∗) = �

(∗,V w∗,V b∗) and (m∗, σw∗, σb∗) =

(m∗, σw∗, σb∗). �Proposition 1. (Second part, equilibrium uniqueness.)

Proof. The mapping � : V → V has the following properties. For ease of exposition write the components of �(, V w, V b) as (�, �wV w, �bV

b).

1. � is monotone as: (a) ∀x, s, m, 1(x) ≤ 2(x), V w1 (x, s, m) ≤ V w

2 (x, s, m), andV b

1 (x, s, m) ≤ V b2 (x, s, m) imply ∀x, s, m �1(x) ≤ �2(x), �wV w

1 (x, s, m) ≤�wV w

2 (x, s, m), and �bVb1 (x, s, m) ≤ �bV

b2 (x, s, m).

2. The mapping � also satisfies �( + c, V w + c, V b + c) ≤ �(, V w, V b) + k · c where

k = max{

2αδ + 11+r

,1 − α,β}

.

Point 2 is proven in the following way. Write ( + c, V w + c, V b + c) = (, ˜V w, V b) and πthe one-period profit function defined following Eq. (5) using the valuation profile (, ˜V w, V b). Then:

• �(x) = π (x) + 11+r

(x′) ≤ π(x) + 2αδ · c + 11+r

(x′) + c1+r

which is �(x) + k · cwith k = 2αδ + 1

1+r.

• � ˜V w ≤ �V w + max{β, 1 − α} · c.

Since α, β ∈ (0, 1), k < 1 if and only if 2αδ + 11+r

< 1, i.e. δ < 12α

· r1+r

. By Blackwell’s [9]

Theorem 5, � is a contraction mapping with modulus k < 1. The space P × V × V defined in the previous proof is compact convex, and a subset of the Banach space V. Hence Banach’s fixed point theorem implies that � has a unique fixed point (∗, V w∗, V b∗) in P×V×V . The strategy profile (m∗, σw∗, σb∗) = �(∗, V w∗, V b∗) ∈ S is an equilibrium strategy profile. � : S → V is onto: two different strategy profiles yield two different valuation profiles. Thus uniqueness of (∗, V w∗, V b∗) implies uniqueness of (m∗, σw∗, σb∗). �A.2. Closed forms for the steady-state white neighborhood

We find households’ valuations when the broker’s action at x = 0 is m(0) = w. Such val-uations follow the steady-state Bellman equation given in Section 3.1, and pw(0, D, w) =V w(0, R, ∅) − Vlw − 1

2ε. Thus:

V w(0,R,∅) =vw + βλ ·

{δαVlw − (1 − 1+α

2 δ)ε}

1 − β [1 − αλδ](A.2)

V w(0, R, ∅) is an increasing function of the outside option Vlw , a decreasing function of the shock ε and its probability λ, and an increasing function of the arrival rate of offers δ, facts used in Sections 4.1–4.4.


A.3. Closed forms for blockbusting profit

We derive closed form expressions of V r(x, s, μ) and blockbusting(x) when m(x) = b, ∀x ∈[0, 1]. These are used in the proof of Proposition 2 (Section A.4).

After the tipping point

We start with white households’ valuation V w(x, s, ∅) of the neighborhood after the tipping point. As described in Section 3.2 (Blockbusting dynamics), V w(x, s, ∅) is a linear function of x, noted V 0w − V 1wx. V 0w and V 1w are functions of the exogenous parameters vw, vb , ρw , ρb , λ, α, δ, ε, β , with V 1w > 0. Also V b(x, s, ∅) = V b0 − V b1x with V b1 < 0. Closed forms available on request. The broker’s blockbusting profit function satisfies the following Bellman equation:

∀x ∈ [x,1] blockbusting(x) = π(x;b) + 1

1 + rblockbusting((1 − δ)x + δ), (A.3)

where π(x; b) = δ ·[λx

(V b0 − V b1x − Vlb − ε

2

) + 1−x2

(V w0 − V w1x − λε − Vlw + V b0 −

V b1x − Vlb

)]. Bellman equation (A.3) on

[x;1

]implies that the solution to the equation is

a second order polynomial in x, i.e. blockbusting(x) = 2x2 + 1x + 0. {2,1,0

}is the

solution of a 3 ×3 system of linear equations. Importantly for the proof of Proposition 2 presented

in this appendix, 2 = (1+r)(V w1+(1−2λ)V b1)

2(r+2δ−δ2), positive when |V w1| > |V b1|. Explicit values of 1

and 0 are not necessary for the proof of Proposition 2 and are available on request.

Determination of the tipping point

Following Eq. (12):

Vlw + (1 − α)1

2

[(V w0 − V w1x − Vlw

)+(V b0 − V b1x − Vlb

)]= V w0 − V w1x (A.4)

when the tipping point x is an interior point of [0, 1]. The equality is > when x = 0 and < when x = 1. The tipping point is decreasing in Whites’ racial preferences ρw and in the arrival rate of offers δ.

Before the tipping point

Before tipping, distressed Whites sell, but relaxed whites do not sell. Whites’ Bellman equa-tion is:

∀x ∈ [0, x] V w(x,R,∅) = vw − ρwx + β · {δλ [Vlw + (1 − α)pw(x′,D,b)]

+ δ(1 − λ)V w(x′,R,∅) + (1 − δ)[V w(x′,R,∅) − λε

]}(A.5)

with x′ = (1 −λδ)x +λδ. Given pw(x, D, b) = 12

[(V w(x,D,∅) − Vlw) + (V b(x,R,∅) − Vlb)

]is a linear function of V w(x, R, ∅), a linear function V w(x, R, ∅) = V 0w − V 1wx satisfies the Bellman equation for values x such that x ′(x) < x, but fails to satisfy the equation for values xsuch that x ′(x) > x. Subtracting V 0w − V 1wx from Bellman equation (A.5), and by backward induction from x(t) to x, ∀x ∈ [0, x]:

V w(x,R,∅)

= V 0w − V 1wx +{β ·

(1 − δλ

1 + α)}t (x) {

V w(xt ,R,∅) −(V 0w − V 1wxt

)},

2


where V 0w and V 1w are positive functions of the exogenous parameters vw, vb , ρw , ρb, λ, α, δ, ε, β .

Before the tipping point, blockbusting(x) is the solution to the following Bellman equation

∀x ∈ [0; x] blockbusting(x) = π(x, b) + 1

1 + rblockbusting(λδ + (1 − λδ)x), (A.6)

with π(x, b) = δλx · (V b(x) − ε/2) + δλ(1 − x) · 12 (V b(x) + V w(x) − ε). As π(x, b) is convex

in x ( d2π(x,b)

dx2 > 0), and blockbusting(x) is convex for x ≥ x, taking the second derivative of

(A.6) and by backward induction from x = x to x = 0 yields that d2

dx2 blockbusting(x) ≥ 0, hence blockbusting(x) is convex over [0, x] ∪ [x, 1] = [0, 1].

A.4. Equilibrium blockbusting

Proposition 2. (Blockbusting of an initially all-white neighborhood.)

Proof. Households’ strategies σ r(x, s, μ) and valuations V r(x, s, μ) when m(x) = b, ∀x ∈[0, 1] have been derived in the previous subsection of Appendix A. We need to prove that the broker’s strategy is optimal at each x, following condition (8) of the equilibrium Definition 3:

∀x ∈ [0,1] π(x, b) + 1

1 + rblockbusting(x

′) ≥ π(x,w) + 1

1 + rblockbusting(x

′′), (A.7)

where π(x; b) (resp., π(x; w)) is the one-period profit when the broker matches sellers to black buyers (resp., white buyers). x′ (resp., x′′) is next period’s fraction of black households when the broker matches sellers to black buyers (resp., white buyers). We define �(x) ≡ π(x; b) −π(x; w)

and ξ(x) ≡ 11+r

[blockbusting(x

′′) − blockbusting(x′)]

so that the no-deviation condition is equiv-alent to �(x) ≥ ξ(x) for all x ∈ [0,1]. Without loss of generality, we set V b(x) ≡ V b(x, D, ∅), V w(x) ≡ V w(x, D, ∅), and Vlw ≡ Vlb ≡ 0 throughout the proof for ease of exposition.

We start by showing the one-deviation property for all xs below the tipping point x. On that interval, �(x) is negative as black buyers have a lower valuation of the neighborhood than white buyers. We can assume that ξ(x) is negative on [0, x] otherwise the one-deviation property is satisfied—there is no trade-off between one period profits and long term profits. We prove that −ξ(x) ≥ −�(x) on the interval x ∈ [0, x] in the following way.

First, the main body of the paper (Section 3.3) showed that the no deviation condition is satisfied at x = 0. We show below that the condition at x = 0 implies the condition on (0; x].

Note that −ξ(x) is equal to 11+r

[blockbusting(λδ + (1 − λδ)x) − blockbusting((1 − δ)x)

]for

x < x (only distressed Whites sell to black buyers), and matching to white instead of black buyers will lead to sales of both distressed and relaxed Blacks to Whites (as V b(x) < V w(x)). We found in Section A.3 that blockbusting(x) is a convex function of x for [0, x]. The difference −ξ(x) is thus increasing in x.

Second, we consider −�(x) = π(x; w) − π(x; b) the one-period excess profit of deviating for x ∈ [0, x] before the tipping point. By plugging in the values of π(x; w), π(x; b) and V w(x), V b(x), we find that −�(x) is the weighted sum of two functions such that −�(x) = x · φ(x) +(1 − x) · ψ(x) where φ(x) = δλ 1

2 [V w(x) − V b(x)] and ψ(x) = δ ·{( 1

2 − λ)V b(x) + 12V w(x)

}.

Elementary calculations show that for x ≤ x, both φ(x) and ψ(x) are decreasing (λ < 12 , Sec-

tion 2) and that φ(x) < ψ(x). Hence − d� < 0 and −�(x) is decreasing over [0, x]. The
dx


Fig. 4. Proposition 2: one deviation property, inequality (A.7) after tipping, x ∈ [x,1].

no-deviation condition is satisfied at x = 0, i.e. −ξ(0) ≥ −�(0), further we just showed that −ξ(x) is increasing over [0, x], and −�(x) is decreasing over [0, x], hence the no-deviation condition is satisfied over [0; x].

Both �(x) and ξ(x) experience a discontinuous jump at x = x. We find that limx→x+ �(x) ≥�(x) = limx→x− �(x) as π(x, w) falls discontinuously26: relaxed Blacks do not sell to Whites when the broker deviates from his equilibrium strategy for x ≥ x. Further limx→x+ ξ(x) ≤ξ(x) = limx→x− ξ(x) as the per-period rate of entry of Blacks in to the neighborhood increases discontinuously from λδ to δ and the per-period rate of entry of Whites in to the neighborhood decreases discontinuously from δ to λδ. Hence the no deviation condition is satisfied at the right of x: limx→x+ �(x) ≥ limx→x+ ξ(x).

We then turn the no-deviation condition for x greater than the tipping point, i.e. x ≥ x. Note x the smallest value of x for which x′′ = x, i.e. the smallest value of x for which Blacks do not sell to Whites when the broker deviates from his equilibrium strategy m(x) = b. In the interval

[x, x

] ⊂ [x,1

], households’ valuations are linear functions of x, as shown in this

appendix’s previous subsection: V w(x, R, ∅) = V w0 − V w1x with V w0 and V w1 positive, and V b(x, R, ∅) = V b0 − V b1x with V b0 positive and V b1 negative. When |V w1| < |V b1|, blockbusting(x) is increasing over [x, 1] and ξ(x) is negative, hence �(x) ≥ 0 ≥ ξ(x). We thus without loss of generality assume V w1 + V b1 > 0. �(x) is a second-order polynomial in x on

[x; x]. The coefficient of x2 in �(x) over that interval is 1

2 (1 − λ) (V w1 + V b1

),

which is positive, hence �(x) is convex over [x; x]. �(x) is decreasing as ddx

�(x) =− 1

2

[(1 − λ)V b0 − (1 − 2x(1 − λ))V b1 + (1 − λ)V w0 − (1 − 2x(1 − λ) − 2λ)V w1

]< 0 since

0 < 1 − 2x(1 − λ) < 1 (Section 2). Without loss of generality, we can assume that blockb.(x)

is decreasing otherwise there is no trade off between one period profit π(·) and long run profits blockb.. Section A.3 showed that blockb.(x) is a convex polynomial. Thus:

∀x ∈ [x, x] dξ(x)

dx= 1

1 + r

[(1 − λδ)

d

dx((1 − λδ)x) − (1 − δ)

d

dx((1 − δ)x + δ)

],

is negative with (x) = blockbusting(x) decreasing convex. Thus ξ(x) is decreasing in x. Both �(x) and ξ(x) are decreasing convex over [x, x], and �(x) ≥ ξ(x). We note �(x) and ξ (x)

respectively the extension of the functional forms of �(x) and ξ(x) over the interval [x, x] to the larger interval [x, 1] (see Fig. 4).

26 We note f (x+) = limx→x+ f (x), and similarly for f (x−).


We compare �(1) and ξ (1). For x ≥ x, black households value an all-black neighborhood more highly than white households: For λ = 0, elementary calculations show that �(1) ≥ ξ (1)

is equivalent to (1 − φ)(V b0 − V b1) ≥ (1 + φ)(V wo − V w1) with φ = α(1+r)δr+δ

≤ α, and the relationship is satisfied as Blacks outbid Whites for x = 1 ≥ x. For λ = 1

2 , similarly �(1) ≥ ξ (1)

is equivalent to V b0 − (1 +φ)V b1 ≥ φ(V w0 −υV w1) + (V w0 −V w1) where υ ≥ 1 a function of r and δ, and is satisfied as V b0 − (1 + φ)V b1 ≥ V b0 − V b1 ≥ (1 + φ)(V w0 − V w1) ≥ φ(V w0 −υV w1) + (V w0 − V w1). �(1) and ξ (1) are decreasing functions of λ hence �(1) ≥ ξ (1) for λ = 0 and λ = 1/2 implies �(1) ≥ ξ (1) for λ ∈ [0, 12 ]. Since �(x) ≥ ξ (x), �(1) ≥ ξ (1), and since �(x) and ξ (x) are decreasing convex, then �(x) ≥ ξ(x) over the entire interval [x, x].

We have shown that �(x) ≥ ξ(x) for all x ∈ [x, x

], and that �(x) ≥ ξ (x) for all x ∈ [x, 1].

Furthermore, for x ∈ [x, 1], we have that �(x) ≥ �(x) ≥ ξ (x) ≥ ξ(x) as black households do not sell to white buyers for x ≥ x. This is illustrated in Fig. 4. Thus the proof has shown that the one deviation property �(x) ≥ ξ(x) holds on [0, x] ∪ [x, x] ∪ [x, 1] = [0, 1]. �Appendix B. Supplementary material

Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.jet.2015.02.006.

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Blockbusting: Brokers and the dynamics of segregation€¦ · models of urban segregation (Schelling [33], Benabou [6,7], Becker and Murphy [5], Frankel and Pauzner [19]) do not explain