Prejudice and Discrimination in
Housing
© Allen C. Goodman 2000
Yinger
• Wants to work with housing price gradient
that we used before.
• Discusses prejudice as disutility of W or B
from living with or near members of the
other race
Uw = Uw [ Zw, Xw, r(u) ],
Where Z is everything else, X is
housing and r is race at u.
Prejudice
ONE FORM OF r(u) IS:
Hw = Xw Dw [r(u)],
WHERE Dw(0) = 1, D' < 0.
1
DW
DB
D*B
D*w
REDEFINES TERMS AS:
Hw = Hw [Xw, r(u)].
R = fraction B
Using a Cobb-Douglas Ftn.
USING A COBB-DOUGLAS UTILITY FUNCTION:
U = Zc1Hc2, OR
Ln U = c1 Ln Z + c2 Ln X + c2 Ln D [r(u)]
SUBSTITUTION, AGAIN, AS IN THE MILLS MODEL, YIELDS
PRICE AND RENT-DISTANCE FUNCTIONS:
21> Pw (u) = Kw (Y - tu) 1/k Dw [ r(u)]; Kw is a constant of int.
_
_
22> Rw (u) = R [(Y - tu)/(Y - tu)] 1/ak {Dw [r(u)]/ Dw [r(u)]}1/a
k = constant, a = land coefficient in housing prod.
_
For W, for B
_
_
_
[r(u)]}1/a
22> Rw (u) = R [(Y - tu)/(Y - tu)] 1/ak {Dw [r(u)]/ Dw
For prejudiced W, function must be lower at locations with higher
concentration of B.
Can come up with similar for B.
Hb = Xb Db [r(u)],
WHERE Db(1) = 1, D' > 0.
_
_
24> Rb (u) = R [(Y - tu)/(Y - tu)] 1/ak {Db [r(u)]/ Db
_
[r(u)]}1/a
Integrated eq’m
_
_
_
22> Rw (u) = R [(Y - tu)/(Y - tu)] 1/ak {Dw [r(u)]/ Dw [r(u)]}1/a
_
_
_
24> Rb (u) = R [(Y - tu)/(Y - tu)] 1/ak {Db [r(u)]/ Db [r(u)]}1/a
Must both hold if there will be an integrated eq’m.
If so:
_
_
Dw [r(u)]/ Dw [r(u)]} = Db [r(u)]/ Db [r(u)]}
For this to hold, r must be the same everywhere  total integration.
Segregated eq’m
_
_
_
22> Rw (u) = R [(Y - tu)/(Y - tu)] 1/ak {Dw [r(u)]/ Dw [r(u)]}1/a
_
_
_
24> Rb (u) = R [(Y - tu)/(Y - tu)] 1/ak {Db [r(u)]/ Db [r(u)]}1/a
If B move in, neighborhood becomes more (less) attractive to B, less
attractive to W. Leads to total segregation.
Final conclusions
• If there is complete segregation, or integration, racial
composition will not affect the rent-distance function.
• Complete segregation is stable eq’m ONLY in the case of
prejudice on the part of all B and all W.
• Complete integration is unstable if everyone is prejudiced,
or if SOME housholds prefer to live in a mixed area.
– To see this see next slide:
Suppose full segregation
r(u) = 1; u < u* all B
r(u) = 0; u > u* all W
RB*(u*) = RW(u*)
Since preference for mixing
implies Db(0) > Db(1),
some blacks will outbid
some whites for white
land  segregation cannot
be stable.
Same for integration.
W
W
B
u*
Courant – Search Model
There seem to be lots of situations (such as racial steering)
which require different models.
Suppose we have 2 groups, B and W. Several assumptions:
1. Housing is heterogeneous
2. Housing  hedonic price functions
3. Usual utility functions
4. Buyers know the distribution of housing available, and
their utility functions. They evaluate each house:
VH

VL
f (v)dv  1
Where VL is lowest utility, and VH is highest.
Searching
5.
6.
Enough houses on market at any one time.
Constant cost of looking at a house is c.
At any stage, the utility of looking for one more house leads
to utility such that:
Vo
V = max {
-c + (1 - F (Vo)) (E(V|VVo) + Vo F (Vo)
Vo is utility associated with best house found so far. Next line
is expected utility after one more search.
Searching
Rational buyer stops searching when first line of expression is
equal to or greater than second.
VH

V0
(V  V0 )dF  c
This gives you the optimal V = V*.
Example
VH

V0
(V  V0 )dF  c
Lowest value is 20,000  annual = 2,000
Highest value is 30,000 annual = 3,000
c = 30.

3, 000
V0
(V  V0 )0.001  30
1
30000  (V  V0 ) 2
2
(3000  V0 )  245
V0  2755
V 3000
V V0
The larger the value of
c, the bigger the difference
between V and VO.
Seller’s aversion
CONSIDER n NEIGHBORHOODS IN THE CITY, 1, ..., n.
FOR EACH NEIGHBORHOOD, THERE IS A
NONZERO PROBABILITY j THAT WHITES WON'T
SELL. ASSUME THAT j FOR BLACKS = 0. PROB.
THAT BLACK WILL FIND AVERSE SELLER = j *
(PCT WHITE) = j.
Assume that j varies across neighborhoods.
Then, order neighborhoods:
1  2  3  4  5 …  n
Seller’s aversion
Then, order neighborhoods:
1  2  3  4  5 …  n
When we optimize, we compare:
Vo
V = max {
-c + (1 - F (Vo)) (E(V|VVo) + Vo F (Vo)
Vo
V = max {
-c1 + (1-1)[(1 - F (Vo)) (E(V|VVo) + Vo F (Vo)] + 1 V0.
cost +
Return w/o averse seller
+ Return w/ averse seller
Seller’s aversion
Vo
V = max {
-c1 + (1-1)[(1 - F (Vo)) (E(V|VVo) + Vo F (Vo)] + 1 V0.
cost +
Return w/o averse seller
+ Return w/ averse seller
-c2, etc.
If there isn’t an averse seller, then  = 0, and we’re back where we
were before.
Otherwise we get:
cj
1 j
Vj
  * (Vj  V j* )dF j
Vj
Seller’s aversion
cj
1 j
Vj
  * (Vj  V j* )dF j
Vj
Must look at 1/(1-j) houses for everyone s/he has option to
purchase.
If housing prices are identical in neighborhoods, B will only
search for housing in those neighborhoods where  is
lowest.
This implies:
Implications
• Sellers’ aversion is consistent with a price differential in
which housing is purchased by B in B neighborhoods at
higher prices than those obtaining in W and integrated
neighborhoods.
• Maximum price differential is an increasing function of the
fraction of W sellers in a neighborhood who are averse to
dealing with B.
• If there is a small price differential, B will never search, so
nothing will eliminate market segmentation.