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Vol. 34, No. 2
ECONOMIC REVIEW
FEDERAL RESERVE BANK
OF CLEVELAND
Vol. 34, No. 2
http://clevelandfed.org/research/review/
Economic Review 1998 Q2
1998 Quarter 2
Bank Diversification:
Laws and Fallacies of Large Numbers
2
by Joseph G. Haubrich
Regional Variations in White–Black Earnings
by Charles T. Carlstrom and Christy D. Rollow
10
1
ECONOMIC REVIEW
1998 Quarter 2
Vol. 34, No. 2
Bank Diversification:
Laws and Fallacies of Large Numbers
2
by Joseph G. Haubrich
Conventional wisdom on bank diversification confuses risk with failure.
This article clarifies the distinction and shows how increasing bank size
may increase bank risk, even though it lessens the probability of failure and
lowers the expected loss. The key result is an application of Samuelson’s
“fallacy of large numbers.”
Regional Variations in
White–Black Earnings
10
by Charles T. Carlstrom and Christy D. Rollow
The authors examine why black Americans’ earnings continue to lag
whites’ and why the problem is especially acute in the southern states.
Better understanding of the factors driving regional pay differentials can
help explain some of the disparities at the national level and would also
be applicable to a wide variety of other public policy issues.
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ISSN 0013-0281
2
Bank Diversification:
Laws and Fallacies
of Large Numbers
by Joseph G. Haubrich
Conventional wisdom states that large banks
are safer than small banks because they can
diversify more. This conventional wisdom,
however, confuses risk with probability of failure. While the law of large numbers does imply
that a large bank is less likely to fail than a
small bank, equating this tendency with lower
risk falls into what Samuelson [1963] termed the
fallacy of large numbers. A $10 billion bank
may be less likely to fail than a $10 million
bank, but it may also saddle the investor with
a $10 billion loss.
In this article, I hope to clarify what this distinction means for banks. Banks diversify by
growing—by adding risks—something distinctly different from the subdivision of risk behind standard portfolio theory. A simple meanvariance example will make the point that a
risk-averse bank owner need not value diversification by addition. After that, I take a regulator’s
perspective and consider how a bank guarantee
fund, such as the deposit insurance agency,
views bank growth and diversification. After a
short review of why diversification by adding
risks decreases the probability of bank failure,
I look at how such diversification alters the
expected value of deposit insurance agency
Joseph G. Haubrich is a consultant and economist at the Federal
Reserve Bank of Cleveland. He
thanks Steve Zeldes, who introduced him to the fallacy of large
numbers over a decade ago but
who bears no responsibility for
the author’s subsequent application of that knowledge.
payments, then turn to diversification’s impact
on the deposit insurance agency’s expected
utility, using recent results from the theory of
standard risk aversion.
To concentrate on the cleanest example, this
article stays with the case of independent and
identically distributed risks. This admittedly
ignores the alleged ability of large banks to
diversify regionally1 or the possibly adverse
incentives of deposit insurance (Boyd and
Runkle [1993], Todd and Thomson [1991]).
I. A Simple Example
Probably the easiest way to understand the
effects of diversification by adding risks is to
consider a bank financed exclusively by an
owner/investor who cares only about means
and variances. With no debt, failure ceases to
be an issue; instead, the question is the utilitymaximizing portfolio for the bank’s owner.
■ 1 Compare Haubrich (1990) with Kryzanowski and Roberts (1993).
Even small banks may diversify, however, by selling loans or participating
in mortgage pools or other forms of securitization.
3
F I G U R E
his wealth into bank loans. Figure 1 shows a
typical case with an interior solution, illustrating
quite clearly that the bank does not always wish
to diversify. Stated another way, the portfolio
return is distributed N ( K µ, K2 σ 2 ), so that
W
W
as the bank invests in more loans, the standard
deviation increases as well as the expected
return. Preferences determine which of them
matters more.
An all-equity bank offers a nice illustration,
but does not provide a very representative case.
Even a stylized bank should have deposits.
1
Bank Opportunity Set
II. Should the
Deposit Insurance
Agency Want Banks
to Diversify?
SOURCE: Author.
Allowing banks to take in deposits means
The owner and sole equity holder has, conveniently for us, sunk his entire wealth W into
the bank. He faces the problem of dividing his
portfolio between holding S safe government
bonds with a certain return of zero and investing in some number K of risky, independent
bank loans with returns Ri normally distributed
as N ( m,σ 2), that is, with mean µ and variance
σ 2. If each loan costs a dollar, the investor’s
budget constraint is W = S + K. These bank
loans are indivisible—the bank cannot diversify by spreading one dollar across many loans.
Then the return on the portfolio is
K
Σ Ri
~p = i = 1
R
W
.
K
Ri is distributed N (K µ, Kσ 2), the
Since i Σ
=1
portfolio has expected return E (R~p) = K m
W
and variance σ 2(R~p) = K 2 σ 2. From this, simple
W
substitution (for this and other standard techniques, see Fama and Miller [1972], chapter 6,
section IV) implies that
(1)
µ
E(R~p) = σ
K(R~p).
In mean-standard deviation space, equation
(1) defines a portfolio opportunity set, or the
different risk and return combinations available
to the investor. This set is illustrated by figure 1
(for W = 5, µ = 1, and σ = 1). The opportunity
set is disjointed, since the decision to add
another loan is discrete. Depending on the
shape of the indifference curves, the bank
owner may choose to put none, all, or some of
allowing banks to fail. The return on assets may
not cover the payments promised to depositors.
Many countries provide some sort of deposit
insurance, which guarantees the deposit. In the
case where bank assets cannot cover the payments promised to depositors, the difference
becomes a liability for the deposit insurance
agency (which may be either public or private).
This provides a natural focal point for our discussion, although what happens in reality is
much more complicated. Actual banks raise
money in many different ways, using several
types of preferred stock, subordinated bonds,
commercial paper, and both insured and uninsured deposits. What happens in bankruptcy is
at best complicated and at worst unknown,
because the courts must determine the validity
of claims as diverse as offsetting deposits and
the source-of-strength doctrine. A detailed consideration of how each class of investors views
diversification, then, is beyond the scope of this
article. Instead, to make what is admittedly a
simple point, I concentrate on the deposit
insurance agency, which ultimately bears the
liability for bank failures.
The deposit insurance agency steps in if the
realization of bank assets Y is too low to repay
the face value of the debt F, that is, if Y < F.
This is a fairly general formulation in that the
assets producing Y may be funded by means
other than deposits, but it is not completely
general because it ignores the possibility that
the deposit insurance agency may have priority
over some investors. For the rest of the article,
however, I will restrict myself to purely depositfinanced banks. This is at the opposite extreme
from the discussion in section I, and hence
provides a nice comparison.
4
F I G U R E
Payout Function and Probability
µ
SOURCE: Author.
What is the face value of the debt, F ? With
no capital, if the bank funds n projects, each
requiring funds f, the face value is the sum of
the deposits, F = n . f . The payout of bank assets
is likewise the sum over the different projects,
n
Yn = Σ xi ,
i=1
where n indexes the number of projects in
which the bank has invested.
The Probability
of Bank Failure
How likely is it that this bank will fail? The
answer is Pr ( Σxi < F ) or
i
As Samuelson points out, a rational utility maximizer maximizes expected utility, not the
probability of success. The probability of each
outcome must be weighted by the utility of that
outcome. As mentioned before, the failure of
a $10 billion bank may cost the deposit insurance agency more to resolve than that of a $10
million bank.
In the simplest case of risk neutrality,
expected utility corresponds to expected value.
The first question, then, equivalent to assuming
risk neutrality on the agency’s part, concerns
the expected value of the deposit insurance
agency’s payout.3 Determining the expected
payout value becomes a question of finding the
expected value of a particular function. The
deposit insurance agency must pay
(4)
Pr (Yn < n . f ).
Assume that the xi ’s are independent
and identically distributed (i.i.d.), with mean
E (xi ) = µ ; further assume that f < µ, so
that the face value of the debt is smaller than
the expected payout of the assets.
We can rewrite expression (2) as
(3)
Y
In particular, since f < m, Pr ( nn < f )
n
< Pr (| Y
n – µ| $ µ – f ). That is, we can represent the values Yn below f as values that are
n
more than µ – f away from the mean µ. Thus,
as Diamond (1984) explicitly states, the weak
law of large numbers implies that diversification by adding risks reduces the probability of
bank failure.2
The Expected Value
of the Deposit
Insurance Agency’s
Liabilities
µ
(2)
Y
Pr {| nn – µ| $ ε } → 0 as n → ∞.
2
Y
Pr ( nn < f )
because the set {y:y < n . f } is the same as the
set {y: ny < f }.
The weak law of large numbers (see Shirayev
[1984], theorem 2, p. 323; for a more elementary discussion, see Hogg and Craig [1978,
chapter 5]) tells us that provided E |xi| < ∞ and
E xi = µ, then for all ε > 0,
5
Y
0
if Yn $ F, that is, if nn $ f
and
Yn
F – Yn if Yn < f, that is, if n < f.
Figure 2 plots the function along with a typical
density function.
It is worth noting that the expected value of
(4) is not a conditional expectation. If the set
A = {Yn: Yn < F }, then the expected value of
(4) is P (A) E (Yn|A) rather than E (Yn|A). A
simple example will make this clear. Take a
four-point distribution, P (1) = P (2) = P (3) =
P (4) = 41 . Then E (X) = 41 (1 + 2 + 3 + 4) = 25.
Now define the function g(x) as g(x) =
{0, if x $ 2.5, and x, if x < 2.5}. Then E [g (x)]
= 41 (1 + 2) = 43 , while E [x|x # 2.5] = 23.
■
■
2 See also Winton (1997).
3 Although the calculation is not particularly difficult, I have not
seen it before in the literature.
5
The question before us is what happens to
the expected value of the deposit insurance
agency’s payments as the bank diversifies.
Recall that the deposit insurance agency pays
(Of course, as Kane [1989] points out, this
dependency may sometimes promote riskseeking behavior, as in the FSLIC case.)
n
Σ xi
Y
off if nn < f or, equivalently, i =n1 < f . By
the strong law of large numbers, the mean of
Conditions for
the Fallacy
n
the partial sums Σ xi converges to a mass point
i =1
on E (x); that is, the sample means approach
the true mean. Intuition suggests that the expected value of anything below the mean (and,
a fortiori, anything below f ) will have very little importance, that is, an expected value
approaching zero. Put another way, as the
bank gets very large the probability gets vanishingly small, and the average loan payoff falls
below the amount promised to depositors;
thus, the probability of a deposit insurer having
to make a payoff gets so low that the expected
value of that payoff approaches zero.
To establish this rigorously and to understand what diversification does to the expected
value of the deposit insurance agency’s payments requires a more formal approach, which
is provided in the appendix. The intuition and
results are less complicated, however. As a
bank makes more loans, the expected value of
deposit insurance agency payouts tends toward
zero, and so the deposit insurance agency
would like to encourage large banks. Diversification by adding loans works.
III. A Risk-Averse
Deposit Insurance
Agency
When risk aversion enters the picture, however,
a deposit insurer can be worse off with larger
banks. Strictly speaking, what Samuelson terms
the fallacy of large numbers enters only with
risk aversion. Applying it to an organization
such as a deposit insurance agency, rather than
to an individual, requires some justification.
One possibility is that a publicly sponsored
deposit insurance agency must obtain its funds
by taxing people, either indirectly through its
assessment on banks or directly by Congressional appropriation. Risk aversion by the
deposit insurance agency may then reflect risk
aversion on the part of those taxed, or nonlinearities associated with distortionary taxation.
Alternatively, the risk aversion may result from
the incentives, constraints, and information facing the organization: The managers running it
may act risk averse, perhaps because their
future income depends on their performance.
Samuelson (1963) shows that if a consumer
rejects a bet at every wealth level, then he will
always reject any independent sequence of
those bets. Under the Samuelson condition, if
the deposit insurance agency found one bank
loan too risky, it would find a portfolio of any
number too risky. It would be no happier to
insure a large bank with many loans than a
small bank with few loans.
Samuelson posits a rather stringent condition.
It rules out, for example, constant relative riskaverse (CRRA) utility, because CRRA exhibits
decreasing absolute risk aversion (DARA), and
so some unacceptable gambles would become
acceptable at higher wealth levels. Pratt and
Zeckhauser (1987, p. 143) improve considerably
on the condition with their notion of proper risk
aversion. The conditions for proper risk aversion answer the question, “An individual finds
each of two independent monetary lotteries undesirable. If he is required to take one, should
he not continue to find the other undesirable?”
In our problem, if the deposit insurer does not
like the risk in a bank with one loan, then it
will not like the risk in a bank with two loans.
Proper risk aversion shares one defect with
Samuelson’s condition, however: It is difficult to
characterize and difficult to determine whether a
particular utility function satisfies the condition.
A slightly stronger condition with a simple
characterization is proposed by Kimball (1993),
whose standard risk aversion implies proper
risk aversion. It thus applies a slightly stronger
condition than is strictly necessary for the fallacy. If a utility function displays standard risk
aversion, then an investor who dislikes a bet
will also dislike a collection of such bets.
Kimball (1993) shows that necessary and
sufficient conditions for standard risk aversion
are (monotone) DARA and (monotone) decreasing absolute prudence. If the utility function in question has a fourth derivative, then
these conditions (where, as before, W indicates
a person’s wealth) become
(5)
2
d (– u ′ ) < 0 or u (3) > (u ′′) > 0
u′
u ′′
dW
and
(6)
d (– u (2)) < 0 or u (4) < (u (3))2 < 0.
u (2)
u (3)
dW
6
A key point here is that the individual finds
each independent risk undesirable. (Kimball has
a slightly weaker, more technical condition that
he calls loss aggravation.) This certainly applies
to the problem as we have defined it, because
the payoff to the deposit insurance agency
is nonpositive—at best, it pays nothing. This is
not the only way to structure the problem, however, because the deposit insurance agency collects premiums from banks. A major strand in
banking research has been to ascertain whether
the insurance premiums are fairly priced, that is,
whether they represent a tax or a subsidy on
the bank (Pennacchi [1987], Thomson [1987]).
The empirical results are mixed, varying by time
period and by bank; in any case, they assume
risk neutrality and so do not directly answer the
question most relevant here. It makes sense,
then, to think about both possibilities—the case
where the deposit insurance agency finds insuring a single loan undesirable and the case
where it finds insuring a single loan desirable.
In the first case, where the deposit insurance
agency dislikes insuring an individual loan, expressions (5) and (6) provide sufficient conditions for the agency to dislike insuring any portfolio of such loans. That is, diversification by
adding risks does not work; adding risks makes
the insurance agency (guarantee corporation)
worse off.
In the second case, where the deposit insurance agency likes insuring an individual loan,
equations (5) and (6) do not help. Their derivation presupposes that the agency dislikes the
risk it bears. For favorable bets, Diamond (1984)
builds on Kihlstrom, Romer, and Williams
(1981) to develop sufficient conditions for when
the fallacy of large numbers is not a fallacy.
Diamond poses the problem in terms of risk
premiums and notes that adding risks provides
true diversification if it reduces the risk premium. That is, diversification works if the incremental premium for adding the second risky
loan to the portfolio is lower than for adding
the first (identical) risky loan. Kihlstrom, Romer,
and Williams show how to handle risk aversion
with two sources of uncertainty by defining a
new utility function, given initial wealth W0 and
initial risky bet x~1, as
(7)
v (x 2) = Eu(W0 + x~1 + x 2 ).
Now v (x 2 ), as defined in equation (7), can be
treated as a utility function, so Diamond’s
question comes down to whether u (.) is more
risk averse than v (.). If it is, then the risk premium for bearing the second risk is lower than
for the first, and the fallacy of large numbers is
not a fallacy.
Diamond derives two sufficient conditions
under which u (.) will be more risk averse than
v (.) Using Jensen’s inequality, he shows that
(8)
u (3) > 0
and
(9)
u (4) > 0
are sufficient conditions when the risk has zero
expected value. When the risk is freely chosen,
he must append decreasing absolute risk aversion, equation (5), to these conditions. The reason is that a freely chosen gamble increases
mean wealth, which requires us to augment the
sufficient conditions.
Notice that inequalities (6) and (9) cannot
hold simultaneously because (6) demands a
negative fourth derivative and (9) demands a
positive fourth derivative. The inequalities apply
in different situations, however. Inequality (6)
concerns unfavorable bets and describes when
bearing one such risk makes the agent less willing to bear another. Inequality (9) concerns
favorable bets and describes when bearing such
a risk makes the agent even more willing to
bear another. The conditions really answer two
quite different questions. Since each inequality
provides a sufficient but not necessary condition, any contradiction between the answers is
more apparent than real.
An important caveat is that this is consciously
a partial equilibrium analysis, concentrating on
the risk of a single bank. If that bank grows by
absorbing smaller ones, the total number of
loans insured by the system does not change.
A bank merger does not change the total loans
insured by the agency, but merely redistributes
them. In a bank with many loans, the profitable
loans may offset the unprofitable, lessening the
guarantor’s liability. Since the deposit insurance
agency does not share in the positive profits, it
cannot undertake a similar offset if the loans are
in different banks. In an extreme case, if each
bank had only one loan, the insurer would
make payments on every nonperforming loan.
If all loans were in one bank, the insurer would
make payments only if the aggregate loan loss
were too large.
This is not the only scenario, however. The
bank may grow at the expense of nonbank
intermediaries or by making loans that would
not be made without the guarantee. Either case
results in an increase of total loans guaranteed
by the deposit insurance agency, increasing
its liabilities as it takes on new loans that must
get insured.
7
T A B L E
as the sum of two independent exponentials,
has a gamma distribution,
1
Premium Computation
Risk
aversion, a
0.1
1.0
10.0
Face
value
p1
p2
1
1
1
0.006
0.066
0.392
0.018
0.193
0.979
SOURCE: Author.
An Exponential
Example
A simple example can serve to illustrate some
of the subtleties involved. To show what can
happen, I use an exponential utility function
and an exponential distribution. The exponential distribution keeps the algebra simple because sums of exponentials are gamma distributions.4 Exponential utility exhibits constant,
rather than decreasing, absolute risk aversion.
It does not satisfy the sufficiency conditions of
Kimball ([5] and [6]) or of Diamond ([8] and [9]).
Whether diversification helps or hurts depends on the risk premium. If the risk premium decreases as the investor adds i.i.d.
risks, diversification helps. If the risk premium
increases, diversification hurts. The simplicity
of the example allows us to calculate the risk
premium explicitly.
Recall from equation (4) that, for one loan,
the deposit insurance agency pays nothing
if the loan’s payoff exceeds its face value;
otherwise it pays the difference. Denoting this
function by g(x) (as in the appendix), the risk
premium is defined as the p1 that satisfies
(10) u{W0 – E [g(x~ )] – π 1} = Eu [W0 – g(x~ )].
With x following the simplest exponential
distribution, e –x, the expected value in (10)
becomes
Eg(x~ ) = ( f – 1) + e –f.
Using exponential utility of the form e –aW
allows us to solve for π 1:
(11) π 1 = (f – 1 + e –f ) – 1 log[e –f
a
– 1 (e –f – e af )].
1+a
For two loans, g(x) is zero if x exceeds 2f
and 2f – x otherwise. The random variable x,
■
4 In general, bank loans are more likely to be negatively skewed,
while the exponential is positively skewed, so this example is not meant as
a realistic description of actual returns.
–x
x ~ xe = x e – x .
Γ (2)
2
The expected value then becomes Eg(x~ ) =
2( f – 1) + 2( f + 1)e –2f . Solving for the risk
premium implicitly defined by u{W0 – E [g(x~ )]
1
– π 2 } = Eu[W0 – g(x~ )] using π 2 = Eg(x) – a
ag(x)
logEe
yields
(12) p2 = 2( f – 1) + 2( f + 1)e –2f – 1 log
a
{e –2f 1 2 e 2af
1+a
[1 – (1 + 2f (1 +a))e – (1 + a)f ]
+ (1 + 2f )e
– 2f }.
To complete the example, set f, the face
value of the debt, to 1, and compute the premium for several values of risk aversion, evaluating (11) and (12).
The example in table 1 illustrates that diversification does not work in every case. The required risk premium for two loans is higher
than for only one: It even exceeds twice the
risk premium for one loan. As risk aversion
increases, the risk premiums also increase.
Although conditions (5) and (6) are not satisfied, the deposit insurance agency dislikes
adding more independent risks to its portfolio.
IV. Conclusion
Discussions of banking have been obscured by
a false analogy with portfolio theory. A bank
diversifies differently than does a mutual fund,
adding risks rather than subdividing them. Using the weak law of large numbers to establish
that diversified banks have a lower expected
failure rate neglects the deeper question of
whether this represents a decrease in economic
risk. To clearly pose that question is the main
point of this article.
Just because a bank is less likely to fail, it is
not necessarily less risky. If the insurer, or
owner, is risk neutral, a more complicated argument shows that the bank is less risky in the
sense of expected value. With risk aversion,
however, the question becomes ambiguous. As
a practical matter, sufficient conditions exist,
and the combination of exponential utility with
exponential distributions provides a tractable
framework for further exploration.
8
Appendix
Let each random variable be defined on the
probability space (Ω, F, P) and identify Ω with
R, the real numbers, without loss of generality.
The random variables are then functions on this
space, Xi (ϖ ), and define Zn(ϖ ) as
n
Zn(ϖ ) = Σ
i=1
Xi (ϖ )
n .
Next, define the function g (ϖ ) as
g(ϖ ) =
5
f – X (ϖ ) if
and
0
if
X(ϖ ) # f
X(ϖ ) > f.
Note that we can think of the expectation
E [X (ϖ )] as a random variable, and so
g (E [X (ϖ )] = g(µ) = 0, since f < µ. Further
define gn(ϖ ) as g [Zn(ϖ )].
The value of diversification can then be
expressed by saying that as n approaches infinity, the expected value of g (Zn ) approaches
zero, or
E
(A1) lim g n (ϖ ) = g(µ) = 0.
n →∞
To prove (A1), we use Lebesgue’s dominated convergence theorem (Royden [1968],
p. 229), which says that if h(ϖ) $ 0 is integrable,
if |g n (ϖ )| # h(ϖ), and if g n (ϖ ) a.s.
→ g (ϖ ), then
lim
n →∞
Eg (ϖ) = g (ϖ).
n
The theorem first requires that we prove
g n (ϖ ) a.s.
→ g ( µ). To do so, we use the strong law
of large numbers for i.i.d random variables (see
Breiman [1992, p. 52, theorem 3.30]), which
says that for i.i.d. X 1, X 2, X 3 ..., if E|X 1|< ∞
ΣX i a.s.
E (X 1), where a.s.
then
→ denotes aln →
most sure convergence, that is, convergence on
all but a set of measure (probability) zero.
Hence, given an ϖ , except for a set of measure zero, we have that for any ε > 0, there
exists an N such that if n > N,|Zn(ϖ ) – µ|< ε.
Choose ε , µ – f, which implies that if |Zn(ϖ ) –
µ|< ε , then Zn(ϖ ) > µ – ε > f. This, with the
definition of g, in turn implies that gn(ϖ ) = 0.
For this ϖ, then, gn(ϖ ) = g (µ) = 0, and, a fortiori,
|gn(ϖ ) – g (µ)|, ε. Since gn(ϖ) → g ( µ) for
each ϖ where Zn → µ, the almost sure convergence of the strong law implies the almost sure
convergence gn(ϖ ) a.s.
→ g (µ).
All that remains to be shown is the existence
of the integrable bound h(ϖ). For this, use
|Xi (ϖ) + µ – f |, which bounds gn and is integrable because E|X1| , ∞ is a hypothesis of
the strong law. Hence, Lebesgue’s dominated
convergence theorem applies.
As a bank makes more loans, the expected
value of deposit insurance agency payouts
tends toward zero. Diversification works.
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10
Regional Variations in
White–Black Earnings
by Charles T. Carlstrom and Christy D. Rollow
Charles T. Carlstrom is an economist at the Federal Reserve Bank
of Cleveland and Christy D. Rollow
is a research associate at the Federal Reserve Bank of Richmond.
The authors thank Ben Craig and
Mark Schweitzer for many helpful
discussions.
The opinions stated in this article
are those of the authors and not
necessarily those of the Federal
Reserve Banks of Cleveland or
Richmond or of the Board of
Governors of the Federal Reserve
System.
Between 1960 and 1980, the United States made
significant progress in narrowing the earnings
gap between whites and blacks. Unfortunately,
the last two decades have not been as successful. In 1997, the median income for whites was
$47,100; for blacks, it was a much lower $26,500.
This difference is not uniform across the country, however. Studies show that although blacks
have made the largest strides in the South, the
pay gap is still widest there (Murphy and Welch
[1990]). This article examines some of the major
reasons why blacks’ earnings continue to lag
whites’ and why this problem is especially acute
in the southern states.
In the broadest sense, we hope that a
greater understanding of the factors driving
white–black pay differentials at the regional
level will help explain some of the disparities at
the national level. Such knowledge is clearly
applicable to a wide variety of important public
policy issues. In the narrower sense of monetary policy, we seek to enlarge the knowledge
base on which policy decisions are made. The
ability to explain patterns of labor compensation is central to many debates about labor
markets, productivity growth, and inflation
dynamics. Furthermore, the regional nature of
much discussion surrounding real-time policy
decisions—a notable public example being the
Federal Reserve System’s regularly published
Beige Book—suggests the importance of understanding how earnings patterns across groups
and geographic areas compare with those typically found in aggregate analyses.
Previous studies of earnings disparities between groups have followed Oaxaca (1973),
which decomposes wage variations into their
explained and unexplained components. Under
the null hypothesis, explained differences are
typically interpreted as reflecting differences in
productivity, while unexplained differences are
loosely associated with discrimination. These
studies account for regional variations through
the use of dummy variables—a method that
presumes identical wage-determination
processes across all regions. As a result, standard wage decompositions provide aggregated
measures, but only limited information about
earnings and discrimination within particular
areas of the country.
This article differs from earlier studies in
that it focuses specifically on decomposing
regional wage differentials. The regions considered here are the South and non-South. Our
approach is a natural extension of the Oaxaca
11
decomposition. Running separate wage regressions for whites and blacks is equivalent to interacting race with the independent variables.
Similarly, estimating separate regressions for
the South and non-South enables the researcher to analyze white–black wage differentials
without assuming the same wage equation for
both regions.
The Oaxaca decomposition has several limitations. While unexplained differences are generally thought to measure discrimination, they
may not do so accurately. Likewise, explained
differences may be due to discrimination rather
than to differences in productivity. Therefore,
caution must be used in classifying the reasons
behind existing wage disparities. A good approach is to look at our findings as a piece of a
much larger puzzle that policymakers and
economists can use to better understand the
determinants of earnings.
The article proceeds as follows: Section I
introduces the Oaxaca decomposition used for
identifying white–black wage differentials.
Section II discusses the limitations of this approach. To help understand the empirical results
covered later, section III provides an overview
of discrimination theory. The final sections then
specify and estimate U.S. and regional wage
models, discuss these results in light of discrimination theory, and summarize our key findings
regarding the determinants of wage differentials
and the nature of the earnings gap.
I. Decomposition
Methodology
One of the reasons for developing the Oaxaca
decomposition was to quantify the amount of
discrimination in the marketplace. Market discrimination occurs when individuals with the
same productivity are paid different wages. Productivity, however, is not observable.
For discrimination to be appropriately captured by the Oaxaca decomposition, the following conditions must be satisfied: 1) The factors
that determine productivity (education, for
example) can be completely identified; and
2) these factors are accurately measured. Given
these conditions, one can estimate what portion
of the wage differential is due to productivity
and what portion is due to discrimination.
The Oaxaca approach involves estimating
separate wage equations for different groups,
such as whites and blacks, and then decomposing the pay gaps into explained and unexplained portions. Explained differences are said
to be attributable to productivity factors, such as
experience and education, while unexplained
differences are said to result from discrimination, as already defined. The conceptual framework for this analysis is presented below.
The average wage gap between whites and
blacks is given by
(1)
^
^
Gap < ln(WW ) – ln(WB ),
where the carets over the W ’s represent the
average wage of whites and blacks. Equation
(1), then, is simply the percentage difference
between white and black wages. Regression
equations for whites and blacks are defined as
(2)
ln(WW ) = XW bW + ε
ln(WB ) = XB bB + ε,
where W represents wages, X is the vector of
the regressors, and b is the corresponding vector of estimated coefficients (Oaxaca and
Ransom [1994]). Within this framework, the
geometric means (approximately equal to the
arithmetic means) of the equations are as follows (b ’s represent the ordinary-least-squares
[OLS] estimates from equation [2]):
(3)
^
^
^
^
ln(WW ) < XW bW
ln(WB ) < XB bB .
To understand the earnings disparity between whites and blacks better, it is important
to distinguish between differences that can be
explained by variations in the explanatory variables and differences that are unexplainable.
There are two ways of decomposing equation (3) into its explained and unexplained
components. The first decomposition, referred
to as “whites as base,” assumes that in the
absence of discrimination, the wage structure
(and hence productivity) for both races is given
by the estimated coefficients for whites.
(4)
^
^
^
^
GapW < ln(WW ) – ln(WB ) < XW bW – XB bB
^
^
^
^
= XW bW – XB bB + XB bW – XB bW
^
^
^
= (XW – XB )bW + XB ( bW – bB ).
The alternative (“blacks as base”) assumes
that the true relationship governing productivity
for both groups is given by the regression estimates for blacks.
(5)
^
^
^
^
GapB < ln(WW ) – ln(WB ) < XW bW –XB bB
^
^
^
^
= XW bW – XB bB +XW bB – XW bW
^
^
^
= (XW – XB )bB +XW ( bW – bB ).
12
By construction, the total wage gap between
whites and blacks is independent of the method selected. The portions that fall under “explained” and “unexplained” may be different,
however. The first term of equations (4) and (5)
is the explained component. It is represented
by differences in the means of the explanatory
variables, X, and measures the part of the wage
difference that arises because blacks are underrepresented in high-paying, skilled occupations
or have less education on average than whites.
These differences are said to be “explained”
because they represent the wage differential
that stems from differences in productivity arising from observed differences in education,
tenure, experience, occupation, and so on.
The second term of equations (4) and (5) is
the unexplained, or potentially discriminatory,
portion. It arises whenever the returns to
whites and blacks differ for given values of an
explanatory variable, including the intercept.
Because measurable characteristics and job
qualities account for only a part of the wage
gap, the remaining differential is often assumed
to result from discrimination.
II. Does the Oaxaca
Decomposition
Capture
Discrimination?
The Oaxaca decomposition employs many
assumptions designed to measure discrimination. Unfortunately, in the real world, some of
these may be invalid. It is important to understand why the explained and unexplained
components of the Oaxaca decomposition
may not capture productivity differences and
discrimination.
One reason for including explanatory variables such as female, firm size, and region is
that they empirically affect wages. Another reason is that differences in the means of some
explanatory variables may reflect discrimination
rather than individual choice. For example,
blacks may be denied promotions or be “segregated” into lower-paying occupations and
industries.1 Similarly, blacks’ shorter tenure
may be the result of more frequent layoffs.
Differences in educational achievement
between whites and blacks may also arise
because of premarket versus workplace discrimination. That is, even before blacks enter
the workforce, their potential to earn a degree
may be limited. If, for example, blacks receive
substandard schooling at the primary and secondary levels, they may have fewer opportunities to attend college.
While differences in the explained component may in fact result from discrimination, the
unexplained component may not reflect discrimination. This can occur if the wage model
is mis-specified; that is, if an important variable
that affects productivity is omitted. Economists
who study male–female wage disparities frequently confront this problem. A researcher
may conclude that women are discriminated
against when, in fact, the gap between men’s
and women’s earnings reflects differences in
some unobservable variable, such as turnover,
or other types of compensating differentials.
To understand these results better, it is useful
to discuss the economics of discrimination. This
gives the researcher some direction as to what
factors may help disentangle the components
of wage variations.
III. The Economic
Theory of
Discrimination
Becker (1971) provided the first analysis of the
economics of discrimination. He posited that, in
general, competition tends to eliminate wage
differences arising from discrimination. The
reason, he said, is that firms must forgo profits
in order to discriminate. Any profits will be
eliminated as nondiscriminatory firms enter the
market to capture these windfalls. Thus, for discriminatory wage differences to exist, there can
be few actual or potential nondiscriminatory
firms. That is, most firms must have a preference to discriminate.
One prediction of this theory is that discriminatory wage differences will be lower in more
competitive labor markets. For example, in
urban areas, where there are many different job
and product opportunities, white–black wage
differentials resulting from discrimination
should be lower than in rural areas, where
fewer opportunities exist.
Even if firms prefer not to discriminate, theory predicts that they might if their customers
wish to do so. If this conjecture is true, discrimination would be more prevalent in jobs
with a high degree of direct customer contact.
Examples of such positions include sales
workers, beauty shop employees, teachers,
doctors, and lawyers. Furthermore, this theory
implies that racial discrimination will be lower
in areas where the fraction of minority customers is large.
■ 1 Sorensen (1989) argues that women and minorities are “crowded
out” of higher-paying jobs.
13
IV. Model
Specification
Our model includes the usual independent
variables that, under the null hypothesis, collectively control for individual productivity
differences: female, subregions, education,
potential experience (exp), experience squared
(exp 2), experience cubed (exp 3), experience
quartic (exp 4), industry (ind ), firm size (size),
union, occupation (occ), tenure (ten), tenure
squared (ten 2), tenure cubed (ten 3), tenure
quartic (ten 4), metropolitan residence (MSA),
and marital status (ms).2
We measure educational attainment by the
highest level of schooling completed: high
school dropout, high school/GED, some
college/vocational training, college, postgraduate (M.A.), professional, and doctorate. Because
of limitations in the Current Population Survey
(CPS) data, information on actual work experience is not available. Therefore, we use potential experience, measured as age minus schooling minus six, as an approximation.3 Murphy
and Welch (1990) argue that the traditional
model, which includes only experience and its
square, is mis-specified and that a quartic specification fits the earnings equation best.
To analyze the wage implications of industries, we define five groups: miscellaneous services (ind 1); manufacturing, transportation, communications, and public utilities (ind 2); hospital
and medical services (ind 3); retail trade (ind 4);
and finance, insurance, and real estate, or FIRE
(ind 5).4 To assess the wage effects of occupation, we define six other groups: executive,
administrative, and managerial (occ1); professional specialty (occ2); sales workers (occ 3);
service workers (occ 4); machine operators,
assemblers, and inspectors (occ5); and transportation and material moving (occ 6).5
We also include a firm size variable representing companies with 1,000 or more employees. Previous studies indicate that large companies pay substantially more than their smaller
counterparts. One possible reason is that large
companies attract more productive workers.
Other research has shown that large companies
offer efficiency wages to maintain their highly
skilled workforce and to reduce worker turnover (Salop [1979]; Bulow and Summers
[1986])6 or that they pay higher wages to avoid
unionization (Brown, Hamilton, and Medoff
[1990]). Union status itself, a potentially important variable, is measured by coverage by a
union contract.
Tenure with one’s current employer is also
included, since human capital theory suggests
that earnings differences can be accounted for
by the types and levels of investment in formal
or on-the-job training. As was the case for
experience, we assume a quartic specification
for this variable.
MSA serves as a proxy for urbanization, providing a measure of regional size and competition. The demographic variable “marital status”
is equal to one if an individual has ever been
married. All else equal, married females tend to
earn less than their single counterparts, while
the opposite holds true for males (Berndt
[1991]). The interaction between female and
marriage measures the differential impact that
marriage has on women’s wages.
The explanatory variables already mentioned are included in the model below. We
have chosen earnings as the dependent variable because policy debate has focused specifically on this factor.7
X = (Subregions, Education, Experience,
Experience 2, Experience 3, Experience 4,
Industry, Occupation, Firm Size,
Union, Tenure, Tenure 2, Tenure 3,
Tenure 4, MSA, Female, Marital Status,
Female 3 Marriage);
and b is the corresponding vector of
coefficients.8
(6) ln(Earnings) = X b + ε.
■ 2 Omitted conditions for all variables are as follows: female: male;
education: high school dropout; industry: mining, construction, wholesale
trade, private household, forestry and fisheries, public administration, educational services, social services, and other professional services; firm
size: fewer than 1,000 workers; union: not covered by a union contract or
no response; occupation: administrative support, including clerical, technical, and related support, handlers, equipment cleaners, and farming,
forestry, and fishing; MSA: not a metropolitan area; and marital status:
never married.
■ 3 For a complete discussion of the CPS measurements and their
limitations, see U.S. Bureau of Labor Statistics (BLS), Employment and
Earnings, any year.
■ 4 The use of industry dummies at the one-digit rather than the twoor three-digit level may introduce an aggregate bias.
■ 5 Listings of detailed occupations used in the CPS are published in
Employment and Earnings, January 1994, for occupations with 50,000 or
more workers. Other listings (unpublished data) can also be obtained from
the BLS.
■ 6 See Katz (1986) for an overview of the literature.
■ 7 Similar results were obtained when the decomposition was done
using both earnings and earnings divided by hours worked.
■ 8 The intercept term is excluded, since each region estimated
includes all of its subregions.
14
F I G U R E
1
Regional Decomposition
Methodology for Blacks
A
R 1BX
R 1WX
R 2BX
R 2WX
bBR1
t-test
bWR1
t-test
bBR2
t-test
bWR2
t-test
B
R2
R1
X
X
BX
BX
( bBR2 – bWR2)
t-test
( bBR1 – bWR1)
t-test
C
(R 1 + R 2 )
R 1BX
( bBR1 – bWR1)
t-test
R 2BX
F-test
( bBR2 – bWR2)
t-test
( bBR1 – bWR1) – ( bBR2 – bWR2)
= ( bBR1 – bBR2) – ( bWR1 – bWR2)
SOURCE: Authors.
Data and
Methodology
Different, but equivalent, regressions can be estimated to calculate the coefficients necessary
for a Oaxaca decomposition. For example, including interactions between race and the independent variables is equivalent to estimating
separate wage regressions for whites and blacks.
Figure 1 illustrates these alternatives as well as
the basic methodology used in this research.
To estimate our model, we pool regions,
whites, and blacks into a single regression with
interactions between region, race, and the
explanatory variables (see figure 1, part C). This
provides a straightforward way to determine
the b’s for the Oaxaca decomposition and to
test whether differences in these estimates,
across both regions and races, are significant.9
The estimates that follow are obtained by
using cross-sectional microdata from the Public
Use Sample of the April 1993 CPS, conducted
by the Census Bureau. Wage equations for
whites and blacks are estimated separately for
the South and the rest of the country.10 The
sample contains full-time (35 or more hours
per week), private, nonagricultural employees
between the ages of 25 and 54.11 The focus on
year-round, full-time workers “minimizes earnings fluctuations due to business cycles” and
“income sources other than earnings” (Levy
and Murnane [1992]).
V. U.S. Wage
Decomposition
Before estimating decompositions for the South
and non-South, we first estimate a standard
U.S. decomposition.12 This will increase our
understanding of wage differentials at the
national level and will provide a reference for
the regional decompositions. Table 1 presents
the average wage gap between whites and
blacks, breaking it down into its explained and
unexplained components. The total wage differential (using whites as base) is estimated to
be 32.7 percent, while the unexplained component accounts for 12.9 percent.
The explained component (variations in the
means) accounts for 19.8 percentage points of
the total differential. One contributing factor is
education, which accounts for nearly 4 percentage points of the wage disparity between
whites and blacks.
The single most important factor contributing to the explained portion of the white–black
wage differential is occupation. The reason is
the lower concentration of blacks in higher■ 9 Although parameter estimates are identical among the three
regressions, standard errors may vary. In this study, differences in standard
errors are minimal.
■ 10 The South, as defined by the U.S. Census Bureau, includes
three divisions: the South Atlantic (Delaware, Maryland, the District of
Columbia, Virginia, West Virginia, North Carolina, South Carolina,
Georgia, and Florida), the East South Central (Kentucky, Tennessee,
Alabama, and Mississippi), and the West South Central (Arkansas,
Louisiana, Oklahoma, and Texas).
■ 11 Individuals earning an average hourly wage of less than $1 or
more than $60 are considered outliers and are eliminated from the sample
(see Anderson and Shapiro [1996]). The age restriction is an attempt to
capture individuals who have completed their education but have not yet
retired (those in their career stages).
■ 12 Regional dummies include all the major U.S. regions: South,
Northeast, Midwest, and West.
15
T A B L E
1
Decomposition of
Wage Differentials: U.S.
Variable
Total wage difference = 32.7%
Percent
Percent
explained
unexplained
Whites
Blacks
Variable
Whites
Blacks
as base
as base
(Race 3)
as base
as base
All
19.8
(0.0001)
Education
3.9
(0.0001)
Occupation 5.7
(0.0001)
Tenure
1.5
(0.0001)
MSA
3.0
(0.0001)
Marital status 2.0
(0.0001)
20.7
(0.0001)
3.9
(0.0001)
5.6
(0.0001)
1.8
(0.0001)
0.9
(0.0001)
0.1
(0.0001)
All
Ind1
MSA
12.9
12.0
(0.0001) (0.0001)
2.4
2.1
(0.001) (0.001)
2.6
4.7
(0.08)
(0.08)
NOTE: Variables presented contribute most to explaining the total wage
difference. P-values are given in parentheses.
SOURCE: Authors’ calculations.
T A B L E
2
Decomposition of Wage
Differentials: South
Variable
All
Total wage difference = 34.2%
Percent
Variable
Percent
explained
(Race 3)
unexplained
19.0
(0.0001)
Education
5.0
(0.0001)
Occupation
6.3
(0.0001)
Tenure
0.4
(0.0001)
MSA
3.4
(0.0001)
Female
1.6
(0.11)
Marital status
2.4
(0.004)
All
Ind1
Occ2
Firm size
Tenure
MSA
Female
15.2
(0.0001)
1.6
(0.07)
–0.7
(0.66)
–9.5
(0.09)
–9.5
(0.43)
2.8
(0.10)
5.5
(0.48)
NOTE: Variables presented contribute most to explaining the total wage
difference (whites as base). P-values are given in parentheses.
SOURCE: Authors’ calculations.
paying jobs. Indeed, executive, administrative,
and managerial positions (occ1) account for
almost half (2.8 percentage points) of the total
explained differential arising from occupation
(5.7 percentage points). This is the highest-paid
group, with workers earning over 6 percent
more than those in the professional specialty
category. Only 12 percent of blacks hold
managerial–administrative positions, compared
to more than 22 percent of whites.
MSA accounts for another 3.0 percentage
points of the explained differential (using
whites as base). On average, whites earn 14
percent more in urban areas than in rural communities. At the same time, 48 percent of
employed whites and 27 percent of employed
blacks live in urban areas. Assuming a black
wage structure, however, the contribution of
MSA to the explained component drops to only
0.9 percent. This occurs because blacks in
urban areas earn only 4.5 percent more than
their rural counterparts.
Nationwide, only MSA and miscellaneous
services (ind 1) contribute significantly to the
unexplained portion of the wage gap. The 9.7
percent less earned by blacks relative to whites
in urban areas is the reason MSA’s contribution
to the explained component depends on the
assumed wage structure (white versus black)
and also explains why MSA is an important factor in the unexplained component. Its 2.6percentage-point contribution to the 12.9 percent total unexplained wage differential is surprising, since one expects metropolitan areas to
provide more competition, which in turn
should discourage discriminatory hiring and
work practices and support more equal wages.
Miscellaneous services (ind 1) accounts for
2.4 percentage points of the nearly 13 percent
unexplained wage difference between the
races. Although its total contribution is not particularly large, blacks in this category earn
nearly 34 percent less than whites. This sizable
gap reflects the small share of blacks (7 percent) working in this industry.
VI. Regional
Variations in
Earnings Inequality
The sources of white–black wage differences
are not the same inside the South as outside.
Tables 2 and 3 show decompositions for both
regions.13 The total wage difference between
■ 13 Interesting differences may exist outside the South, but the
small number of nonsouthern blacks in the CPS sample makes further disaggregation inappropriate.
16
T A B L E
3
Decomposition of Wage
Differentials: Non-South
Variable
Total wage difference = 25.2%
Percent
Variable
Percent
explained
(Race 3)
unexplained
All
15.8
(0.0001)
Education
3.8
(0.0001)
Occupation
2.3
(0.0001)
Tenure
4.6
(0.0001)
MSA
1.5
(0.0001)
Female
2.3
(0.002)
Marital status
1.8
(0.0001)
All
9.4
(0.01)
2.5
(0.09)
3.9
(0.02)
–2.8
(0.68)
–22.5
(0.07)
2.8
(0.38)
20.1
(0.07)
Ind1
Occ2
Firm size
Tenure
MSA
Female
NOTE: Variables presented contribute most to explaining the total wage
difference (whites as base). P-values are given in parentheses.
SOURCE: Authors’ calculations.
T A B L E
4
Representation of Whites
and Blacks in Sample (percent)
South
Variable
Non-South
Whites
Blacks
Whites
Blacks
Education
High school or less
Some college
College grad or higher
37.6
28.6
33.8
50.2
23.0
26.8
34.8
29.7
35.5
28.9
49.3
21.8
Occupation
Highest paying
Lowest paying
19.3
6.8
8.7
17.4
23.4
5.2
19.0
2.8
Tenure (mean years)
(9.2)
(9.1)
(8.7)
(7.5)
MSA
44.7
21.9
49.7
36.6
Female
45.0
60.0
44.5
62.7
Marital status
86.0
69.4
84.4
71.1
NOTE: Variables presented contribute most to explaining the total wage
difference. Mean years of tenure are given in parentheses.
SOURCE: Authors’ calculations.
whites and blacks averages 34.2 percent in the
South, compared to 25.2 percent outside. Nearly
half of this difference can be traced to the explained components. In the South, 19.0 percent
of the earnings disparity is explained, while in
the non-South that share drops to 15.8 percent.
Reporting just the sum of the explained
components, however, masks important regional differences. For example, occupation contributes 6.3 percentage points to the explained
component in the South, but only 2.3 percentage points in the non-South. This situation is
reversed for tenure, which contributes 0.4 percentage point in the South but 4.6 percentage
points outside the South. These patterns occur
because of important differences in the representation of blacks in certain occupations and
in the average length of tenure between whites
and blacks in the two regions.
Table 4 illustrates these disparities. Southern
blacks are underrepresented in the highestpaying occupation (administrative–managerial)
and disproportionately represented in the
lowest-paying one (operatives). These differences are virtually nonexistent in the rest of the
country. On the other hand, tenure levels in the
South are nearly identical between the races,
while in the non-South blacks average over a
year less on their current jobs.
Tables 2 and 3 indicate that educational differences between the races explain 5.0 percentage points of the white–black wage gap
in the South and 3.8 percentage points outside
the South. Despite the similarity of these numbers, table 4 shows that blacks have strikingly
different patterns of educational attainment in
the South versus the non-South. These differences are not captured if one considers just the
mean years of education for blacks in the two
regions (13.3 years in the South, 13.5 years in
the non-South).
Substantially more blacks attend college outside the South than in the South—71 percent
versus 50 percent. Interestingly, though, more
blacks in the South obtain their four-year college
degree (26.8 percent, compared to 21.8 percent
outside the South). This occurs because 54 percent of blacks who attend college in the South
obtain at least a bachelor’s degree, while only 31
percent of blacks outside the South do. If blacks
in the non-South completed college at the
same rate as their counterparts in the South, our
model indicates that their wages would increase
by 2.3 percentage points, narrowing the white–
black earnings gap to 22.9 percent. The share of
the white–black wage differential that is unexplained is 15.2 percent in the South and 9.4 percent in the non-South—both highly significant.
17
T A B L E
Unexplained Wage
Differentials
5
Regional Parameters
and Differences
Tables 2 and 3 also report the breakdown for
Parameter estimates
Variable
(Race 3)
South
NonSouth
Difference
Ind 1
–0.25
(0.07)
–0.30
(0.09)
0.05
(0.84)
Occ 2
0.06
(0.66)
–0.37
(0.02)
0.43
(0.04)
Firm size
0.12
(0.09)
0.04
(0.68)
0.08
(0.49)
MSA
–0.13
(0.10)
–0.077
(0.38)
–0.063
(0.66)
Female
–0.09
(0.48)
–0.32
(0.07)
0.23
(0.30)
NOTE: Variables presented contribute most to explaining the total wage
difference (whites as base). P-values are given in parentheses.
SOURCE: Authors’ calculations.
T A B L E
6
Unexplained Wage Differences
for Various Groups
South
NonSouth
Difference
All
15.2
(0.0001)
9.4
(0.01)
5.8
(0.24)
All – Occ 2
15.9
(0.0001)
5.6
(0.18)
10.3
(0.06)
All – Ind 2
13.6
(0.0002)
6.9
(0.09)
6.7
(0.20)
Males
20.1
(0.0001)
6.5
(0.28)
13.6
(0.08)
Females
11.9
(0.004)
11.2
(0.02)
0.7
(0.90)
Urban
22.1
(0.0005)
10.3
(0.09)
11.8
(0.18)
Rural
13.1
(0.0003)
4.2
(0.73)
8.9
(0.51)
Large firms
10.4
(0.17)
7.0
(0.12)
3.4
(0.70)
Small firms
25.7
(0.0001)
15.6
(0.03)
10.1
(0.28)
Variable
NOTE: Variables presented contribute most to explaining the total wage
difference (whites as base). P-values are given in parentheses.
SOURCE: Authors’ calculations.
the independent variables in which unexplained wage differences are significant in one
or both regions. Only miscellaneous services
(ind 1), which was important for the U.S. as a
whole, is significant in both the South and nonSouth, accounting for 1.6 and 2.5 percentage
points of the unexplained differences, respectively. Within that occupational category, blacks
earn 25 percent less than their white counterparts in the South and 30 percent less than
whites outside the South (see table 5).
The biggest difference between the two
regions occurs within the professional specialty
category (occ2). In the South, occ2 does not
account for any of the unexplained component
of the white–black wage differential, whereas
in the non-South it accounts for 3.9 percentage
points of the 9.4 percent unexplained wage
gap. This is because within this occupation
blacks outside the South earn an average of 37
percent less than whites (see table 5). In the
South, the sign is reversed but insignificant.
The difference in the way whites and blacks
are rewarded within professional specialty occupations is highly significant (p -value = 0.04)
and impacts the two regions’ race-based pay
gaps. Excluding this category, the unexplained
earnings disparity in the non-South drops to 5.6
percent and becomes insignificant. This is in
sharp contrast to the 15.9 percent unexplained
pay differential seen in the South when this
group is excluded (see table 6).
In the South, the variable with the greatest
impact on the unexplained pay component is
firm size. Whites working in large firms earn
3.9 percent more than those working in small
firms, but the earnings differential for blacks is
much greater—16.3 percent. The pay difference between whites and blacks in large firms
is 10.4 percent and is insignificant (see table 6);
in small firms, that share jumps to 25.7 percent.
One commonly held belief is that the narrowing of the white–black wage gap between 1960
and 1980 is attributable to antidiscrimination
laws and the genesis of affirmative action programs. The substantial premium associated with
firm size in the South could occur if large firms
there are policed much more closely than small
firms or if the laws are enforced unequally.
MSA is significant only in the South, but contributes 2.8 percent to the total unexplained
wage differential in both regions (see tables 2
and 3). This occurs because a much greater
share of working blacks outside the South live
18
T A B L E
7
Non-South Parameter Estimates for
Black 3 Tenure and 3 Female
Original
Variable
(Black 3)
Parameter
estimates
Variable
(Black 3 male)
Parameter
estimates
Variable
(Black 3 female)
Parameter
estimates
All tenure a
–0.225
(0.07)
All tenure a
–0.0153
(0.80)
All tenure a
–0.3431
(0.04)
Ten
0.1246
(0.10)
Ten
0.0015
(0.99)
Ten
0.1969
(0.04)
Ten 2
–0.0155
(0.14)
Ten 2
–0.0034
(0.88)
Ten2
–0.0226
(0.09)
Ten 3
0.0007
(0.20)
Ten 3
0.0003
(0.81)
Ten3
0.00087
(0.19)
Ten 4
–0.000009
(0.27)
Ten 4
–0.000006
(0.78)
Ten4
–0.0000106
(0.30)
Black 3
female
–0.672
(0.02)
Female
–0.32
(0.07)
a. This estimate is the unexplained component for tenure.
NOTE: Variables presented contribute most to explaining the total wage difference (whites as base). P-values are given in parentheses.
SOURCE: Authors’ calculations.
in urban areas (36.6 percent versus 21.9 percent; see table 4). The importance of MSA in the
South may arise because blacks make up more
than 50 percent of the rural population there,
but only 22 percent of the urban population.
The next section takes a closer look at the
reasons behind these regional differences. It
also attempts to ascertain whether some of the
explained and unexplained differences are due
to productivity and discrimination, respectively.
Regional
Differences:
A Closer Look
By analyzing the apparent differences between
the two regions, this section illustrates why
teasing out discrimination is so difficult. Tables
2 and 3 show that in the non-South, blacks
gain significantly more than whites from an
additional year on the job (p -value = 0.07),
and black women earn 32 percent less than
comparable black men. In the South, there is
no significant difference for either variable. Although our strict interpretation of the null is
that discrimination in the non-South is associated with black women only, there are alternative explanations.
Table 4 shows that the non-South is also the
region with important tenure differences be-
tween the races. In the South, tenure levels for
whites and blacks are almost identical, with no
differential return. This suggests that certain unobservable characteristics may cause higher
turnover and hence lower wages for blacks in
the non-South. As blacks accumulate time on
the job, the importance of this unobservable
diminishes.
Evidence suggests that, on average, women
have weaker attachments to the labor force
than do men and as a result earn less. Firms,
however, probably cannot tell at the time of
hire which women are more likely to stay.
Since turnover is costly, firms must be compensated whenever they hire someone they believe
will have a greater chance of leaving. A woman
may initially start out at a lower salary than a
man doing the same job, but should correspondingly gain more per additional year of
tenure. Women should then be compensated as
they slowly reveal themselves to be “stayers.”
The large return to blacks from tenure, coupled with the large negative return to black
women, suggests a connection between the
two. For example, black women may have a
weaker attachment to the labor force than do
black men, and this difference may be greater
than it is for whites. To investigate whether this
explanation can account for the unusually large
returns to tenure and the small returns to black
women, we break the black-tenure interaction
into its male and female components. The
19
T A B L E
8
South Parameter Estimates for
Black 3 Tenure and 3 Female
Variable
(Black 3)
Original
Parameter
estimates
Variable
(Black 3 male)
Parameter
estimates
Variable
(Black 3 female)
Parameter
estimates
All tenure a
0.095
(0.43)
All tenure a
–0.473
(0.02)
All tenure a
0.147
(0.46)
Ten
–0.023
(0.73)
Ten
–0.187
(0.06)
Ten
0.272
(0.05)
Ten2
0.002
(0.78)
Ten 2
–0.022
(0.08)
Ten 2
–0.033
(0.10)
Ten3
–0.0001
(0.78)
Ten 3
0.001
(0.09)
Ten 3
0.0014
(0.18)
Ten 4
–0.000002
(0.80)
Ten 4
–0.00001
(0.11)
Ten 4
–0.000002
(0.31)
Black 3
female
–0.742
(0.03)
Female
–0.092
(0.48)
a. This estimate is the unexplained component for tenure.
NOTE: Variables presented contribute most to explaining the total wage difference (whites as base). P-values are given in parentheses.
SOURCE: Authors’ calculations.
expectation is that black tenure will be significant for women and insignificant for men.
Table 7 confirms this suspicion, showing that
black females gain more than whites for extra
years on the job, while the returns to tenure for
black males and whites are nearly equal.14 In
other words, tenure helps equalize the wage
difference between white and black females
but not between white and black males. As a
result, the black female coefficient falls even
further and becomes more significant. Before
interactions with tenure, black women earn 32
percent less than white women; afterward, they
earn 67 percent less. This is because of the 34
percent that tenure adds to the wages of black
females compared to whites. The average wage
difference is about 33 percent, close to the 32
percent estimated before allowing interactions.
Because the importance of tenure and female
in the non-South could be due to model misspecification, the unexplained differences may
not reflect labor market discrimination.
Table 8 shows the same experiment for the
South. Unlike earlier (table 5), the black female
coefficient is now highly significant and nearly
identical to that of the non-South. Therefore, the
surprising finding that black women earn less
than black men in the non-South could reasonably be attributed to model mis-specification.
Tables 7 and 8 reveal that the major difference
between the two regions with regard to tenure
is that tenure offers a larger return to blacks in
the non-South than in the South. This may
reflect the greater tenure differences between
whites and blacks in the non-South.
If the return to black females is indeed due
to model misspecification, then perhaps the
unexplained returns to black males is a better
indicator of potential discrimination. If these differentials are interpreted as reflecting discrimination, we would conclude that discrimination
is important in the South (the unexplained component is 20.1 percent and is highly significant
[p -value = 0.0001]) but not outside the South. If
this interpretation is correct, there is little evidence to suggest that non-South discrimination
is important, since the unexplained return is
only 6.5 percent and is insignificant.
The single most important factor contributing to the explained wage differences between
whites and blacks is occupation—6.3 percent
in the South and 2.3 percent outside the South.
Although these differences are “explained,” as
mentioned before, they may reflect discrimination if blacks are segregated into lower-paying
occupations. Probit estimates indicate that educational differences between whites and blacks
in the non-South can explain why 19 percent of
blacks are in the highest-paying occupations
■ 14 We also tried an interaction between female and tenure for whites,
but it was insignificant; therefore, we use whites as the reference group.
20
T A B L E
9
Non-South Professional
Specialty Occupation
Variable (Race 3)
Parameter estimates
Public
–0.635
(0.006)
Nonpublic
–0.182
(0.345)
NOTE: P-values are given in parentheses (whites as base).
SOURCE: Authors’ calculations.
(occ 1), compared to 23.4 percent of whites
(p-value = 0.49). This is not the case in the
South. Even after controlling for educational
differences, blacks are significantly less likely
than whites to work in occ 1 (p-value =
0.0002). This provides some ammunition for
the argument that blacks are more likely to be
segregated into lower-paying jobs in the South
than in other areas of the country.
Another argument is that differences in
white–black tenure levels in the non-South
may reflect discrimination (that is, the 4.6percentage-point “explained” tenure difference
is due to discrimination). This could arise if
blacks in the non-South are fired “without
cause” more often than are whites. Basically,
the question is whether wages between whites
and blacks adjust more freely in the South,
whereas quantities adjust more freely in the
non-South. It is impossible to determine conclusively whether tenure differences in the
non-South result from discrimination; however,
the following argument casts doubt on discrimination’s ability to explain the lower tenure levels for blacks in the non-South.
If blacks are being laid off without cause
more frequently in the non-South than in the
South, then one would expect the difference
between potential experience (which is observable) and actual experience (which is not
observable) to be greater in the non-South.
This is because unemployment spells are much
longer for laid-off workers than for those who
quit. If blacks in the non-South are indeed
more likely to be laid off, then they should also
have more interruptions in their work histories.
Thus, the ratio of actual experience to potential
experience for blacks in the non-South would
be lower, or equivalently, the measured returns
to potential experience would be lower. No
such difference should exist in the South, however. The data show that there is no significant
difference between the returns to potential
experience for whites and blacks in either
region; furthermore, the sign is the opposite of
that expected. This casts doubt on the proposition that discrimination is reflected in wage differences in the South and quantity differences
outside the South.
Another important difference between the
regions is in how whites and blacks are compensated within professional specialty occupations. Outside the South, whites within this category earn significantly more than blacks; in the
South, however, no such difference exists.
Faced with this observation, one might appeal
to discrimination theory and speculate that the
source of the effect is customer-driven discrimination. The next regression addresses this possibility by examining whether white–black wage
differences within professional specialty occupations outside the South arise because the category comprises public jobs. (See the appendix
for groupings of public and nonpublic jobs.)
Rerunning the equation for the non-South
suggests important differences in how blacks
are compensated within this category. In public occupations (those with more customer
contact), blacks earn 63.5 percent less than
their white counterparts (p -value = 0.0056); as
table 9 shows, this difference is insignificant in
nonpublic occupations (p -value = 0.35). The
difference between these two figures is only
marginally significant (p -value = 0.11).
VII. Concluding
Thoughts
There is little evidence to suggest that white–
black wage disparities outside the South cannot
be explained by differences in observable characteristics. The situation is much less clear in
the South, where the significance of the unexplained components of the race-based pay gap
is robust. Furthermore, even some of the “explained” differences in the South suggest the
possibility of workplace discrimination. For example, unlike the rest of the nation, the South
cannot attribute blacks’ underrepresentation in
high-paying occupations to differences in observed characteristics like education.
The other major factors driving the lower
wages of blacks in the South are firm size and
location: Blacks working in small firms or urban areas earn significantly less than comparable whites. It is difficult to determine whether
discrimination is behind either of these unexplained wage differences. Although the lesser
amount earned by urban blacks is consistent
with discrimination, it may also arise because
21
of unobservable differences between rural versus urban blacks. The difficulties heterogeneity
causes in trying to measure discrimination cannot be overemphasized; discrimination exists
for both whites and blacks, even after controlling for observable characteristics. And evidence suggests that this problem is getting
worse. Income inequality over the last 30 years
has increased for both whites and blacks; however, it has been especially pronounced for
blacks. Understandably, identifying the true
source of these trends and differentials represents a major challenge for economists and
policymakers alike.
Appendix:
Public and
Nonpublic
Occupational
Groupings
Miscellaneous Services Industry
Public
Hotels and motels
Lodging places, except hotels and motels
Laundry, cleaning, and garment services
Beauty shops
Barber shops
Funeral services and crematories
Nonpublic
Advertising
Services to dwellings and other buildings
Personnel supply services
Computer and data processing services
Detective and protective services
Business services
Automotive rental and leasing,
without drivers
Automobile parking and car washes
Automotive repair and related services
Electrical repair shops
Miscellaneous repair services
Theaters and motion pictures
Video tape rental
Bowling centers
Miscellaneous entertainment
and recreation services
Professional Specialty Occupation
Public
Health diagnosing occupations
Health assessment and treating occupations
Teachers, college and university
Teachers, except college and university
Counselors, educational and vocational
Librarians, archivists, and curators
Social scientists and urban planners
Social, recreation, and religious workers
Lawyers and judges
Nonpublic
Engineers, architects, and surveyors
Mathematical and computer scientists
Natural scientists
Writers, artists, entertainers, and athletes
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