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4.1 Models of discrimination
In Chapter 4 I test one economic model of labor market discrimination,
the occupational segregation model, formalized by Bergmann.1 Berg-
mann™s model assumes constraints on which workers can be hired for
which jobs. Suppose there is a production function,
Y ¼ f °E1 ; E2 ; K Þ

where E1 and E2 are two different jobs. Occupational segregation occurs
in the form of constraints. If M is the number of male workers and F is
the number of female workers, the constraints are:
E1 ¼ M E2 ¼ F
Females cannot be hired for the male job and males cannot be hired for
the female job. The occupational segregation model has implications for
wages if women are con¬ned to a small number of jobs. Because of the
diminishing marginal product of labor, con¬ning women to a small
number of jobs results in ˜˜overcrowding™™ and reduces their marginal
product. I test for the presence of discrimination in this form by exam-
ining whether employers were willing to substitute men and women
workers. If there were rigid employment constraints, that would imply
that men and women were not substitutable. If employers did in fact
substitute men for women, and vice versa, such a practice implies that
they were not completely limited by occupational segregation constraints.
Discrimination may also exist in the form of wage discrimination; in
this case employers are willing to hire women but pay them less than
their marginal product. Because wage discrimination and occupational
segregation can operate independently, showing that the labor market


1
Bergmann, ˜˜The Effect on White Incomes of Discrimination,™™ pp. 294“313. Bergmann
described occupational segregation between blacks and whites; I have applied the same
model to segregation between men and women.

342
Appendix to Chapter 4 343

was perfectly competitive requires showing that neither form of dis-
crimination prevailed. The labor market may be characterized by one
form and not the other. Women may be allowed to enter any occupation
but be paid less than their marginal product, so that the market is
characterized by wage discrimination but not occupational segregation.
Alternatively, women may be paid their marginal product but be con-
¬ned to certain occupations.
The wage discrimination model, also called taste discrimination, was
set forth by Gary Becker.2 In Becker™s employer-discrimination model,
the employer has a taste for discrimination in his or her utility function,
which leads the employer to offer a lower wage to the group discrim-
inated against. The employer™s utility function might look like this:
U ¼ p°M ; F Þ À dF

where M is the number of men hired, F is the number of women, p is the
¬rm™s pro¬t, and d is the taste for discrimination against women. Utility
maximization implies:
p2 À d ¼ wf ;

or the wage of a woman worker will equal her marginal product less the
employer™s taste for discrimination. The employer will only hire a woman
if he is compensated for the disutility of doing so by a suf¬ciently low
wage. Under Becker™s wage discrimination model, the employer is willing
to substitute men and women. Thus a test for substitution between men
and women cannot detect the presence of wage discrimination.
Tests of wage discrimination suggest that women were paid their
marginal product. For example, Cox and Nye ¬nd that women™s wages
were equal to their marginal productivity in French industry.3 Such a
test, however, cannot detect occupational segregation constraints.
Occupational segregation lowers women™s wages precisely by lowering
their marginal product. The statistical test described in the next section
adds to our knowledge of discrimination by testing for occupational
segregation.

4.2 Cross-price elasticity as a test for gender segregation
While we can easily observe that men and women worked in different
occupations, the reason that they did so is more dif¬cult to determine.

2
Becker, The Economics of Discrimination.
3
See Chapter 2 for a more detailed discussion of the evidence on relative productivity.
344 Gender, Work and Wages in Industrial Revolution Britain

Noting that men and women worked at different jobs will not tell us
whether the market was characterized by occupational segregation. We
need a test that can distinguish whether men and women were hired in
two segregated labor markets, or whether there was one uni¬ed labor
market in which men and women competed against each other. One
method to distinguish between segregated and competitive markets is to
test whether men and women were substitutes. If the labor market was
truly segregated by gender, then the wage of one sex would not affect the
employment of the other sex. If the results indicate that men and women
were substitutes, then this implies that they competed against each other
in a single labor market.
Substitutability would be indicated by a positive cross-price elasti-
city, or:
dxf wm
>0
dwm xf

which tells us that the employment of women increased in response to
an increase in the wages of men, and, similarly,
dxm wf
>0:
dwf xm

While both elasticities must have the same sign, they will in general have
different magnitudes, depending on the relative factor shares. In a two-
factor model, the cross-price elasticity can be expressed as:
dx1 w2 w2 x2
¼ ° r À gÞ
dw2 x1 °w1 x1 þ w2 x2 Þ

where r is the elasticity of substitution and g is the elasticity of demand
for the output.4 Since r and g are the same for both cross-price elasti-
cities, the input that has the larger factor share will have the smaller
cross-price elasticity.

Within-task versus across-task substitution
To establish that gender constraints on occupations were not present, I
need to show that men and women were substitutable within tasks rather
than merely across tasks. Within-task substitution exists if employers are
willing to hire either men or women for any job, i.e., there are no

4
R. G. D. Allen, Mathematical Analysis for Economists (New York: St. Martin™s Press,
[1938] 1964), p. 373. However, since male and female labor were clearly not the only
factors, this is only an approximation.
Appendix to Chapter 4 345

constraints on which sex can be hired for any task. Across-task substi-
tution exists if ˜˜women™s jobs™™ and ˜˜men™s jobs™™ are substitutable in
production, and may or may not be coupled with gender constraints on
occupations. For example, if a farmer was willing to increase the size of
his or her dairy herd when women™s wages were lower, but was not
willing to hire women for any job except dairying, then women and men
were substitutable across task but not within task. Across-task substi-
tution means that women™s wages will respond to men™s wages, but does
not imply equality of opportunity.
A speci¬c model will help to make the issues clearer. Suppose there
are two jobs on the farm, and employment levels in these jobs are des-
ignated L1 and L2. The farmer can hire two factors, M (male labor) and
F (female labor). The production function is nested, with

y ¼ f °L1 ; L2 Þ

and
Li ¼ Mi þ ui Fi :

The parameter u measures the relative productivity of female workers. If
u ¼ 0.5, then each female worker can do half as much work as a male
worker, and if u ¼ 1 the sexes are equally productive.
I will de¬ne occupational segregation to be the constraints that assign
each sex to one of the tasks, i.e., the constraints M1 ¼ 0 and F2 ¼ 0. In
the absence of these constraints, the production function is:
y ¼ f °M1 þ u1 F1 ; M2 þ u2 F2 Þ;

and men and women are substitutable within task. If the constraints
hold, however, the production function reduces to
y ¼ f °u1 F1 ; M2 Þ;

and no within-task substitution is possible. Across-task substitution is
independent of the constraints, and exists if L1 and L2 are substitutes.
Across-task substitution is not suf¬cient to establish an unsegregated
market. Suppose the occupational constraints hold but men™s and
women™s jobs are perfectly substitutable. The employment of women
will expand in response to a decrease in the ratio wf /wm, but women can
still be hurt by the constraints, since they could be con¬ned to less
productive tasks. The production function in this case is:
y ¼ a u1 F1 þ bM2
346 Gender, Work and Wages in Industrial Revolution Britain

where a and b are constants indicating the relative productivity of each
task. If in addition to u1 < 1 we also have a < b, then female marginal
productivity will be lower due to the constraint, as well as due to their
natural disadvantages. In order to establish a non-discriminatory mar-
ket, I need to establish that men and women were substitutable within
tasks rather than merely across tasks.
Unfortunately, across-task substitution can potentially mask within-
task constraints. If the occupational constraints hold, we have:

y ¼ f °u1 F1 ; M2 Þ

while if there are no constraints:

y ¼ f °M1 þ u1 F1 ; M2 þ u2 F2 Þ:

If L1 and L2 are substitutable, men and women will be substitutes even if
the constraint is binding. Only if L1 and L2 are not substitutes can we be
con¬dent that a positive cross-price elasticity indicates the absence of
occupational constraints. For example, farmers might respond to an
increase in the male wage by switching from grain to dairy production,
and thus hiring more women because of a change in farm tasks. I want to
know whether farmers were willing to hire women instead of men for
grain production when male wages rose. To address this question, I need
to control for changes in the production tasks. For a given level of L1 and
L2, the cross-wage effects will be zero if the constraints hold and positive
if they do not. Controlling for across-task substitution, then, allows us to
use the cross-price elasticity to test for within-task substitution. If I can
control for the tasks done on the farm and still ¬nd that men and women
are substitutable, I can be con¬dent that the substitution is within-task
rather than across-task substitution.
Unfortunately, I cannot directly control for the levels of each pro-
duction task. In the regressions in Chapter 4, I use farm animals, which
are correlated with different types of output, to control for farm output.
While these controls will not eliminate all variation in tasks, they will
account for most of the variation. Dairy agriculture was quite different
from arable agriculture but, given the type of output, the production
tasks required were relatively ¬xed. An eighteenth-century farmer could
choose between different types of output, perhaps producing less butter
and more grain, and such a change would alter the production tasks
required. Given the choice of product, however, there was little scope for
substituting one task for another. More planting could not substitute
for less harvesting. Thus, controlling for the type of output will control
Appendix to Chapter 4 347

for most of the variation in production tasks. The remaining substitution
between men and women workers will be within-task substitution.

4.3 Wage correlation as a test for gender segregation
The model presented here will generate a testable conclusion that can,
under certain assumptions, distinguish between a segmented and a uni¬ed
labor market. I will model the hypothesis of discrimination in the form of
occupational segregation as a constraint in the employer™s maximization
problem that does not allow him or her to hire women for certain jobs. I will
assume the constraint, when imposed, is exogenous to the employer™s
problem, so that the employer simply maximizes pro¬ts. Note that this
differs from the taste discrimination model, which adds a discriminatory
taste factor to the employer™s utility function. If the no-women constraint is
not binding, no decisions are distorted, and wages are the same as what
they would be under perfect competition. If the no-women constraint is
binding, women™s jobs may become crowded, causing their wages to be
below those that would prevail under perfect competition.
I assume that employers cannot accurately measure individual prod-
uctivity strength and that they use sex as a signal. Individuals of the same
sex cannot be distinguished. While individuals may differ in product-
ivity, the employer cannot observe individual levels of productivity. In
tasks that require strength, women are less productive than men. I also
assume that the work is unskilled, so that individuals do not differ with
respect to human capital.
Assume that there are two ¬rms, each of which hires workers for a
single task. One of the tasks requires strength, and the other does not. At
¬rm A women are less productive than men, but at ¬rm B men and
women are equally productive. Firm A maximizes

p ¼ PA a ln°MA þ uFA Þ À wm MA À wf FA ;

where u re¬‚ects the fact that male and female workers have different
productivities, and is known by the employer but not by the researcher.
Since, as I have shown in Chapter 2, strength was an important com-
ponent of productivity in unskilled labor, I will assume that u < 1. Firm
B maximizes

p ¼ PB b ln°MB þ FB Þ À wm MB À wf FB :

Men and women are equally productive at ¬rm B, re¬‚ecting the fact that
some jobs in the economy did not require strength.
348 Gender, Work and Wages in Industrial Revolution Britain

If there is gender discrimination in the form of occupational segre-
gation, additional constraints are added:
FA ¼ 0; MB ¼ 0;

i.e., there is some arbitrary standard assigning men and women to cer-
tain jobs. These constraints represent the claim that social forces rather
than pro¬t maximization determined employment.
Women have a comparative advantage in working at ¬rm B, so we
would expect to ¬nd women specializing in that task. Given the pro-
duction functions above, perfect competition without discrimination will
result in segregation of workers by sex. The ¬rms, taking wages as
exogenous, will set

wf
MB ¼ 0 <1
if
wm
wf
MA ¼ 0 >u
if
wm
wf
FB ¼ 0 >1
if
wm
wf
FA ¼ 0 > u:
if
wm

Note that if FA > 0 and MB > 0, then one of the ¬rms is not maximizing,
since u < 1. In equilibrium, one or both of the ¬rms will hire only one
sex. Occupational sorting results from differences in strength, and thus
is not necessarily an indication of an exclusionary constraint. The model
also implies that if strength is a scarce factor in the sense that FA > 0,
then female wages will be less than male wages whether or not there is an
occupational constraint. Neither evidence of occupational sorting nor
evidence of lower female wages will tell us whether there is a constraint
on female employment.
I can combine these ¬rm maximization problems with labor supply
functions and solve for the market wage. If strength is a scarce factor
in the sense that a competitive market would have an equilibrium with
FA > 0, then we can detect the presence of the exclusionary constraint
by examining whether male and female wages are correlated. Male and
female wages will be correlated only in the absence of the constraint.
Let the labor supply functions be:

MA þ MB ¼ c0 þ c1 wm
FA þ FB ¼ d0 þ d1 wf :
Appendix to Chapter 4 349

Take the case where strength is scarce, so that MB = 0.5 In equilibrium:
q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
À°c0 þ ud0 Þ þ °c0 þ ud0 Þ2 þ 4°c1 þ u2 d1 Þ°PB b þ PA aÞ
wm ¼
2°c1 þ u2 d1 Þ

wf ¼ uwm :

This gives:
dwf @wf
dwm
¼ u @wA
>0 >0 m
dPA dPA @PA @P
dwf @wf
dwm
¼ u @wB
>0 >0 m
dPB dPB @PB @P
@wf dwf
@wm
¼ u dwm :
<0 <0
@do @do ddo ddo

As long as ¬rm A ¬nds it pro¬table to hire positive amounts of female
workers, the wage ratio must equal the productivity ratio. A shock to the
demand for either ¬rm™s product (a change in PA or PB), or a shock to
either of the labor supply functions (such as a change in d0), will move
both male and female wages in the same direction, so male and female
wages will be correlated.6
However, if ¬rm A is constrained to hire only males, male and female
wages will not be correlated. The male wage will solve

1
PA a ¼ wm
MA


MA ¼ co þ c1 wm ;

so
p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
2
Àco þ co þ 4c1 PA a
wm ¼ :
2c1


5
This requires that a, ¬rm A™s production function shifter, be large. An example of an
equilibrium with MB ¼ 0 is: a ¼ 170, b ¼ 30, u ¼ ½, MA ¼ 5 þ wm, FA þ FB ¼ 2wf. Then
wm ¼ 10, wf ¼ 5, MA ¼ 15, MB ¼ 0, FA ¼ 4, FB ¼ 6.
6
The model also implies a test for time-series data:

dwf wf
¼u¼ :
dwm wm
Such a test, however, would require the assumption that the technology did not change
over time, which would not be an appropriate assumption for the time period.
350 Gender, Work and Wages in Industrial Revolution Britain

Similarly,
p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
do2 þ 4d1 PB b
Àdo þ
wf ¼ :
2d1
In this case male and female wages will not be correlated:
dwf
dwm
>0 ¼0
dPA dPA
dwf
dwm
¼0 >0
dPB dPB
dwf
dwm
¼0 > 0:
ddo ddo

In other words, if the constraint binds, a shock to the system will alter
either the male wage or the female wage, but not both. If shocks are
random, wages will be uncorrelated. The difference in the predicted
outcome here, as opposed to the non-discriminatory case, allows me to
test for constraints by examining whether male and female wages are
correlated.
There is, however, a potential problem with this method. If shocks
(for example, PA and PB) are positively correlated, then the wage cor-
relation will be positive whether or not there is an occupational segre-
gation constraint. Only if the shocks are uncorrelated can the wage
correlation be used to distinguish between the two situations. This
limitation is potentially a serious problem and might prevent using wage
correlations as evidence of an integrated market if I could not correct for
location-speci¬c effects. Fortunately, I can correct for this problem
whenever I have multiple wages for the same location. In a cross-sec-
tional sample, wages at the same location may be correlated simply
because of price level or any other location-speci¬c effect. When I have
seasonal wages, I can difference the wages across the seasons to correct
for these ¬xed effects. I can then examine the correlation of changes
in wages across the seasons. In other words, I will measure whether
female wages are likely to rise more at harvest time in locations where
male wages rise more. Unfortunately, differencing wages increases the
attenuation effect of measurement error, which may bias my results.
However, since the bias is downward, I can be con¬dent about the
results if I ¬nd a strong positive correlation.
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Index




Abram, A., 279 Barber-Surgeons™ Company, 293
Agricultural laborers, 311 Bardsley, Sandy, 105
female, 56, 310À13 Barristers, 291
unionization, 253, 260À4 Barritt, Mary, 300
wives of, 54À6 Bateman, Fred, 81
see also Swing riots Bathgate, Janet, 302
Agriculture Becker, Gary, 242, 343
female self-employment, 54À6 Bedford
gender division of labor, 155À8 lace industry, 45, 46
hours of work, 94À5 straw industry, 48
regional patterns, 57 Beer, as portion of wage, 206
see also Allotments, Dairying, Enclosures, Ben-Amos, Ilana, 120
Farmers, Farm servants, Harvesting, Benjamin, Dwayne, 101, 326
Hoeing, Plowing Berg, Maxine, 122
Alexander, Sally, 172 Bergmann, Barbara, 136, 342
Allen, Robert, 114, 190, 192 see also Crowding
Allotments, 55 Best, Henry, 155
Anderson, Michael, 22À3 Biology, gender differences, 10À13
Apothecaries, 294À5, 297 see also Child bearing, Strength
Apprenticeship, 161, 234À5 Birmingham,
attorneys, 292 commercial directories, 31À2, 38
costs, 118 index of segregation, 316
gender differences, 117 Black, Anne, 249
handloom weaving, 255 Blackstone, William, 278, 279
impact on occupational segregation, Bobbinet machine, 46
120À1 see also Lace industry
of females, 235, 236À8 Bondager, 57, 97À8
parish, 117, 238 Bookbinders, 245
to females, 236 Borjas, George, 189
woolcombing, 269 Boserup, Ester, 155
Arch, Joseph, 262 Boyer, George, 270
Atack, Jeremy, 81 Bradley, L. Barbara, 249
Attorneys, 291À2 Brandt, Loren, 101, 326
Auctioneers, female, 67 Brenner, Johanna, 12, 172, 182
August, Andrew, 306 Brewing, 286À8
Austin, Alfred, 53, 72, 113, 179, Brokers, female, 67
232, 313 Browne, Kingsley, 10, 12, 13
Austin, Anthony, 92, 256 Brunt, Liam, 191
Authors, female, 70 Business owners, percent female, 307
Butchers, 316
Bailey, Martha, 334 Button industry, 50, 169
Bailey, Roy, 18, 270, 307 Bythell, Duncan, 188, 255


370
Index 371
Cadbury family, 309 Couverture, 278À9
Cadbury, George, 242 in US, 278
Calico printers, 264 Coventry
Campbell, R., 225, 285 commercial directories, 33, 309
Capital, index of segregation, 315
requirements for business, 281, 285 Cox, Donald, 102, 343
effects on women™s occupations, 281À8 Craig, Lee, 101, 102
Carver, Frederick, 4 Croppers, 264
Censuses, 18À25, 306 Crosley, Sarah, 299
1841, 7À8, 18, 61 Crowding, occupational, 136À7,
1851, 22, 56, 298 187, 199
measurement problems, 18À25 Cunningham, Hugh, 202
treatments of tradesmen™s wives, 307 Cunningham, James, 189
Charring, 59, 60

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