Hi again vitaminic!
OK, let's assume that both B1 and B2 are constants, although I've
never seen such model specification. Anyway, I'll tell you the method
you have to use in order to answer these questions, so that if B2
turns out to be multiplied by something (could be a typo in the book)
you'll be able to redo the questions easily. So here are the asnwers:
a) These are the effects of the changes described in the questions for
each group: male, female, black, white.
Male:
===========
We know that, for males, G=0. Therefore, the equation for the
log-wages received by males becomes:
W=B1+B2+B3*S*D+B4*E+B5*E^2+B7*R (1)
(all the other terms are multiplied by G, which for males takes the
value of zero).
- What's the effect of an extra year of experience for workers with 7
years of experience?
What we have to find here is basically the derivative of the log-wage
(W) with respect to experience (E). Recall that the idea of a
derivative is to find the response of a function to a change of one
unit in one of its arguments. Here we have to find the response of the
log-wage to a change in E. The derivative of (1) with respect to E is:
B4 + B5*2*E
(I'm assuming here that you have knowledge of calculus and thus know
how to compute these derivatives. If you do not, please let me know
through a clarification request)
So, as you see, the effect depends on the actual number of years of
experience. Therefore, for a worker with seven years of experience,
the effect of an extra year is:
B4 + B5*2*7 = B4 + B5*14
An extra year of experience adds B4 + B5*14 to the log-wage, for a
male with 7 years of experience.
- When (in years of experience) are the returns to experience the
highest?
Depending on the values of the coefficients, there may or may not be a
"highest" return to experience. For example, if both B4 and B5 are
positive, you can see that B4*E+B5*E^2 (the part of the log-wage
explained by experience) is increasing in E; so as experience
increases, returns to experience also do. The most usual case is when
B5 is negative; so there is a maximum. In this case, the maximum is
attained when the derivative of that expression is 0, so:
B4+B5*2*E = 0
ans solving for E gives:
E = -B4/(B5*2)
Despite the negative symbol, this E is positive, because we assumed
that B5 was negative. So that's the expression for the years of
experience where the returns to experience is highest.
- What's the effect of an extra year of schooling for college
graduates?
Recall the equation for males:
W=B1+B2+B3*S*D+B4*E+B5*E^2+B7*R (1)
Since for college graduates D=1, this equation becomes:
W=B1+B2+B3*S+B4*E+B5*E^2+B7*R (2)
Now, taking the derivative of W with respect to S (years of schooling)
gives B3. Therefore, an extra year of schooling for college graduates
increase the log-wage by an amount of B3.
- What's the effect of attending colleage?
In this case, we can't take the derivative with respect to D, because
D is a discrete-valued variables (it is either 0 or 1). In order to
answer this question, we compare the wage equations for people who are
college graduates and people who are not:
Graduate: W=B1+B2+B3*S+B4*E+B5*E^2+B7*R
Non-grad: W=B1+B2+B4*E+B5*E^2+B7*R
The difference is in B3*S; therefore, being a college graduate adds
B3*S to the log-wage (notice that the amount by which the log-wage
increases for going to college depends on the number of years of
schooling).
Female
========
The procedure to obtain the effects of the changes described in the
question are exactly the same as for males (and the same for black or
white people), so I will not write here all the details. I'll also
leave the groups black and white undone; but again, the procedure is
exactly the same. If you have any question concerning any of these
groups, please request a clarification, I'll be more than happy to
explain.
For females, since G=1, the equation becomes:
W=B1+B2+B3*S*D+B4*E+B5*E^2+B6+B7*R+B8*S+B9*S^2+B10*E+B11*E^2
- An extra year of experience for workers with 7 years of experience.
The derivative with respect to E is:
B4+B5*2*E+B10+B11*2*E
=(B4+B10) + 2*(B5+B11)*E
So that's the effect of an extra year of experience for women
- When (in years of experience) are the returns to experience the
highest?
Using the same procedure as before, the maximum (if it exists) is
attained at:
E = -(B4+B10)/2*(B5+B11)
- An extra year of schooling for college graduates.
The equation becomes:
W=B1+B2+B3*S+B4*E+B5*E^2+B6+B7*R+B8*S+B9*S^2+B10*E+B11*E^2
and the derivative with respect to S is:
B3+B9*2*S
Notice that the model allows the effect of an extra year of schooling
for women to depend on S (for males it was only B3)
- Attending college
Graduate: W=B1+B2+B3*S+B4*E+B5*E^2+B6+B7*R+B8*S+B9*S^2+B10*E+B11*E^2
Non-grad: W=B1+B2+B4*E+B5*E^2+B6+B7*R+B8*S+B9*S^2+B10*E+B11*E^2
Thus the effect is again B3*S
As I said before, the effects for black and white people are
calculated in the same fashion. Please request a clarification if you
have any doubt regarding their computation.
B) Hypothesis testing:
"There is gender discrimination in the labor market."
In order to check if there is gender discrimination in the labor
market, we should test wether the coefficients B6, B8, B9, B10, B11
are all equal to 0. Why is this so? Compare the equation for men (G=0)
and women (G=1):
Men: W=B1+B2+B3*S*D+B4*E+B5*E^2+B7*R
Women: W=B1+B2+B3*S*D+B4*E+B5*E^2+B6+B7*R+B8*S+B9*S^2+B10*E+B11*E^2
If there weren't gender discrimination in the labor market, we should
see that the equation for wages is exactly the same for men and women.
That is, people with the same characteristics (schooling, experience,
etc) should earn the same wage regardless of gender. As you can see
from both equations, they are equal only when B6, B8, B9, B10 and B11
are not statistically different from zero. If any of these is
significatively smaller than zero, then there is discrimination
against women; because for a man and woman that have exactly the same
characteristics, the log-wage of the woman would be smaller.
"There is race discrimination in the labor market."
The procedure is the same as before. Let's compare the log-wage
equation for white people (R=0) and black people (R=1):
Wh: W=B1+B2+B3*S*D+B4*E+B5*E^2+B6*G+B8*G*S+B9*G*S^2+B10*G*E+B11*G*E^2
Bl: W=B1+B2+B3*S*D+B4*E+B5*E^2+B6*G+B7+B8*G*S+B9*G*S^2+B10*G*E+B11*G*E^2
The only difference is the coefficient B7. Therefore, to test wether
there is race discrimination in the labor market, we should test the
hypothesis that B7 is not statistically different from 0. If it's not
different from zero, then there is no race discrimination in the labor
market. If B7 is negative, then there is race discrimination against
black people; again because for a white and a black person with the
same characteristics, the white worker would have a higher wage.
- "Women's returns to schooling are lower than men's returns to
schooling."
In order to see this, we have to examine first the part of the
log-wage that is explained by years of schooling, for men (G=0) and
women (G=1). These are the coefficients that are multiplied by S.
Men: B3*S*D
Women: B3*S*D+B8*S+B9*S^2
Clearly, returns to schooling are equal for men and women only when B8
and B9 are not statistically different from zero. Women's returns to
schooling are lower when B8 and B9 are significatively less than zero.
This can also be seen by taking the derivative of the equation for men
and women with respect to S (we would be measuring the effect of one
extra year of schooling for men and for women). This gives:
Men: B3*D
Women: B3*D+B8+B9*2*S
As you can see, we arrive to the same conclusion. We have to check
that B8 and B9 are less than zero.
- "Women's returns to experience are lower than men's returns to
experience."
This question can be answered in the same fashion. The part of wages
that is explained by experience (E) for men and women is:
Men: B4*E+B5*E^2
Women: B4*E+B5*E^2+B10*E+B11*E^2
Using the same reasoning as above, we would have to check here that
B10 and B11 are less than zero.
I hope this helps! If you have any doubts regarding my answer, please
request a clarification before rating it. Otherwise I await your
rating and final comments.
Best wishes!
elmarto |
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