Hello,
Okay, let's start at the start here and see if we can't figure this
out.
First, let's lay our our variables (I know you've done this already,
but I find it helpful to do it again).
Lw is the amount of white labour
Lb is the amount of black labour
Ww is the white wage
Wb is the black wage
p is your output price and is equal to $100 per unit
q is the amount produced, and the production function is q = 10 *
sqrt( Lw + Lb)
Marginal product of labour MPL = 5 / ( sqrt( Lw + Lb) )
Most of these make sense intuitively, but MPL might not. MPL is the
change in the firm's output resulting from using additional worker.
This is a diminishing value (L is in the denominator, so as L
increases MPL decreases) because as you add more labour the increase
in output of the firm is going to fall. A factory with only one
worker could probably increase its output quite significantly by
adding another worker, but a factory with a thousand workers probably
won't see nearly the same degree of increase.
The first question is how many black and white workers would a
non-descriminatory firm hire? You're right that the firm is going to
hire the black workers because they are the cheapest. Since black and
white workers have the same level of productivity, the firm will hire
based on the following decision rule:
If Wb < Ww then hire all black
If Wb > Ww then hire all white
So basically hire whoever is cheaper.
To figure out how much to hire, we need to consider VMPL. This is
simply the increase in revenue generated by hiring an additional
worker. So this is pretty obviously related to MPL. The relationship
is just this:
VMPL = p * MPL
So in this case since we know MPL and p, VMPL = 100 * ( 5 / ( sqrt( Lw
+ Lb ) ) )
You probably remember from earlier studies that a firm outputs until
marginal revenue is equal to marginal cost (MR = MC). Same idea
applies here. The firm hires more labour as long as hiring another
unit of labour will generate more revenue than it costs. It stops
hiring when the wage is equal to the revenue generated, which is just
this equation:
W = VMPL
We know the firm is hiring only black workers, so W = Wb. We also
know then that Lw (the amount of white labour) is going to equal zero,
since the firm isn't hiring any white workers. Now we can work the
equation out.
W = VMPL
Substitution in Wb for W and breaking VMPL into its constituent parts
yield:
Wb = p * 5 / sqrt( Lb ) remember that Lw = 0, so Lb+Lw can be
written as just Lb
Filling in the variables we already know:
10 = 100 * 5 / sqrt( Lb )
10 = 500 / sqrt( Lb )
10 / 500 = 1 / sqrt( Lb )
500 / 10 = sqrt ( Lb )
50 = sqrt( Lb )
square(50) = Lb
2500 = Lb
So the amount of black labour is 2500 units.
The next question is about profits of the non-descriminating firm.
Profit as you know is just revenue minus costs. Now this firm has no
capital costs, only labour, so profit can be written as:
Profit = p*q - W*L
Revenue is price p times output q, and costs are the wage w (either Wb
or Wl) times the amount of labour L (either Lb or Lw). So let's call
profit P, and figure it out:
Remember that q = 10 * sqrt( Lw + Lb) or really just 10 * sqrt( L )
so q = 10 * sqrt( 2500 )
q = 10 * 50 = 500
P = p*q - W*L
P = (100 * 500) - (10 * 2500)
P = 50,000 - 25,000
P = 25,000
Question 3 introduces the notion of discrimination. The
discrimination coefficient is simply the premium the employer adds to
the wage rate of the group in quesion. In this case the employer
values black workers lower than white workers. The discrimination
coefficient (let's call it d) is 0.3, which means that even though the
black worker's wage Wb is $10 the employer acts as if it's Wb*(1 + d)
or Wb * 1.3, which would be $13.
Returning to our decision rule about who to hire we cas see that even
with a discrimination coefficient of 0.3 Wb < Ww. So the firm will
still hire an all black workforce. In this case the profit function
will remain the same because even though the firm discrinates against
black workers and hires them _as if_ they had a wage of $13, their
wage is really only $10, so L is still 2500, Wb is still $10, and
profit P is still $25,000.
In the last question things change a bit. Now the discrimination
coefficient is 0.7, so the firm acts as if the black wage is Wb * (1 +
0.7) which would be $17. Now the decision rule kicks in and Wb > Ww
so the firm hires all white workers. But the wage for white workers
is $15, so we need to recompute everything:
Ww = VMPL
15 = VMPL
500 / 15 = sqrt( L )
L = 1111.11...
Now we also need to figure out q again:
q = 10 * sqrt( L )
q = 333.33...
So profit is:
P = p*q - W*L
P = (100 * 333.33) - (15 * 1111.11)
P = 33,333 - 16,666
P = 16,667
So as you can see, discriminating against the black workers raised the
labour cost of the firm, so decreased the amount of labour it could
hire. This in turn decreased the output of the firm. And put it all
together and the discrimatory hiring cost the firm profits.
Those should answer your homework questions, but if you're still
unclear on this you may want to check out the lecture notes of Prof J.
Kluve of Berkely which can be found at:
http://emlab.berkeley.edu/users/webfac/kluve/e151_su02/econ151lecturenotes12.pdf
I hope this was helpful. |
