Hi derekwtp!
Here are the guidelines to answer your problems, and the solutions in
case you had any trouble. Please remember that this question will not
be closed until you're fully satisfied with it. You may use the
Request for Clarification feature if you feel there's something still
unclear.
1. When the firms form a cartel, this cartel acts like a monopolist.
That is, the cartel doesn't take the price as given (as in a perfectly
competitive market); rather it chooses a price-quantity combination
(which depends on the demand curve and its cost function) such that
the profit is maximized. There is, however, a 'twist' on this problem
with respect to the monopolist's problem: in this case the cartel must
also choose the quantity that each of the firms has to produce in
order to maximize profits. So the problem here is to choose the
quantities Q1 (quantity of Cokes) and Q2 (Pepsis) such that the
following function is maximized:
Max P(Q1+Q2)*(Q1+Q2) - C(Q1) - C(Q2)
Q1, Q2
where C(Q1) and C(Q2) are the cost functions of each of the firms, and
P(Q1+Q2) is the inverse demand function; which tells, given that the
market quantity is Q1+Q2, what is the price consumers are willing to
pay for this quantity. Clearly, this function represents the combined
profits of both firms. We already know C(Q1) and C(Q2). We find P(Q)
by isolating P in the demand function:
Q = 119 - 0.5P
0.5P = 119 - Q
P = 238 - 2Q
So P(Q1+Q2) is:
P = 238 - 2*(Q1+Q2)
So the function we must maximize is:
Max [238-2(Q1+Q2)]*(Q1+Q2) - 3Q1^2-48Q1-572 - 6Q2^2-18Q2-849
Q1, Q2
In order to solve this, we must the first order conditions: we must
find the derivative of this function with respect to Q1 and equate it
to 0, and then do the same derivating with respect to Q2. With this,
we'll have two equations with two unknowns (Q1 and Q2). These will be
the quantities that each firm must produce (the answer to question b).
The sum of these quantities, and the price that comes from plugging
this sum into the inverse demand function is the answer to question a.
Finally, plugging these quantities into the above function will give
you the cartel's profits; and calculating the income of each firm
(quantity of each firm times market price) minus each firm's cost will
give you the profits earned by each firm. This constitutes the answer
to c.
The solutions to this problem and the following ones are given at the
end of the whole answer, so that you can try to solve it yourself
before seeing the solutions.
2. This problem is a bit simpler than the previous one. The firm must
choose its output (Q) such that profits are maximized. So the firm
must maximize the following function:
Max P(Q)*Q - C(Q)
Q
We already know P(Q) and C(Q). The function becomes:
Max (475-11Q)*Q - Q^3 + 21Q^2 - 500Q
Q
Again, in order to find the optimal Q, we calculate the first order
condition. We must find the derivative of this function with respect
to Q and equate it to zero. From this equation we should be able to
calculate the optimal Q. Finally, when plugging Q into the inverse
demand function, we find the market price. This is the answer to a.
In order to find the solution to b, we must find the minimum point of
the LAC curve. Since this is a convex curve, we can again use the
procedure of derivating and equating to zero, and in this case we will
have found the minimum point of the curve. We know that the LAC curve
is:
LAC = 500 - 21Q + Q^2
Therefore, we must take the derivative of this function with respect
to Q, equato it to 0 and isolate Q from there. This will be the
quantity that minimizes the long-run average cost. Finally, in order
to find the market price at this quantity, we just plug this Q into
the inverse demand function.
3. There seems to be an error in the statement of this question,
because the same text is both in the main statement and in question c.
I'll assume then that question 'a' asks you to find the equilibrium
wage and labor when there is no tax (or tax is equal to zero). If my
interpretation is wrong, please let me know through a clarification
request.
Answering questions a and b is a matter of solving a system of 2
equations and 2 unknowns, where the 2 unknowns are the equilibrium W
and L, and the 2 equations are the demand and supply curves. There is
a further complication to this problem with the inreoduction of the
tax, but as we shall see it will not be difficult to solve anyway. I
will shows you here the equation you must solve as a function of a tax
"t". This equation will be good for both question b (by setting
t=0.25) and question a (by setting t=0).
Let's define W to be "wage paid by the firms", which is not
necessarily equal to the wage that workers receive (because there may
be an income tax). Of course, when firms decide how much labor to
hire, they only care about the wage they pay and not about how much of
it the workers actually receive. Therefore, keeping in mind that W is
"wage paid by the firms", we have that the equation of demand for
labor is simply
L = 1000 - 100W
Let's get now the workers' side. When deciding how much labor to
offer, a worker does care about the wage he will receive. If the firm
pays a wage W, the worker will receive a wage equal to (1-t)W.
Therefore, again recalling that W is "wage paid by the firms", we have
that labor supply as a function of W is:
L = -400 + 100(1-t)W
Now, in order to find the equilibrium wage and labor, we just have to
solve this system of two equations and two unknowns:
L = 1000 - 100W
L = -400 + 100(1-t)W
We will have the answer as a function of t. So when we set t=0 we'll
have the answer to a, and when we set t=0.25 we will have the answer
to b. Recall that here we will find W, which is the wage paid by the
firms. This is not important for question a, because the wage paid by
the firms is equal to the wage received by the workers, but it is
important for question b. After finding W, just multiply it by (1-t)
in order to find the wage received by workers.
Now let's see how to answer question c. It's easy to find the tax
revenue of the government. After we find W and L, the tax revenue is
simply t*W*L.
The deadweight loss can be computed in many ways. For more complex
demand and supply functions, finding the deadweight loss requires ths
use of integrals. However, since in this case the demand and supply
functions are lines, we can compute this loss by calculating areas of
triangles. First of all, you can see graphically which is the
deadweight loss in the following Powerpoint presentation.
The costs of taxation
http://cc.kangwon.ac.kr/~kimoon/pr/mankiw/Ch08.ppt
You can see a graphic of this problem I made for you at the next link.
The graph is not necessarily to scale (although the fact that the
supply line "starts" at wage 4 and the demand line "starts" at wage 10
is accurate - you can check this by setting L=0 in the supply and
demand functions and isolating w).
http://www.angelfire.com/alt/elmarto
(if you have trouble viewing it, try right-clicking on the link that
you'll found in that page and choosing Save Target as...)
The deadweight loss can be found by taking the are of the "big"
triangle (the one that encompasses the producer and consumer surplus,
the tax revenue and the deadweight loss) and then substracting the
areas of the producer and consumer surplus (which are triangles) and
the area of the tax revenue (which is t*W). Clearly, in order to do
this we must first compute the answers to a and b, in order to know
the equilibrium labor with no taxes (which would be the height of the
"big" triangle), the equilibrium L with taxes which is the "base" of
the small triangles, and the equilbrium W with taxes, which defines
the the height of these triangles.
SOLUTIONS TO ALL QUESTIONS
We have to choose Q1 and Q2 in order to maximize
Max [238-2(Q1+Q2)]*(Q1+Q2) - 3Q1^2-48Q1-572 - 6Q2^2-18Q2-849
Q1,Q2
The derivative of this function with respect to Q1 is:
-2(Q1+Q2) + [238-2(Q1+Q2)] - 6Q1 - 48 = 0
238 - 4(Q1+Q2) - 6Q1 - 48 = 0
238 - 10Q1 - 4Q2 -48 = 0
190 - 10Q1 - 4Q2 = 0
The derivative of the function with respect to Q2 is:
-2(Q1+Q2) + [238-2(Q1+Q2)] - 12Q2 - 18 = 0
238 - 4(Q1+Q2) - 12Q2 - 18 = 0
238 - 4Q1 - 16Q2 - 18 = 0
220 - 4Q1 - 16Q2 = 0
Therefore, we have the system:
190 - 10Q1 - 4Q2 = 0
220 - 4Q1 - 16Q2 = 0
That is a system of 2 equations with 2 unknowns which can be easily
solved. The solution is then:
Q1 = 15
Q2 = 10
So Coke produces 15 units and Pepsi produces 10. The cartel as a whole
produces 25 units. The price that is paid for these units is
P = 238 - 2Q
P = 238 - 2*25
P = 188
The profits of Coke are:
Profits = Price*Quantity - Costs
= 188*15 - 3*15^2 - 48*15 - 572
= 853
The profits of Pepsi are calculated in a similar fashion. Pepsi's
profts are 251. The cartel's profits are 853+251=1104
2a. We must find Q that maximizes
Max (475-11Q)*Q - Q^3 + 21Q^2 - 500Q
Q
The derivative of this function with respect to Q is:
-11Q + (475-11Q) - 3Q^2 + 42Q - 500 = 0
475 + 20Q - 3Q^2 - 500 = 0
-3Q^2 + 20Q - 25 = 0
This is a quadratic equation with two solutions: 5 and 5/3. Let's see
which of these two points yield the highest profits. If we plug Q=5 in
the original function:
(475-11Q)*Q - Q^3 + 21Q^2 - 500Q)
(475-55)*5 - 5^3 + 21*5^2 - 500*5
=0
If we plug 5/3 instead, we get -18.51
Therefore, profits are maximized (and are equal to zero) when Q=5. The
market price that is paid for 5 units is
P = 475 - 11Q
= 475 - 55
= 420
2b. The LAC curve is:
LAC = 500 -21q + q^2
When we take the derivative of this curve and equate it to zero, we get:
-21 + 2Q = 0
Q = 21/2
= 10.5
And the price paid for this quantity is:
475 - 11*10.5 = 359.5
3a and b. The system we have to solve is
L = 1000 - 100W
L = -400 + 100(1-t)W
Solving this simple system, we find that:
W = 1400/[100+100(1-t)]
L = 1000 - 100*1400/[100+100(1-t)]
So for question a, when t=0, we have that:
W = 7
L = 300
For question b, when t=0.25, we get:
W = 8
L = 200
Here W means "wage paid by the firms". Therefore, the wage received by
the workers is (1-0.25)*8 = 6
3c. Tax revenue is simply 0.25*8*200 = 400. Let's compute the
deadweight loss step by step.
Area of "big" triangle = 6*300/2 = 900
Tax revenue = 400
In order to calculate the consumer surplus, notice that the height of
this triangle is (10-8)=2. The base of this triangle is 200.
Therefore, the are is 200*2/2=200. The area of the consumer surplus
can be calculated in exactly the same fashion and is also 200.
Therefore, the deadweight loss amounts to:
900 - 400 - 200 - 200 = 100
Google search strategy
deadweight loss
://www.google.com/search?sourceid=navclient&ie=UTF-8&oe=UTF-8&q=deadweight+loss
I hope this helps! Again, if you have any doubts, please don't
hesitate to request a clarification before rating my answer. Otherwise
I await your rating and final comments.
Best wishes and best of luck in your final!
elmarto |
