Elements of Economic Analysis-2
Econ 20100
Fall 2015
Problem Set 3 Solutions
Sebastien Gay
Louis Serranito
Yu Ushioda
Question 1 - Perfect Competition in the Short Run
In the market for widgets the aggregate demand is given by:
Q = 300 − 4p
To produce widgets firms need to combine capital (K) and labor (L) using a Cobb Douglas technology with the
following form:
1
1
q = K 2 L2
The cost of 1 unit of capital or labor is $10.
a) Calculate the long run cost function.
Solution:
The cost minimization problem is:
min
K,L
wK K + wL L
st.
1
1
q = K 2 L2
After writing the langrangean and taking the FOC (assuming the SOC are verified) you would easily see that
the factor demands are given by:
K
LR
LR
L
(wK , wL , q)
(wK , wL , q)
=
=
wL
wK
12
wK
wL
12
q
q
And then the cost function would be given by:
1
C LR (wK , wL , q) = 2q (wK wL ) 2
Obviously you do need to show us the algebra in your PS as every answer must have a good justification.
1
b) Suppose that there is free entry and exit in this market and that there are 22 firms that produce the same
quantity. Calculate the long run market equilibrium1 .
Solution:
From the profit maximization problem and after substituting the values of the factor prices you would easily see
that the individual and aggregate supply is given by2 :


, P > 20
∞
LR
Q (P ) = [0, ∞[ , P = 20


0
, P < 20
Representing this curve and the demand curve we have the following graph:
Figure 1: Supply and demand in the long run
Then you could easily see that the market equilibrium is given by P = 20 and Q = 220. With 22 firms (that
produce exactly the same quantities) we have q = 10 and as P = AC we have π = 0. Obviously we expect your
written answer to be a bit more elaborate than this. For example start with the individual firm, describe its supply,
then discuss the aggregate supply on the market, then intersect aggregate supply and demand to find the equilibrium
market price and aggregate quantity, finally use these results to calculate all the small variables associated with the
individual firm.
c) Now suppose that in the short run capital cannot be changed i.e. we are fixing the capital level at the level
that would have been optimally chosen in question (b). What is the short run cost function?
Solution:
First of all note that the level of capital will be fixed at K = 10, (use the long run demand for capital, the factor
prices and the equilibrium quantity that individual firms produce).
Then the cost minimization problem is:
min
L
wK K + wL L
st.
1
1
q = K 2 L2
1 Note that when we ask for the equilibrium in exercises you must provide the equilibrium price and quantity in addition to the
profits of each firm and the number of firms in the market (although in this exercise we already specify the amount of firms).
1
2 Once again I want to bring it to your attention that as the profit function is π = P q − 2q (w w ) 2 you want to just take the
K L
derivative with respect to q
∂π
∂q
1
= P − 2 (wK wL ) 2
and when you see that it is a constant (doesn’t depend on the value of q) then
you should take your conclusions by seeing what happens when this derivative is positive or negative or zero. Our first instinct in
optimization problems is to automatically take the derivative and equate it to zero which is not exactly the correct approach when we
have this case.
2
1
and the short run cost function is:
From this we have that L wK , wL , q, K = q 2 K
1
C SR wK , wL , q, K = wK K + wL q 2
K
Using the values we have that:
C SR (q) = 100 + q 2
d) Using a simple and clear graph where you depict the AC, AVC, AFC and MC in the short run, show what
the individual supply curve will look like. Where would the firm choose not to produce? Where would the firm
produce and incur a loss? Where would the firm have profits?
Solution:
Rendering the actual curves for this problem in Mathematica we have the following graph:
Figure 2: Cost Curves
From the profit maximization problem you know that, forgetting about the average variable cost for now, the
firm will limit itself to producing optimally by setting P = M C as its’ supply curve3 .
Now note that here we are in the short run and have already acquired and paid for K making it a sunk cost so
the firm is not thinking of whether he will exit market but rather of whether he produces or not (in other words
the fixed costs are gone and he cannot do anything about that). The firm will choose to not produce any quantity
whenever P < AV C or more specifically when M C < AV C given the condition that gives us our supply curve.
Whenever P < AV C the firm’s revenues cannot even cover the wages for workers that work that day so there is no
point in producing. In this problem the marginal cost curve is always above the average variable cost so the firm
fill always produce (except for the obvious case of P = 0) and the supply curve will be the full M C curve.
The firm will have a loss whenever P < AC, this is an obvious condition that comes from the profit function4 .
Thus the firm will have a loss on the part of the M C curve that is below the average cost curve.
Finally, the firm will have a profit when P > AC, which will happen when the firm is producing on the part of
the marginal curve that is above the average cost curve.
3 Careful, there have been cases in the past where I’ve seen students set q = M C... I believe that is due to the fact that we say that
"the supply curve is given by the marginal cost curve". P = M C (q) is the condition that gives us the supply curve and q is implicitly
defined in that expression. Thus all the (q, P ) points that respect P = M C (q) is our supply curve.
4 You can easily see this by dividing the profit function of a firm by q:
P q − C (q)
C (q)
P−
q
P
<
0
<
0
<
C (q)
= AC (q)
q
3
e) An economics specialist said that the AC in the short run cannot be lower then the AC in the long run for
any level of production. Elaborate on his comment using at most 3 lines5 .
Solution:
Quite simply, the long run AC is the lower envelope of the short run AC curves. So every point on the long run
AC is actually a point on some short run AC curve that uses the amount of K that minimizes the costs for that
quantity.
f) Give an expression for the individual supply curve and calculate the market equilibrium if demand suddenly
shifted to Q = 400 − 4p.
Solution:
As we saw in (d) the supply will be given by P = M C = 2q when P ≥ AV C = q. As 2q ≥ q for all q ≥ 0 then
the short term supply curve is given by:
P
q (P ) =
2
With 22 firms the aggregate supply is given by:
Q (P ) = 11P
Representing this graphically we have:
Figure 3: Supply and Demand
880
With this the market equilibrium is given by P = 80
3 and Q = 3 . Each firm produces q =
are given by:
2 !
80 40
40
700 ∼
π=
− 100 +
=
= 77.778
3 3
3
9
40
3
and their profits
g) Suppose that demand did not shift and that we are still in the short run but the government decided that a
higher level of consumption of widgets would benefit the economy. Therefore they decided to provide a subsidy of
10% over the sales price. Calculate the new equilibrium.
5 It
would be an even more interesting question if we told you to draw the long term average cost function and the short term cost
function for several levels of K, but we will let you do this exercise by yourselves.
4
Solution:
Applying the subsidy on the supply curve we see that the new supply is given by:
121
P
10
q (P ) =
If you want to motivate this curve then let us start with the firm’s profit maximization problem where s.P.q is
the subsidy:
maxP.q + s.P.q − 100 − q 2
q
Or using our subsidy of 10%:
max1.1P.q − 100 − q 2
q
Solving the FOC assuming that the SOC are verified:
[q] : 1.1P − 2q = 0
Simplifying this expression we would obtain the individual firm’s supply function:
q (P ) =
11
P
20
Aggregating this over quantity for all 22 firms you would find that the supply is given by:
Q = 22.
11
121
P =
P
20
10
Thus graphically we have:
Figure 4: Supply and Demand with Subsidy
Where I represent the new supply curve with a dashed line... the reason is that it is not a completely new supply
curve but rather a representation of how the firms behave with the subsidy and this new curve will help us find the
new equilibrium but when it is time to see what is the full value that the firm receives you need to look at the blue
curve. Nonetheless the dotted curve and green demand curve will give us the price you will see on the market, the
rest of the money will be given to the firms by the government.
Calculating the equilibrium you would get:
Then you can easily see that Q =
36 300
161
QD
=
300 − 4P
=
P
=
and q =
1815
161
QS
121
P
10
3000
161
and the profits of the firms are:
3000 1815
π = 1.1 ×
.
− 100 −
161 161
5
1815
161
2
=
103 175 ∼
= 3.98
25 921
Note that question g and f are very different because question f uses an expanded demand while question g
returns to the original demand and question g has a subsidy while question f has no subsidy. Do not compare the
values of these two questions.
h) The tax burden from the previous measure was too high for the government’s budget, thus they decided to
open this market to trade widgets on the international market where the price of widgets is $10. What is the new
short run equilibrium?
Solution:
Given that consumers can buy on the international market at $10 and firms can sell there at that price then the
equilibrium price in the market is P = 10. Let’s see the problem graphically:
Figure 5: Supply and Demand with International Market
With this the consumers will buy 260 units (that we called XD ), firms will produce 110 units (that we called
XS ), in which each firm produces 5 units, and 140 units will be imported. Additionally, in the short run we will
see that the firm’s profit are given by:
π = 10 × 5 − 100 + 52 = −75
i) A politician stated that trading on the international market of widgets was affecting the economy negatively
and thus they should close the market to international trade. But to gain support from the consumers without
affecting the government’s budget he also defended that the government should set a price ceiling that maintained
the international price. What is the short run equilibrium? How did the producer surplus change? How did the
consumer surplus change? Is this equilibrium better?
Solution:
Graphically:
Figure 6: Supply and Demand with Price Ceiling
6
Here we maintain the price of 10 thus consumers still want to consume 260 units, firms only produce 110 units
and we cannot import any quantity... so in the end consumers can only consume 110 units (but we will not specify
how the goods are attributed to each consumer). Naturally we can see that firms will still have a loss 75. You
could also easily see that consumers are worse off, the quickest way would be to look at the surplus on the graph
and define that consumer have in each case:
Figure 7: Comparison of Surplus
In question h the consumers received areas A,B and C in surplus while in question i they only receive A and B.
Question 2 - Fixed Factors
Suppose Wendy decides to open a sandwich shop in Hyde Park. After extensive market research she finds out
that people in Hyde Park only want to eat chicken caesar wraps, all chicken caesar wraps are exactly the same,
people will happily go anywhere in Hyde Park to buy their chicken caesar wraps, each chicken caesar wrap costs $4
to manufacture (this include all the costs associated with manufacturing and selling them) and all shops can only
produce a maximum of 100 chicken caesar wraps.
a) What would her supply curve look like?
Solution:
Looking at this problem, Wendy only thinks that she has a constant marginal cost of $4 and she still doesn’t
know about the renting issues... Given this her profit maximization problem is given by:
max
CW
P.CW − 4.CW
st.
CW ≤ 100
But let us set aside the inequality (because we have only seen how to do optimization with equality constraints)
so that our problem is:
max
CW
P.CW − 4.CW
Now, assuming SOC are verified, if we take the derivative wrt CW we see that:
∂π
=P −4
∂CW
Once we reach this stage we see that we cannot equate this to zero. Why? Because from the perspective of
the firm P is a constant (in perfect competition). Obviously 4 is also a constant. So for Wendy the P − 4 is just
∂π
a number, you cannot equate that to zero. How do we solve this problem? Well... think about what ∂CW
means.
∂π
If ∂CW > 0 then the firm earns more money as we increase production so obviously Wendy should produce more
∂π
chicken wraps and keep on increasing its production while the derivative is positive. Similarly if ∂CW
< 0 then
7
Wendy decreases her profits as she increases production so she should decrease production to reduce her loses.
∂π
Finally if ∂CW
= 0 then changing the level of production has no effect on profits so Wendy can choose to increase
or decrease her production as much as she wants as long as the derivative is zero.
∂π
= P − 4 then: when P < 4 she wants to decrease
Adding up this whole discussion and using the fact that ∂CW
production as much as possible so she will choose to not produce anything (besides that, note that the firm would
also not produce because with P < AC = 4 the firm would exit the market); when P > 4 then she wants to produce
as much as possible to increase her profits and given the quantity constraint she will produce 100 chicken caesar
wraps; finally, when P = 4 she can increase or decrease production as much as she wants for any quantity (as long
as its not over 100) because her profits don’t change. In the end we can represent her supply as:


P <4
100
CW (P ) = [0, 100] P = 4


0
P <4
The issue here is that the marginal cost is constant for any quantity and this essentially leads to a constant
supply curve. If the marginal cost depended on quantity then we could take the FOC equate it to zero and solve it
like you would normally expect. In your graded work you should provide an adequate explanation on how you you
found the supply curves otherwise we will assume that you do not know what you are doing.
Representing Wendy’s supply curve graphically we would have:
Figure 8: Wendy’s Supply Function
b) Now suppose that the inverse market demand was given by:
P = 10 −
1
CW
1000
Calculate the long run competitive equilibrium.
Solution:
In the long run with perfect competition firms enter and exit until their profits go to zero. We know that this
happens when P = min AC, as it is the only point where the marginal cost curve intersects the average cost curve,
which is just when P ∗ = 4 because the average cost function is just a horizontal line. Then using the demand
equation it is easy to see that CW ∗ = 6000.
So in a long run equilibrium P ∗ = 4, CW ∗ = 6000. Then when P = 4 the firms can be producing 1 unit or 10
or 100 because at that price they are indifferent from producing any quantity given the format of the supply curve.
This also means that the number of firms is indeterminate because we cannot pin down what each firm produces
(although we know that it cannot be more then 100 units).
8
You should realize that summing supply curves horizontally to get the aggregate supply is a bit different now
because of the capacity constraint. So if we have two firms, you would have a graph almost identical to the one
above but instead of a limit of 100 you would get a limit of 200. As you add firms this limit increases successively.
This means that you would need at least 60 firms otherwise if you were to intersect the aggregate supply (using 59
or less firms) with the demand equation then you would get a price above 4 and that would tell you that firms are
having profits and another firm would want to enter. From there on, we do not have any way of saying if we have
60 or 600 or 6000 firms.
c) Wendy then made a startling discovery: people in Hyde Park were actually paying $6 for their chicken caesar
wraps! How much profit was each firm making by selling chicken caesar wraps? (hint: remember that they all had
a quantity constraint)
Solution:
Looking at the beginning you would realize that each firm can produce a maximum of 100 units. You also know
that if P > 4 then they want to produce as much as possible, reaching their capacity constraint. Thus the firm’s
profits will be π = (6 − 4) × 100 = 200. This also gives us another piece of information... there are less then 60
firms on the market (in b. you saw that 60 firms was the lower limit that lead to zero profits). In fact, using the
demand equation you would see that for P = 6 you would have CW = 4000 and if each firm is producing 100 units
then you know that n = 40.
d) So she enthusiastically started to look for a shop to rent... but then realized that there were only 40 stores
available where you could install a sandwich shop and all of them were taken by competing sandwich shops. She
also noticed that the university owned all of those properties and rented them out to the sandwich shops. What is
the maximum amount of rent that the university could charge for renting out these stores?
Solution:
Here Wendy discovers that there is this fixed factor which is the total number of stores available. In the previous
problems she did not realize that this was a problem and so ignored the importance of the rent in her decision
process. In c) we already noticed that only 40 firms were on the market but now we see why we only have 40 firms
on the market. Given that this is the same equilibrium as in c) we see that before paying the rental fee each firm is
earning $200 and this represents the maximum amount that the university could charge in rent (so effectively with
that rent the firms end up having zero profit).
In fact we should be more careful in this statement, $200 is the highest rent that the university could charge
such that all their stores would be rented out to sandwich shops. They could always choose a higher rent but then
some firms would have to leave the market and these stores would remain unoccupied. We could also play with this
option to see what is the highest revenues that the university could get from renting... they might even prefer to
have some stores unoccupied.
e) Suppose that the university decided to build 15 new stores and reduce the rent to $100 in all stores. What
would be the new long run competitive equilibrium?
Solution:
Now, looking at each firm before anybody else enters the market you would see that:
π = P × y − 4 × y − rent = (6 − 4) × 100 − 100 = 100
9
So with a positive profit firms want to enter the market. As they enter they increase the total quantity supplied
which will then decrease the market price and so decrease the profit of all the previous firms. Therefore firms will
enter until profits go back to zero.
Now the amount each firm supplies is fixed at 100 because for any price above 4 each firm wants to produce as
much as possible. So our problem is to try and find the P that makes all the firms have zero profits. Looking at
the above equation you would quickly say that this happens when P = 5. Looking at the demand equation we see
that this would only happen if Y = 5000 and if each firm produces 100 units then we have 50 sandwich shops in
the market. Once you come up to this solution P = 5, Y = 5000, n = 50 and y = 100 then you know that you have
an equilibrium where no one else wants to enter (or exit). Do notice that we did not use up all the stores that the
university constructed. If the university would want to occupy those spaces then they would need to reduce the
rent even further.
Question 3 - Licenses in the Taxi Market
Every day the demand for cab rides is given by:
Q = 500 − 40P
In each ride the driver earns P , the market price, and cost of the fuel spent on that ride is $1. Additionally, the
cab drivers calculated their monthly expenses and discovered that they can state that they spend a fixed amount
of $60 a day on maintenance and cab leasing fees. Every day the driver makes a maximum of 20 rides.
The market so far was a free market, i.e. free entry and exit of the market. Now, the city decided to introduce
a system of licenses for cab drivers to increase their revenue. The fee for this license is charged monthly and the
goal of the city is to maximize the income obtained from these licenses. The final goal of this problem is to find
this income maximizing price for a license (from the city’s perspective). For your exercise state that L represents
the daily expense with the licensing fee (i.e. the monthly licensing fee is 30 × L).
If it helps you understand the market has the following timing: every morning the cab driver decides if he wants
to enter the market, he pays all the necessary fees, enters the market and produces rides. If he chooses not to enter
he does not pay L or have to lease the taxi cab for that day.
a.i) What is the profit maximization problem of a given driver if he enters the market? (use P for price and L
for license and represent this as a day by day problem)
Solution:
If we represent this problem assuming that they have already entered and paid the fixed fees then the profit
maximization problem of a given driver, with a fixed cost of 60 + L a marginal cost of c = 1, will be given by:
maxP.q − (c.q + 60 + L)
q
Is this a short run or a long run problem? Depends on where you are in your timing. If they have already paid
the license fee and lease costs for taxi that morning then we are in the short run as we have this set of factors (car
and license) that we have already paid for and cannot change (or at least this is true as long as they cannot sell of
the license and car). If we are in the period before they pay for these items then we are in the long run because
everything is variable and can still run away from these fixed fees. In this case he has already entered the market
and so has already incurred in those fixed costs. In the books you will find a similar type of fixed costs (that you
only pay if you have a positive production level) referenced as “quasi-fixed” costs (in our case after they enter and
pay the fixed costs they still have the option of not producing so it doesn’t exactly fit the exact definition, that us
10
unless they can sell of the license quickly, but our analysis will match that form of cost function exactly because
this us a daily market they can runaway from very quickly).
a.ii) What is the optimal choice of q for a cab driver facing price P ? (hint: you might have to distinguish 2
cases)
Solution:
First things first, if we calculate the derivative of the profit function we get:
∂π
=P −c
∂q
Just like in the perfect complements exercise you would state that P − c is a constant from the perspective of
the firm and so: if this number is less then 0 (i.e. if P < c) then the firm wants to decrease production all the way
to zero; if its zero (i.e. if P = c) then the firm is indifferent from producing any quantity; and if its positive (i.e. if
P > c) then the firm wants to produce as much as possible, thus reaching its capacity constraint of 20 rides. But
we cannot forgot a crucial part, the profits must be positive to pay off the maintenance and license fees. Therefore
we must have:
π
≥
0
...
L
≤
P.q − c.q − 60
Now the issue is that if P = c then P.q − c.q would certainly be zero and so the firm would not have any funds
to cover the license and maintenance fees. So P has to be bigger than c and with this the firm will produce 20 rides
every day. Knowing that the firm will do q = 20 for certain if they choose to produce, let us look at that condition
again (and also use the fact that c = 1):
L ≤
P.20 − 20 − 60
L ≤
P.20 − 80
Thus, all the cab drivers will either produce q = 20 when L respects this condition or they will choose not to
enter the market and set q = 06 :
(
20 L ≤ 20P − 80
S (P ) =
0 L > 20P − 80
b) Define the long run competitive equilibrium. Denote the equilibrium outcome as a function of the license fee
L (i.e. P (L) , n (L) , Q (L) , ...).
Solution:
In the long run competitive equilibrium there is free entry and exit thus firms will enter or leave until the profits
become zero in equilibrium. If you wanted a more exact formalization you would say that there is an equilibrium
price, aggregate quantity, number of firms and levels of production for each firm such that the aggregate demand
function is satisfied, the firm’s behave as in a.ii the aggregate quantity supplied is n (L) times the individual supplies
and supply equals demand. But we did not request a formalization here so we can disregard that. Coming back to
6 Careful,
we are stating that when q = 0 they do not enter the market. So they do not need to pay L or the maintenance fee.
11
the problem, if we use the condition specified above (which is a condition on the profits) we would see that to have
zero profits we must have:
L =
P (L)
=
P.20 − 80
L
+4
20
Using the market demand you would see that the quantity supplied in the market is:
Q∗ (L) = 340 − 2L
And the number of firms is given by:
n∗ (L) = 17 −
L
10
c.i) What is the city of Chicago tax revenue maximization problem?
Solution:
The city’s problem is just to maximize the revenue from the license given the equilibrium number of cabs:
max
L
L × n∗ (L)
st.
n∗ (L) = 17 −
Or:
L
maxL. 17 −
L
10
L
10
c.ii) Solve it. (hint 1: you might need to consider 2 cases from question a.ii; hint 2: find n (L) and P (L) through
the LR competitive equilibrium system in question b, then solve the maximization problem for the city of Chicago).
Solution:
Calculating the FOC (assuming SOC are verified):
L
L
∂R
= 17 −
−
=0
∂L
10 10
And we see that L∗ = 85. For now we will leave it at this even though we will see in c.iii) that it does not give
a discreet number of firms.
c.iii) Find the resulting equilibrium values of n∗ , p∗ and Q∗ . (n∗ might not be an integer... it is fine)
Solution:
For L∗ = 85 we see that n∗ (L) = 8.5... if we believe in this number then Q = 8.5 × 20 = 170 and P = 8.25.
Although I feel squirmish with it so I would also like to see what happens if we have 8 and 9 firms. With 8 firms
12
we would have Q = 8 × 20 = 160, P = 8.5 and the firms profits before paying the licensing fees (P.q − (q + 60))
are 90 so the government can charge this amount to each firm and get a total of 720 (= 90 × 8) in revenues. If you
calculated the equilibrium with 9 firms you would see that Q = 180, P = 8, (P.q − (q + 60)) = 80 and then the
government still collects 720 in total revenue. So the government is indifferent between having 8 or 9 firms. For
any other discreet number of firms these revenues would certainly be lower.
But in your solutions we will accept fractional firms so as to not to complicate your work.
d) Find the equilibrium if there was no tax. How did the introduction of the license fee affect the number of
cabs in Chicago? The price of a ride? Does that make sense?
Solution:
Without a tax, i.e. L = 0, we have that P0∗ = 4, Q∗0 = 340 and n∗0 = 17. We see that the introduction of a
license increased prices and decreased the amount of cabs. This is an expected result given that the cab drivers
need the additional margin between the price and the marginal cost to pay the higher fixed cost.
So, if you increase the fixed costs of the firm with the license fee then the firms in the market would start to
have negative profits and some of them decide to not enter the market that day. This increases prices and with this
firms obtain more revenue per ride so they will eventually reach a breakeven point where nobody makes profits.
e) Derive consumer surplus with and without the license. How did the introduction of a license for cab drivers
affect consumers?
Solution:
You will easily see that the surplus before the tax is 1445 and after the implementation of the license... if you
have the equilibrium with 8 firms you would see that the consumer surplus is 320 while the surplus with 9 firms is
405 and if you used 8.5 firms you would get 361.25. If we put all this in a table we would see that:
L
n (L)
Q (L)
P (L)
CS (L)
0
17
340
4
1445
85
8.5
170
8.25
361.25
80
9
180
8
405
90
8
160
8.5
320
In the end the lower surplus is expected as the license fees end up becoming a barrier that reduces the amount
of taxi cabs that enter the market and as each taxi cab has a quantity constraint this will reduce the amount of
service provided to the consumers.
Question 4 - Monopoly Basics
Suppose that the inverse demand in a given market is:
P (Q) = A − bQ
This market has a monopolist with the following cost function:
C (Q) = DQ2
13
a) Show, for this linear demand curve, that the marginal revenue curve is half of the demand curve (hint: first
write down the revenue function and then calculate the marginal revenue). Graph the marginal revenue and demand
curve on the same graph.
Solution:
By definition the monopolist’s revenue is:
R = Q × P (Q) = Q (A − bQ) = AQ − bQ2
With this the marginal revenue is:
M R (Q) =
∂R
= A − 2bQ
∂Q
If you want to represent this in terms Q you would have:
Q=
1 A − MR
2
b
And you could easily show that the demand equation is:
Q=
A−P
b
Either representation is OK for us.
Then representing this graphically you would see that:
Figure 9: Marginal Revenue with a Linear Demand
b) Now we want to look at the elasticities...
b.1) Calculate the elasticity of demand. What is the elasticity when Q = 0, Q =
14
A
b
and when Q =
A
2b ?
Solution:
If we calculate the elasticity of demand we would see that:
ε=
∂Q P
1P
=−
∂P Q
bQ
If you wanted you could still plug in P to get a nice representation in terms of Q:
ε (Q) = −
1 A − bQ
b Q
Then we see that:
Q
Q→0
A
2b
A
b
ε
−∞
−1
0
Where we use Q → 0 to symbolize that we are looking for a number that approximates zero because the elasticity
is not defined when Q is exactly 0.
b.2) Draw a graph with the demand equation and the marginal revenue. Include the numbers you calculated
above at their relevant points and then tell us what happens to the elasticity on all the other points on the demand
curve.
Solution:
The graph would be:
Figure 10: Elasticity Along a Linear Demand
Essentially you would show that the elasticity of demand decreases as quantity increases. For |ε| < 1 we are in
the inelastic part of the demand curve and for |ε| > 1 we are on the elastic part of the demand curve.
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A
b.3) You probably noticed that at Q = 2b
the M R is equals to zero and so we are maximizing the revenue
extracted from the market (do not confuse revenue with profits). Using the elasticity discuss what happens to
A
the firm decreases its price. Do the same for a point to the right of
revenue if from a point to the left Q = 2b
A
Q = 2b and for a increase in prices
(hint: the elasticity says how quantities change percentually when there is a percentual variation in the price...
so a 1% change in P has a ε percent change in quantity demanded... and for small percentual changes you could
state that the percentual change in M R is approximately the percentual change in prices plus the percentual change
in quantities)
Solution:
A
On a point to the left of Q = 2b
we have |ε| > 1 and so a 1% decrease in prices leads to a more than 1% increase
in Q (remember the sign on the elasticity). So quantity increases percentually more than the percentual decrease
in prices and so the revenues increase.
A
On a point to the right of Q = 2b
we have |ε| < 1 and so a 1% increase in prices leads to a less than 1% decrease
in Q (remember the sign on the elasticity). So quantity decreases percentually less than the percentual increase in
prices and so the revenues increase.
So graphically:
Figure 11: Revenue Along a Linear Demand
b.4) By now you have realized that firms maximize their revenues when |ε| = 1. You have probably also seen
in Econ 200 that when |ε| < 1 we call this the inelastic part of the demand equation and when |ε| > 1 we call this
the elastic part of the demand curve. Will the monopolist ever produce in the inelastic part of the demand curve?
(hint: you know the firms profits are equals to revenue - cost... in b.3 you saw how revenues changed when you
increased prices in the inelastic part of the demand curve... when you increase prices you reduce the quantity sold
and so the firm produces less goods and so reduces its costs...)
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Solution:
No. If the firm is in the inelastic part of the demand curve then decreasing quantities (increasing prices) will
increase the revenue. Then with a lower quantity the costs are also lower, so the firm benefits from this. Essentially
you could say that decreasing quantity when you are on the inelastic part of the demand curve leads to:
π = Revenue
| {z }−Costs
↓
↑
| {z }
↑
↑
b.5) Will the firm strictly prefer to produce where |ε| = 1 over any point in the elastic part of the demand curve
(hint: you’re going to see that you cannot really conclude where the firm produces without looking at actual values
but you can think of what would happen if D was a really big number)
Solution:
Using the same idea as before, suppose we are to the left of Q =
to |ε| = 1 where we maximize revenues:
A
2b
and decide to increase quantity to get closer
π = Revenue
| {z }−Costs
?
↑
| {z }
↑
↓
So without any information on the cost function we cannot state where the firm wants to produce on the elastic
part of the demand curve because decreasing/increasing Q will have contradictory effects on the gains/losses from
revenue and cost. At most we can state that Q∗ will be given by M C (Q∗ ) = M R (Q∗ )
c) Now let’s solve the problem for some values. Suppose that A = 120, b = 2 and D = 1.
c.1) Formalize the monopolists problem.
Solution:
The monopolist’s problem is:
max (120 − 2Q) Q − Q2
Q
c.2) Calculate the monopoly equilibrium (hint: you must have price, quantity and profits).
Solution:
Assuming SOC are verified then the FOC lead to:
[Q] : 120 − 2Q − 2Q = 2Q
With this we have QM = 20, P M = 80 and π M = 1200.
c.3) Suppose the monopolist acts like a competitive firm (i.e. sets P = M C). Calculate the equilibrium price,
quantity and profits.
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Solution:
When a firm behaves competitively we have that:
P = MC
So:
120 − 2Q = 2Q
With this we have that QC = 30, P C = 60 and π C = 900.
c.4) Graph both situations, illustrating the consumer’s surplus, firm’s surplus and deadweight loss in each case.
Solution:
Graphing these two situations you would have:
Monopoly Behavior
Competitive Behavior
Figure 12: Surplus and Deadweight Loss
Question 5 - Monopoly With Two Factories
Wendy decided to start producing cheesecakes and she found a market where she can act as a monopolist (assume
that the transport costs are so prohibitive that consumers cannot buy cheesecakes from a different supplier). The
demand in this market is given by:
D (p) = 200 − 4P
Suppose that her production technology leads to a cost function of:
C1 (Q) = Q2 + 5Q
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a) Formalize the monopolist’s problem and find the equilibrium in the market.
Solution:
Notice that the inverse demand is P = 50 − Q
4 . Then the monopolist’s problem is:
Q
max 50 −
Q − Q2 + 5Q
Q
4
With this, assuming the SOC are verified, the FOC are:
[Q] : 50 −
Q Q
− − 2Q − 5
4
4
5Q
45 −
2
=
0
=
0
Then you would very quickly solve the rest and tell us that Q = 18, P = 45.5 and π = 405.
Suppose now that she opens a second factory with the following technology:
C2 (Q) =
Q2
+ 2Q
2
b) Formalize the monopolist’s problem and find the equilibrium where she uses both factories at the same time.
Why does your result make sense (hint: It has something to do with the relation between M C1 , M C2 and M R)
Solution:
Now the monopolists problem becomes:
2
Q
q2
50 −
Q − q12 + 5q1 −
+ 2q2
max
q1 ,q2
4
2
st.
Q = q1 + q2
Or:
2
q2
q1 + q2
(q1 + q2 ) − q12 + 5q1 −
+ 2q2
max 50 −
q1 ,q2
4
2
Assuming the SOC are verified, the FOC are:
[q1 ]
:
[q2 ]
:
q1 + q2
q1 + q2
−
− 2q1 − 5 = 0
4
4
q1 + q2
q1 + q2
50 −
−
− q2 − 2 = 0
4
4
50 −
In other words we get that M R = M C1 , M R = M C2 and more importantly M C1 = M C2 . Then once you
195
559
13 275
solve this problem you would obtain q1 = 87
' 948.21.
7 , q2 = 7 , P = 14 and π =
14
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Question 6 - Monopoly and Regulation
Suppose that a monopolist has the following demand:
D(p) = 50 − p
and that its cost function is given by:
C (Q) = 39 + 10Q
a) Formalize the monopolist’s problem and calculate the equilibrium in this market (i.e. price, quantity and
profits).
Solution:
The monopolist’s problem is:
max (50 − Q) Q − 39 − 10Q
Q
Assuming SOC are verified the FOC are:
∂π
∂Q
=
50 − Q − Q − 10 = 0
⇔ Q = 20
With this the price is P = 30 and profits are: π = 30 × 20 − 39 − 10 × 30 = 261.
b) Find the equilibrium if the monopolist simply wanted to maximize quantity while staying in the market (i.e.
not losing money). (hint: although you can formalize and solve a maximization problem here, we recommend that
you just look at the AC and the demand...)
Solution:
The problem here is:
max
Q
Q
st.
(50 − Q) Q − 39 − 10Q ≥ 0
But instead of solving a maximization problem with inequality constraints let us use a simpler method... you
already know that π ≥ 0 when P ≥ AC, so let us represent the demand and AC on a graph:
20
Figure 13: Average Cost and Demand
Then as we are maximizing quantities we will end up choosing a quantity and price where P = AC. Thus:
P
= AC
39
+ 10
50 − Q =
Q
Q2 − 40Q + 39 = 0
Solving this quadratic equation we get that Q = 1 ∨ Q = 39, so the solution is to choose Q = 39. Then P = 11
and π = 0.
c) Normally we define competitive behavior as P = M C (although you know by know that firms will only do
this if it respects the shutdown or exit condition... nonetheless you should be warned that lots of times economists
will simply state that competitive behavior is just doing P = M C without being concerned about shutdown/exit
conditions so keep that in the back of your mind). Find the equilibrium where the monopolist has a competitive
behavior.
Solution:
With the competitive behavior we will have P = 10, Q = 40 and π = −39.(Naturally you should explain a bit
more then I do)
d) Calculate the social surplus (consumer surplus + (profits+FC)) of each case. Which of these 3 cases would
a social planner (like the government) prefer?
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Solution:
Taking the elements from all three questions you could easily create the following table:
Question
a
b
c
Consumer Surplus
(50−30)×20
= 200
2
(50−11)×39
1521
=
2
2
(50−10)×40
=
800
2
π + FC
261 + 39 = 300
39
0
Social Surplus
500
799.5
800
So the social planner would prefer the third situation, competitive behavior, which we already expected. But do
note that to enforce this equilibrium they need to make a lump sum transfer to the monopolist to cover the fixed
cost and that means introducing taxes...
e) Suppose the government could set a fixed price on the market but if the monopolist has a loss it will leave
the market. Could it impose the equilibrium in c)? What about the equilibrium in b)?
Solution:
With the fixed price, the social planner would like to set P = 10 to get the competitive price but in this
situation the monopolist would leave the market because P < AC... so this option (equilibrium in c) ) is not
possible. Therefore it will have to choose the second best alternative and set P = 11 which the firm would accept
given that it ensures zero profits.
f) Suppose the government could set a price on the market and then impose a lump sum tax or give a lump
sum subsidy to the monopolist. Now, is it possible to implement the equilibrium in c)?
Solution:
In question e the problem we faced was that the monopolist would have a loss of 39 if we simply set P = 10...
but now if we set P = 10 and give the monopolist a lump sum subsidy of 39 then the firm will have zero profits
and will stay in the market, thus insuring that the equilibrium in c can be realized. Notice that we could have even
extracted the taxes from the consumers with lump sum taxes and used these to pay the monopolist and they would
still increase the consumer’s surplus.
Some of you are probably going to say "why are we not going farther"... so why not set P = 0 and get the
highest consumer surplus possible... Well, if you want to get that solution then you just need to collect the taxes
somewhere and then give them to the monopolist. But would you want that solution? The marginal cost reveals
how costly it is to produce that additional unit and the demand curve reveals how much consumers value those
units... so if you set a price lower then marginal cost you will be supplying the good to agents who give it less value
then what it really costs (thus it would be an inefficient solution). But this all depends on the objectives of the
social planner (maybe he would want to make people buy more computers to speed up innovation in the country...
this is a completely different problem that we are not analyzing here).
Question 7 - 1st Degree Price Discrimination
In class we saw a mostly verbal discussion of 1st degree price discrimination. Nonetheless we can easily create
an algebraic example to represent this form of price discrimination. Assume that every agent i has this demand
equation:
Pi = Ai − bi qi
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Where Ai and bi can be different between consumers. Also assume, for simplicity, that the firm has a constant
marginal cost of c and no fixed costs. Now from what you saw in class you know exactly what the answer should
be... but we are not going to take shortcuts this time. Instead, we are going to use a two part tariff where the
monopolist will set a different tariff for every agent (i.e. the consumers will not have the power to choose a different
plan). This is distinct from second degree price discrimination because there the agents had the power to choose
between different sets of two part tariffs (while now we impose it on each person). You can also use two part tariffs
for third degree price discrimination but the issue is that you can only identify market segments and the aggregate
demand in each segment while individual consumers might have minor differences between themselves... so if the
fixed part of your two part tariff is too high then he could lose some consumers. A two part tariff can be used in
any context but it becomes particularly useful with 2nd degree price discrimination as you can create two separate
plans that makes consumers self select into the two plans and reveal their type.
Now let us return to our first degree price discrimination problem:
a) Suppose that the monopolist supplies quantity qi to agent i. Show that the consumer surplus, as a function
of qi , is: CSi (qi ) = b2i qi2 .
Solution:
If we draw this out:
Figure 14: Consumer Surplus
You will see that consumer i’s surplus is given by a triangle with height Ai − Pi and width qi . Therefore,
consumer i’s surplus is given by:
1
CSi = (Ai − Pi ) qi
2
Then if we plug in the expression for consumer i’s demand equation then we get:
CSi =
1
bi
(Ai − (Ai − bi qi )) qi = qi2
2
2
b) Now take a two part tariff (i.e. Ti (qi ) = Fi + pi qi ) and formalize the monopolist’s problem when he analyzes
his profit maximization problem for this agent i. (hint: the T (qi ) represents the total revenue collected from
consumer i)
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Solution:
The monopolist’s problem is:
maxFi + (Ai − bi qi ) qi − cqi
Fi ,qi
c) What is the highest Fi that the monopolist can charge?
Solution:
When the monopolist sells qi units to the consumer at price Pi = Ai − bi qi then the consumer obtains b2i qi2 units
of consumer welfare. Thus the maximum that the monopolist can charge for the fixed fee is exactly the amount of
the consumer welfare. If the monopolist charged anything higher than this then the consumer would prefer not to
buy anything at all. In the end you would state that Fi = b2i qi2 .
d) Substitute the Fi with the expression you specified in c) and solve the monopolist’s problem. (after you take
the FOC and solve the expression you will get exactly what we saw in class)
Solution:
After plugging in Fi the monopolist’s problem is:
max
qi
bi 2
q + (Ai − bi qi ) qi − cqi
2 i
Assuming that SOC are verified then the FOC are:
[qi ] : bi qi + Ai − bi qi − bi qi − c = 0
Solving this we obtain that:
qi =
Ai − c
bi
Then you would easily see that:
pi
= c
Fi
=
2
(Ai − ci )
2bi
The monopolist will keep on selling more and more until M C = Pi where the Pi gives us the value that the
consumer has for quantity qi . This is akin to setting Pi = c so that the monopolist sells as much as efficiently
possible while not collecting any profit from the pi × qi component and then getting as much profit as possible from
the fixed component.
e) What are the consumer’s surplus and the firm’s profits? Is this result efficient? (i.e. is there any way we can
increase the social welfare in this relationship between consumer i and the monopolist)
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Solution:
Given that the monopolist is extracting all the consumer surplus with Fi then we obviously have:
CSi = 0
Then the profits extracted from consumer i are simply given by:
πi
= Fi + pi qi − cqi
= Fi + cqi − cqi
= Fi
2
=
(Ai − ci )
2bi
Which was also an obvious value. All these results are efficient as the monopolist is ensuring that pi = c which
leads to no deadweight loss when he sells to consumer i. Nonetheless the consumer would most likely prefer an
alternative scheme as he is getting zero surplus when the monopolist has full information on his preferences and is
allowed to set a different pricing scheme for every agent. Whether this is fair or not is another issue (that we do
not analyze here).
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