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Professor Scholz
Economics 101, Problem Set #8, brief answers
Monopolies, Oligopolies, and Monopolistic Competition
Posted: 11/03/2009
Due: 11/10/2009
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Problem 1: Profit Maximization for a Monopolist
In Madison, Charter Communications can be considered a monopolist in providing cable
television services to local consumers. The market demand for cable television in Madison, WI
is given by P=1,000-4QD. Charter’s cost curves are as follows: MC= 2q ATC=q.
For the following subsections, assume that Charter is a rational profit maximizing monopolist.
a.) Find the optimal quantity and price pair for profit maximization. By doing the following steps
1.) Find Charter’s associated Marginal Revenue Curve. (Hint: MR has the same equation
as demand, except the slope is doubled)
MR=1,000-8QD
2.) Find the quantity where Marginal Revenue equals Marginal Cost.
Since there is only 1 firm in the industry, the monopolist, Q=q.
MC=MR=> 2Q=1,000-8Q
Q=100
3.) Calculate how much consumers are willing to pay for this quantity. This is what
Charter will charge.
Plug Q=100 into the demand curve. P=1,000-(4)(100)=1,000-400= $600
b.) Find the associated profit at the profit-maximizing (PM, QM) pair.
Profit=TR-TC=PMQM-ATCQM = (600)(100)-[100]100=
100[600-100] = $50,000
c.) What is the consumer surplus at (PM, QM)? What is the producer surplus at (PM, QM)?
Consumer Surplus = (1,000-600)100(1/2)=$20,000
Rather than producer surplus, we will talk about monopolist profit (in the monopoly case) ,
which you found in part b.
d.) Now, assume Charter is no longer a monopoly in the local cable industry and instead acts
like a perfectly competitive firm. What would price quantity pair (PPC, QPC) be in equilibrium?
Price comes from the intersection of S and D. Think of MC as the supply curve. The
intersection of MC=Demand Curve will therefore determine P. So 2Q=1,000-4Q, or
6Q=1000, Q=166.7
Substituting back in for price gives us, P=2(166.7)=333.3
e.) What is Charter’s associated profit when it acts as in perfect competitor of each firm at (PPC,
qPC)?
We can see that profits will be positive since P=MC is greater than ATC.
Profit=TR-TC=P*Q-ATC*Q=(333.33)(166.7)-(166.7)(166.7)= $27,777.22
f.) What is the consumer surplus at (PPC, QPC)? What is the producer surplus at (PPC, QPC)?
Consumer Surplus = (1/2)(1,000-333.33)(166.7)=$55,555.6
Producer Surplus=$27,777.22=Profit
g.) Is there a deadweight loss when we go from perfect competition to monopoly? If so,
calculate this. Explain what happens to price and quantity when a monopoly, like Charter takes
control of a local cable service.
Yes. DWL=(600-200)(1/2)(66.7)=13,340
Question 2: Natural Monopoly
Madison Gas and Electric (MGE) can be considered a natural monopoly in Madison, WI for
electricity. MGE’s cost functions are as follows: ATC=105/q+24 and marginal costs are
constant regardless of the quantity produced, MC=24. Market Demand for gas and electricity in
Madison is given by P=80-4 QD. Q is measured in Kilowatt-hours (kWh) of electricity.
a.) Graph the cost functions, demand curve, and marginal revenue curve all on one graph.
b.) Now solve for the profit-maximizing quantity chosen by the monopolist. (Hint: Find the
equation for Marginal Revenue and set this equal to Marginal Cost to find Q). What price will
MGE charge per kWh of electricity?
MR=80-8Q
MC=MR => 24=80-8Q so Q=7
Use the demand curve to find the price MGE can charge. P=80-4(7)=52
c.) Does MGE earn economic profit or incur a loss? Show this amount graphically.
ATC=24+105/7=39<P => Economic profits will be positive. Profits = TR-TC=52(7)39(7)=13(7)=$91. This is shown by the shaded area in the graph above.
Now, let’s assume that MGE discovers some of its meters are faulty and they need to reinstall
proper devices in over 100 Madison homes. This increases total costs for the company, and thus,
MGE’s average total cost will also increase. The new ATC=350/q+40 but marginal costs remain
constant (MC=24) as does aggregate demand.
d.) Now graph the new average total cost curve with the original marginal cost, demand, and
marginal revenue curves.
See graph below. We note that MC in this problem is not consistent with the total cost
curve (MC would be 40, not 24).
e.) Calculate the new profit-maximizing quantity chosen by the monopolist, along with the price
MGE will charge per kWh of electricity. Indicate these amounts on the graph.
There is no change here, assuming that MC stays at 24.
f.) Now, does MGE earn an economic profit or incur a loss? Show this graphically.
Since P=52<ATC=50+40=90, the firm incurs an economic loss. This loss is represented by
the shaded region in the graph above.
g.) Explain how natural monopolies differ from regular monopolies. What is in the main
difference in terms of costs?
Natural monopolies generally form because there are high fixed costs to production,
making it easier for one firm to produce the total quantity in a market cheaper than
multiple firms could. For example, installing monitoring devices to detect how much
heat/electricity has been used by a particular building. In these cases, especially when the
monopoly is operating at a loss, the government may intervene and aid the monopoly in
order to ensure it continues to produce.
Problem 3: Marginal Revenue Curves for the Monopolist a.) Fill in the table
Quantity
0
1
3
4
6
7
9
10
Price
11
10
8
7
5
4
2
1
Total
Revenue
0
10
24
28
30
28
18
10
Marginal
Revenue
-10
7
4
1
-2
-5
-8
b.) How much would this firm produce if their goal was to maximize total revenue?
6 units.
c.) Graph the Marginal Revenue and Total Revenue Curves on two separate graphs. Do this in a
way that the Marginal Revenue graph is directly above the Total Revenue graph. Find the
equation for the Marginal Revenue and then the equation for the Demand Curve. Add the
Demand Curve to the graph with the Marginal Revenue curve. What do you observe about the
relationship between Marginal Revenue when the Total Revenue curve is at its maximum?
You can find Marginal Revenue by choosing two points from the table, calculating the
slope, and putting the equation into point slope form. This is because the Marginal
Revenue Curve is linear. Then we can find the demand curve’s equation by dividing the
slope of the marginal revenue equation by 2. When we graph the Total Revenue function,
we see that it hits its maximum at q=6, which is what we found in part b. We also see that
at this point, on the second graph, Marginal Revenue is switching from positive to negative.
To summarize, when total revenue is maximized, marginal revenue is equal to zero. (Note
this is not exact in this case since we are finding marginal revenue in the table by
approximating the slope of a curve.)
MR=-2q+12
D=-q+12
d.) Go back to the original table. Now assume there is a constant marginal cost equal to $4. How
many units will this monopolist produce? What price will it charge? How much profit will it
make?
Set MC=MR. The monopolist will sell 4 units and charge $7 per unit. Since Marginal costs
are constant, the Total Costs are the sum of all the marginal costs, or TC=4(4)=$16. When
P=7 and Q=4 TR=$28 (you can get this from the table.) Then we can find profit=TRTC=$12.
Question 4:
You have a monopoly over the supply of your autograph (i.e. signature), which has suddenly
become valuable to other people due to your newfound celebrity status as an economics 101
wizard. Assume that it costs you nothing to provide an autograph (i.e. no fixed or marginal
costs) and that you would like to maximize your profit from giving your autograph out to people.
Both men and women want your autograph, but each group has a different demand function for
your autograph. Specifically, the demand function for men is given by Q M =8-P and the demand
function for women is given by Q W =4-P.
a.) What is the maximum profit you can obtain for your autograph?
I would price discriminate, charging different prices to men and women. First calculate
the marginal revenue for men, MR=8-2Q. Since MC=0, set MR=MC, and find optimal
Q=4 and P=4 for men. Profits from men are 16. Women have a marginal revenue of
MR=4-2Q. Setting MR=MC, you find the optimal Q=2 and P=2 for women. So the
maximum profit from women is 4, and the total profit is 20.
b.) Now assume that you cannot charge different prices for men and women (i.e. you must
aggregate women’s and men’s demand and find the associated marginal revenue curve.) Now
how much profit would you make? Compare this to the profit in part a.
When you aggregate demand you get that P=6-(1/2)Q where P<4 and P=8-Q where 4<P<8
So when you calculate your marginal revenue function, it is also piecewise.
MR=6-Q where P<4 and MR=8-2Q where 4<P<8
Since MC=0, we want to look at the second portion of the MR curve. Set MR=MC to find
the quantity you will produce 0=6-Q, and find Q=6. Now, put this back into the demand
curve to get the price you will charge. Let’s put it into the second part of the demand
equation and then verify that we are on the correct portion of the demand curve. P=6(1/2)(6)=3. And since 3<4, we know we are on the correct portion. So we have P=$3, Q=6.
Now let’s find the profit. Since we know the total costs of signing your name are 0, we just
have to find total revenues. Profit=TR=P*Q=3*6=$18. This is less than you made when
you charged different prices to different groups by $2.
Question 5: Monopolistic Competition
Assume that the market for luxury hotels in Las Vegas is monopolistically competitive. There
are numerous firms in the market, but each has a slightly differentiated product. In particular, this
means that each hotel has provides similar, but slightly different services. For instance, while
both hotels provide the same general accommodations, the Bellagio features free gym access and
wireless internet and the Palms offers complementary beverages and particularly attractive
bellhops.
If each hotel faces a demand curve of QD = 100-(1/2)P and a new hotel, The Bank wants to enter
the market. Its Total Cost function is given by: TC=320+4Q+5Q2 and its Marginal Cost is given
by MC=4+10Q.
a.) Write down the equation for Marginal Revenue and the Average Total Cost for the new hotel.
Demand: P=200-2Q
MR=200-4Q
ATC= 320/Q+4+5Q
b.) Graph the demand curve, marginal revenue curve, and marginal cost curve.
c.) What is the optimal quantity for The Bank to produce in the short run?
Set MC=MR. 4+10Q=200-4Q
14Q=196 => Q=14
d.) What is the associated price in the short run?
Substituting back into the Demand curve gives us P=200-2(14) = $172
e.) With the price and quantity you found in c) and d) how much profit is The Bank making?
Profit=TR-TC= 172(14)-(320+4(14)+5(14)2)=$1,052
f.) What do you think is going to happen in the long run in this industry? Draw this on a new
graph.
Since The Bank is earning positive profits in the short run, we expect more firms to enter.
This will shift the firm’s demand curve inward until the firm is earning zero profits (i.e.
where Average Total Cost is tangent to the demand curve where MR=MC).