Intermediate Microeconomic Theory (3550)
Homework 6 - Answers
Due: Thursday, November 20, 2008
1.
If you were an actor in a major motion picture, would you prefer to earn a percentage of the
profit made by that film, or the same percentage of total revenue generated by the film?
Explain.
Because profit is total revenue minus some factor, profit will always be lower than
total revenue. One should prefer a portion of total revenue to the exact same portion of
profit.
2.
Explain in words or illustrate graphically why a contractual arrangement for actors and
directors to receive a percentage of a film’s total revenue rather than a percentage of the
profit generated by a film might create a conflict in determining what price should be
charged for a movie ticket.
Those receiving a percentage of total revenue want to maximize TR; those wanting to
receive a percentage of profit maximize profit. The monopolist always operates on
the elastic portion of his demand curve. At each output (Q), the MR curve’s height
shows how much TR changes when one more unit is sold. The height of the MR curve
therefore equals the slope of the TR curve at that output. TR is maximized when MR
= 0 (demand elasticity = 1). Therefore, PTR < P.
Elasticity = 1
$ / unit
MC
Pprofit
Elasticity < 1
PTR
D
Price
TR curve
Qprofit
QTR
Quantity
3.
At the profit maximizing output the price of Tommy jeans is twice as high as marginal cost.
What is the elasticity of demand?
Using the formula MR = P[1-(1/)], solve for  and remember MC = MR. Elasticity of
demand = 2.
The following scenario applies to questions 4 through 6. Denton is considering alternative
ways of providing cable service to its citizens. Economists have determined that the total
cost and inverse demand curves for a cable company in Denton are:
TC = 2Q – 0.1Q2 + 0.005Q3
P = 20 – 0.5Q
4.
City Councilor A believes the city should own and operate a cable system for the purpose
of making as much profit as possible. The profit would be used to lower the city
government’s deficit. If Councilor A gets his way, what will be the price and output of
cable, and by how much will the city-owned system be able to reduce the city’s deficit?
Find total revenue by multiplying price (20-0.5Q) by Q. Then find MR and equate
that with MC from the given total cost function. Q = 17, P = $11.50, Profit = $165.83.
5.
Councilor B believes the city should produce as much cable service as possible without
losing money (i.e., provide cable on a non-profit basis). If Councilor B gets his way, what
output and price will result?
Set P = ATC and solve. Q = 32, P = $4.
6.
Councilor C believes that the private sector should provide cable to the city but that the
single, private firm that gets the city’s franchise should pay 10 percent of its total revenue
back to the city in the form of an annual franchise fee. If Councilor C gets her way and the
franchise is awarded to the firm promising to pay the largest franchise fee, what price and
output will result? What will be the size of the annual franchise fee?
If Q = 17 (a profit maximizing firm’s production), TR = $195.50. Ten percent of that
is $19.50.
However, if the firm knows it must give up 10 percent of profits, it should take that
into account before making its profit-maximizing output decision.
7.
Suppose that a monopolist sells in two markets with demand curves:
QA = 100 – 10PA
QB = 8 – 2PB
Show that for any given quantity, demand is more elastic in market A than in market B.
We know that in whichever market elasticity is higher, MR is higher. Calculate MR.
QA = 100 – 10PA
QB = 8 – 2PB
10PA = 100 – QA
2PB = 8 - QB
PA = 10 - .1QA
PB = 4 - .5QB
TR = PQ
TR = PQ
TR = (10 - .1QA)QA
TR = (4 - .5QB)QB
TR = 10QA - .1QA2
TR = 4QB - .5QB 2
MR = 10 - .2QA
MR = 4 - QB
Since 10 - .2QA > 4 - QB , demand in market A must be more elastic.
8.
Suppose that a monopolist produces at zero marginal cost. How much does he supply in
each market, and what prices does he charge?
If MC = 0, monopolist sets MR = MC to find quantity to produce in each market.
10 - .2QA = 0
4 - QB = 0
10 = .2QA
4 = QB
50 = QA
4 = QB
PA = 10 - .1(50)
PA = $5
PB = 4 - .5(4)
PB = $2
This may seem counterintuitive, given the statement in the text that the group with
more elastic demand is always charged the lower price. Here, the elasticity in each
market is equal to 1 at the quantity being sold (we know this because we set MR = 0,
and at MR = 0, demand is unitary elastic). Since the elasticity in both markets is
equal, one might expect the prices in both markets to be equal. This is true for
elasticity other than 1. But when elasticity = 1, producers cannot sell one more unit
by lowering price – they would increase cost and decrease total revenue by doing so.
The following scenario applies to questions 9 through 12. On Thanksgiving Day, CBS is
the monopoly provider of televised football games. Demand for televised football games is
given by Q = 26/6 – 1/6P. The cost to broadcast a game to a household is given by C(Q) =
20 + 2Q.
9.
Graph the deadweight loss associated with CBS being the monopoly provider of televised
football games.
Price
DWL is this triangle.
PM = $14
Profit is this
square.
MC =PC =
$2
MR
QM = 2
10.
QC = 4
D
QFootball games
Indicate on the graph CBS’s profit.
Use ATC = TC/Q = 24/2 = $12 to illustrate the profit.
11.
Calculate the deadweight loss (given that the loss = ½(Pmonopoly – Pcompetitive)(Qcompetitive –
Qmonopoly).
To calculate deadweight loss one first must find the monopoly price and quantity and
the competitive price and quantity.
To find the monopoly price and quantity, first, find inverse demand:
Q = 26/6 – 1/6P
1/6P = 26/6 – Q
P = 26 – 6Q
TR = (26 – 6Q)Q
TR = 26Q – 6Q2
MR = 26 -12Q
TC = 20 + 2Q
MC = 2
Setting MR = MC: 26 – 12Q = 2
24 = 12Q
2 = QM
If QM = 12, PM = 26 – 6(2) = $14.
For a competitive firm, P = MC: 26 – 6Q = 2
24 = 6Q
4 = QC and P = $2.
DWL = ½ (14 – 2)(4 - 2) = $12.
12.
What is CBS’s profit per household on Thanksgiving Day?
Profit = TR – TC
Profit = (14)(2) – 20 – 2(2) = $4 per household.
13.
Use the following information to illustrate whether consumers should prefer a competitive
industry or a monopolistic industry.
Q = 100 – 10P
TC = 50 + .4Q2
First, find inverse demand: P = 10 - .1Q
TR = (10 - .1Q)Q
MR = 10 - .2Q
Setting MR = MC: 10 - .2Q = .8Q
QM = 10.
If QM = 10, PM = 10 - .1(10) = $9.
For a competitive firm, P = MC.
(10 - .1Q) = .8Q
10 = .9Q
QC = 11.11
PC = .8(11.11) = $8.89
Quantity is greater and price is lower when a market is competitive.
14.
Many hotels allow children to stay in their parents’ room for free. Why?
Families with children have a higher elasticity of demand for hotel rooms than other
travelers. They are far less likely to be on business trips and therefore more apt to
change their travel plans if they find the hotels are too expensive. Families traveling
with children also tend to have more expenses and consequently are more sensitive
to prices then other travelers.
15.
In many cities, when three people share a taxicab to exactly the same address, the fare
depends on whether the three were traveling together at the time they hailed the cab.
Riders who know each other are charged less than those who don’t. Why?
Riders with traveling companions must have more elastic demand for cab rides than
riders who are alone. Why? Maybe because three rides can decide to split the cost of
renting a car instead three ways, whereas riders who are alone must pay the entire
cost of the car. Maybe people traveling in groups of three are more willing to get on
buses or subways, which take longer than cabs, since they have friends to pass the
time with and can protect each other from getting lost.