Reference no: EM131000

Question 1)

The following relations describe monthly demand and supply for a computer support service catering to small businesses:

QD = 6 000 - 10P
QS = -2 000 + 10P

where Q is the number of businesses that need services and P is the monthly fee, in dollars.

a. At what average monthly fee would demand equal zero? (2p)

b. What is the equilibrium price/output level? (4p)

c. Suppose demand increases and leads to a new demand curve:

QD = 7 000 - 10P

What is the new equilibrium P and Q? (4p)

Solution:
a.

600

b.

Demand: P = 600 - 0,1Q
Supply: P = 200 + 0,1Q
Set them equal:
600 - 0,1Q = 200 + 0,1Q
Q = 2000
P = 400

c.

Demand: P = 700 - 0,1Q
Supply: P = 200 + 0,1Q
Set them equal:
700 - 0,1Q = 200 + 0,1Q
Q = 2500
P = 450

Question 2) The manufacturer of high-quality flatbed scanners is trying to decide what price to set for its product. The costs of production and the demand for the product are assumed to be as follows:
TC(Q) = 500 000 + 0,85Q + 0,015Q2
Q = 14 166 - 16,6P.

a) Determine the short run profit-maximizing price? (5 p)

b) Give expressions for AC, AVC, MC, P and MR (5p)

Solutions:

a.

To solve this problem, we find the MR and MC functions, set them equal to each other, and solve for the optimal Q. Using this Q, we then find the optimal P.

P = 853,37 - 0,06Q
TR = 853,37Q - 0,06Q2
MR = 853,37 - 0,12Q
MC = 0,85 + 0,03Q

853,37 - 0,12Q = 0,85 + 0,03Q
Q* = 5683,466
P = 853,37 - 0,06(5683,466)
P* = $512.36

This price can then be rounded to a more even number (e.g., $500).

b.
AC = 500000/Q +0,85 + 0,015Q
AVC = 0,85 + 0,015Q
MC = 0,85 + 0,030Q
P = 853,37 -0,06Q
MR = 853,37 - 0,12Q

Question 3) An amusement park, whose customer set is made up of two markets, adults and children, has the following demand schedules:

QA = 20 - 1PA (adults)
QB = 30 - 2PB (children)
QT = 50 - 3PT (the two markets combined)

The marginal operating cost of each unit of quantity is $5. (Hint: Because marginal cost is a constant, so is average variable cost. Ignore fixed cost)

Solve these equations for the maximum profit that the amusement park will attain when it charges different prices in the two markets and when it charges a single price for the combined market. (10p)

Solution:
First, each of the demand curves is converted, so that price is the independent variables:

PA = 20 - 1QAS
PC = 15 - 0,5Q 5QC
PT = 16.667 - .333QT

Marginal revenue will be calculated for each:

Adult market: TRA = 20QA - 1QA2
MRA = 20 - 2QA

Children's market: TRC = 15QC - .5QC2
MRC = 15 - 1QC

Combined market: TRT = 16.667QT - .333QT2
MRT = 16.667 - .667QT

Now, set marginal revenue equal to marginal cost ($5), and solve for Q and P:

Adult market: 20 - 2QA = 5 PA = 20 - 1(7.5)
QA = 7.5 = 12.5

Children's market: 15 - 1QC = 5 PC = 15 - .5(10)
QC = 10 = 10

Combined market: 16.667 - .667QT = 5 PT = 16.667 - .333(17.5)
QT = 17.5 = 10.83

Total Profit:

Market with differential pricing (price discrimination)

Adult market: 7.5(12.5-5) = 56.25
Children's market: 10(10-5) = 50.00
Total $106.25
Combined market: 17.5(10.83-5) = $102.03

Question 4) You own a large collection of fine wines. You now decide that the time has come to consider liquidating this valuable asset. However, you predict that the value of your collection will rise in the next few years. The following are your estimates:

Year Estimated value ($)
Today 70 000
1 88 000
2 104 000
3 119 000
4 132 000
5 142 000
6 150 000

If you assume your cost of capital to be 10%, when should you sell your

a. If you assume your cost of capital to be 10%, when should you sell your collection to maximize your NPV? (6p)
b. Calculate the growth rate each year (4p)

Solution: Sell after the 4th year

Year Estimated value ($) NPV (a) Growth (b)
Today 70 000 70 000
1 88 000 80 000 25,71%
2 104 000 85 950 18,18%
3 119 000 89 406 14,42%
4 132 000 90 158 10,92%
5 142 000 88 171 7,58%
6 150 000 84 671 5,63%

Question 5) The Compute Company store has been selling its special word processing software, Aceword, during the last 10 months. Monthly sales and price for Ace word are shown in the following table. Also shown are the prices for a competitive software, Goodwrite, and estimates of monthly family income.

Price Quantity Family Price
Month Aceword Aceword income Goodwrite
($) ($) ($)

1 120 200 4 000 130
2 120 210 4 000 145
3 120 220 4 200 145
4 110 240 4 200 145
5 114 230 4 200 145
6 115 215 4 200 125
7 115 220 4 400 125
8 105 230 4 400 125
9 105 235 4 600 125
10 105 220 4 600 115

a. Calculate the arc price of elasticity of demand for Aceword between months 3-4 and between months 4-5. (4p)
b. Calculate the arc cross-price elasticity of demand between Goodwrite and Aceword between months 1-2 and months 5-6. (3p)
c. Calculate the arc income elasticity between months 2-3 and months 6-7. (3p)

Solution

a.

Price elasticities

Months 3-4 -1,00 {20/(220+240)/2} / {-10/(120+110)/2}

Months 4-5 -1,19 {-10/(240+230)/2} / {4/(110+114)/2}

b.
Cross elasticities

Months 1-2 0.45 {10/(200+210)/2} / {15/(130+145)/2}

Months 5-6 0.46 {-15/(230+215)/2} / {-20/(145+125)/2}

c.
Income elasticities

Months 2-3 {0.95 10/(210+220)/2} / {200/(4000+4200)/2}

Months 6-7 0.49 {5/(215+220)/2} / {200/(4200+4400)/2}