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Math Problem

Math Problem

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Taylor, T., & Greenlaw, S. A. (2014). Principles of microeconomics. Retrieved from http://cnx.org/content/col11627/latest
Review Figure 7.3 (p. 161). Determine the marginal gain in output from increasing the number of barbers from 4 to 5 and from 5 to 6? Does it continue the pattern of diminishing marginal returns?
b. Compute the average total cost, average variable cost, and marginal cost of producing 60 to 72 haircuts. Draw the graph of the three curves between 60 and 72 haircuts. (Taylor & Greenlaw, 2014, p. 177)
c.The AAA Aquarium Co. sells aquariums for $20 each. Fixed costs of production are $20. The total variable costs are $20 for one aquarium, $25 for two units, $35 for three units, $50 for four units, and $80 for five units. In the form of a table, calculate total revenue, marginal revenue, total cost, and marginal cost for each output level (one to five units). What is the profit-maximizing quantity of output? (Taylor & Greenlaw, 2014, p. 203)
d. A computer company produces affordable, easy-to-use home computer systems and has fixed costs of $250. The marginal cost of producing computers is $700 for the first computer, $250 for the second, $300 for the third, and $350 for the fourth, $400 for the fifth, $450 for the sixth, and $500 for the seventh.
i. Create a table that shows the company’s output, total cost, marginal cost, average cost, variable cost, and average variable cost.
ii. At what price is the zero-profit point? At what price is the shutdown point?
iii. If the company sells the computers for $500, is it making a profit or a loss? How big is the profit or loss? Sketch a graph with AC, MC, and AVC curves to illustrate your answer and show the profit or loss.
iv. If the firm sells the computers for $300, is it making a profit or a loss? How big is the profit or loss? Sketch a graph with AC, MC, and AVC curves to illustrate your answer and show the profit or loss. (Taylor & Greenlaw, 2014, p. 204)

1. Complete the following problems:
a. Using Figure 9.2 (p. 207), suppose P0 is $10 and P1 $11. Suppose a new firm with the same LRAC curve as the incumbent tries to break into the market by selling 4,000 units of output. (Taylor & Greenlaw, 2014, p. 223).
i. Estimate from the graph what the new firm’s average cost of producing output would be.
ii. If the incumbent continues to produce 6,000 units, how much output would be supplied to the market by the two firms?
iii. Estimate what would happen to the market price as a result of the supply of both the incumbent firm and the new entrant.
1. Approximately how much profit would each firm earn?
b. Using Figure 9.6 (p. 217) draw the demand curve, marginal revenue and marginal cost curves. (Taylor & Greenlaw, 2014, p. 223).
i. Identify the quantity of output the monopoly wishes to supply and the price it will charge.
ii. Suppose demand for the monopoly’s product increases dramatically. Draw the new demand curve.
1. What happens to the marginal revenue as a result of the increase in demand?
2. What happens to the marginal cost curve?
3. Identify the new profit-maximizing quantity and price.
c. Use the following table for this problem: (Taylor & Greenlaw, 2014, p. 242).
Price Quantity TC
$25.00 0 $130
$24.00 10 $275
$23.00 20 $435
$22.50 30 $610
$22.00 40 $800
$21.60 50 $1,005
$21.20 60 $1,225

Andrea’s Day Spa began to offer a relaxing aromatherapy treatment. The firm asks you how much to charge to maximize profits. The demand curve for the treatments is given by the first two columns. The total costs are given in the third column. For each level of output, calculate total revenue, marginal revenue, average cost, and marginal cost.
i. What is the profit-maximizing level of output for the treatments and how much will the firm earn in profits?

Complete the following problems:
a. A $10,000 ten-year bond was issued at an interest rate of 6%. It is now year 9 and you are thinking about buying the bond from its owner. Interest rates are now 9%. (Taylor & Greenlaw, 2014, p. 403).
i. Would you now expect to pay more or less than $10,000 for the bond? Why?
ii. Calculate what you would be willing to pay for the bond.
b. Consider the following stock ownership situation (Taylor & Greenlaw, 2014, p. 403).

Investor 1: 20,000 shares
Investor 2: 18,000 shares
Investor 3: 15,000 shares
Investor 4: 10,000 shares
Investor 5: 7,000 shares
Investors 6 – 11: 5,000 shares each

ii. What is the minimum number of investors it would take to vote to change the top management of the company?
iii. Which investors would this involve?
iv. In 1-2 paragraphs discuss if any one investor would be able to make corporate changes without the agreement of the other investors.

SAMPLE ANSWER

Math Problem

1. Marginal Gain in Output

Marginal gain = change in total output/change in quantity

4-5 – ([80 * 5] – [72 *4])/ (80 – 72) = 15 units per unit of labor

5-6 – ([84 *6] – [80 * 5])/ (84 – 80) = 26 units per unit of labor

The above marginal output follows the principles of diminishing marginal returns.

1. Average Total, Variable and Marginal Costs 

 The profit maximizing output is when; marginal cost = marginal revenue. Since there is no equilibrium in MC and MR, the closest maximizing quantity is 4.

1. Output and Costs

1. ii)

The zero profit point is at the minimal point of the average curve at point c while the shutdown point is where price equals the average variable cost.

If the firm sells the computers at $300, it will be making a loss when compared to prices at $500. This is because profits at $300 will earn profits at 03cb while profits made at $500 will be 03da.

3. Complete the Following Problems

Figure 9.2

Selling 4,000 units of output will attract a price above p₁.

1. II) The total amount of production supplied by the two firms would be 10,000 units.

III) The market price will be lowered since there would be more supply of goods from both firms.

1. b)

900. The monopoly firm wishes to supply its commodities at 4 units and at a price of 900.
901. If the demand increases, the demand curve will shift outwards.

Therefore, the marginal revenue will increase while the marginal cost curve shifts inwards. The new profit maximizing quantity and price will be where the new marginal cost curve intersects with the marginal revenue curve.

Price    Quantity           TR       TC       AC       MR      MC
$25.00             0                      0          $130    $0        –           –
$24.00             10                    $240    $275    $27.5   24        14.5
$23.00             20                    $460    $435   $21.75 22        16.0
$22.50             30                    $675    $610    $20.33 21.5     17.5
$22.00 40                   $880    $800    $20      20.5     19.0
$21.60             50                    $1080  $1,005 $20.1   20.0     20.5
$21.20             60                    $1272  $1,225 $20.4   19.2     22.0

The profit maximizing level is where MC equals MR which is either arrived at by producing 40 or 50 units. Therefore, the profit at 40 units will be $80 and the profit produced at 50 units will be $75. For that reason, the firm’s maximizing profit level will be 40 units.

4. Complete the Following Problems
5. I) one would expect not to pay more for the bond since an increase in interest rates indicates an increase in the value and risk of bonds.
6. II) Future value = principle amount * (1 + rate) ^time ; 10,000 * (1 + 0.09)⁹ = 21, 718

Bonds i – 4.5%, n – 20 semiannual periods, interest (PMT) – 300 semiannual (6% * 10, 000 * 6/12),

Therefore, present value of the bond – 300 * [1/(1 + 0.045)²⁰]

3. b) The minimum number of investors required to change the top management of a company is 3.

III) The investors would be 1, 2, and 3 who make 53% in ownership and can gather votes.

1. IV) Making corporate changes requires one to have more shares than the rest. This means that a shareholder who has the most number of shares can gather more votes (Taylor & Greenlaw, 2014). Therefore, one investor can make corporate changes. In this case, investor number 1 can make organizational changes without agreeing with other investors. This is because he has 20,000 shares which are 20% higher than all the rest. After all, having more shares means that one is capable of gaining more votes and has the highest ownership percentage.

References

Taylor, T., & Greenlaw, S. A. (2014). Principles of microeconomics. Retrieved from http://cnx.org/content/col11627/latest

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