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# Determination of optimal price and output for two firms

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Two firms face a demand equation given by P=200,000 -6(Q1 + Q2) where Q1 and Q2 are the outputs of the two firms. The total cost equations for the two firms are given by: TC1 = 8000Q1 and TC2 = 8000Q2.

If each firm sets its own output rate for profit maximization, and the other firm holds constant, what is the optimal output of each firm, as well as the optimal price for each firm.

Please show the work and/or formulas to derive the solution.

https://brainmass.com/economics/output-and-costs/determination-optimal-price-output-two-firms-212442

#### Solution Preview

Solution is also attatched in MS word file.

Solution:

Q1= output of one firm
Q2 = output of other firm

Total Revenue of one firm = P*Q1
= {200,000-6(Q1+Q2)}*Q1
= 200000Q1-6Q1^2-6Q1Q2
Profit for firm 1 (Pi-1) = Total Revenue - ...

#### Solution Summary

The solution describes the steps for determining optimal output and price for given two firms.

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## Managerial Economics: Price/Output Determination, Firm Supply, Markup on Cost, point price elasticity of demand, Incremental Analysis, Costs of Regulation, optimal price, output, and profit levels, Common Electric Products, Rainbo Luncheon Deli, Appalachian Coal Company,

1. Price/Output Determination. The City of Ithaca, New York is considering two proposals to provide its city government the service of computer maintenance. First, a national computer maintenance and sales franchise has offered to purchase the city's computer equipment at an attractive price in return for an exclusive franchise on computer maintenance. A second proposal would allow several small companies to provide the service without any exclusive franchise agreement or competitive restrictions. Under this plan, individual companies would bid for the right to provide service in a given department. The city would then allocate business to the lowest bidder.

The city has conducted a survey of its department to estimate the amount they would be willing to pay for various amounts of computer maintenance. The city has also estimated the total cost of service per department. Service costs are expected to be the same whether or not an exclusive franchise is granted.

A. Use the indicated price and cost data to complete the following table.

Hours of
Computer
Maintenance
per Month
Price
per
Hour

Total
Revenue

Marginal
Revenue

Total
Cost

Marginal
Cost
0 \$75.00 \$ 0 --
1 72.50 \$50.00
2 70.00 50.00
3 67.50 50.00
4 65.00 50.00
5 62.50 50.00
6 60.00 50.00
7 57.50 50.00
8 55.00 50.00
9 52.50 50.00
10 50.00 50.00
______________________________________________________________________________
2. Firm Supply. Common Electric Products, Inc., and Lighthouse Manufacturing, Inc., are domestic suppliers of halogen gas light bulbs used in roadside lamps. Given the vigor of domestic and foreign competition, P = MR in this market. Marginal cost relations for each firm are:

MCC = \$15 + \$0.0005QC (Common Electric Products, Inc).

MCL = \$45 + \$0.000125QL (Lighthouse Manufacturing, Inc).

where Q is output in units, and MC > AVC for each firm.

A. What is the minimum price necessary in order for each firm to supply output?

3. Markup on Cost. Oil-n-Go, Inc., offers automobile oil changes at a number of outlets in the greater Ann Arbor area. The company recently initiated a policy of matching the lowest advertised competitor price. As a result, Oil-n-Go has been forced to reduce the average price for oil changes by 5%, but has enjoyed a 15% increase in customer traffic. Meanwhile, marginal costs have held steady at \$20 per oil change.

A. Calculate the point price elasticity of demand for oil changes.

B. Calculate Oil-n-Go's optimal price and markup on cost.

_____________________________________________________________________________
4. Incremental Analysis. The Rainbo Luncheon Deli is considering offering a homestyle dinner menu from 6:00 to 10:00 P.M. on weekends. Currently, the deli only operates Monday through Friday between the hours of 10:00 A.M. and 4:00 P.M. At an average price of \$14 per diner, the deli's manager projects that 100 diners per evening would take advantage of the new service. Projected costs for each night the deli is open are:

Cost Category Costs
(per hour)
Four waitresses' and busboy's salaries (each \$10
Two cooks salaries (each) 20
Allocated fixed overhead (lease expenses, insurance, etc.) 80

A. Would the new weekend service be profitable?

B. Calculate the breakeven price per dinner for evening dining at a projected attendance of 100 dinners.

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5. Costs of Regulation. The Appalachian Coal Company sells coal to electric utilities in the southeast. Unfortunately, Appalachian's coal has high particulate content and, therefore, the company is adversely affected by state and local regulations governing smoke and dust emissions at its customer's electricity-generating plants. Appalachian's total cost and marginal cost relations are:

TC = \$250,000 + \$5Q + \$0.0002Q2

MC = &#61508;TC/&#61508;Q = \$5 + \$0.0004Q

where Q is tons of coal produced per month and TC includes a normal rate of return on investment.

A. Calculate Appalachian's profit at the profit-maximizing activity level if prices in the industry are stable at \$25 per ton, and therefore P = MR = \$25.

B. Calculate Appalachian's optimal price, output, and profit levels if a new state regulation results in a \$300,000 fixed cost increase that cannot be passed onto customers.

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