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# The answer to calculating optimal output and profit

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Calvin's Barbershop is a popularly-priced hair cutter on the south side of Chicago. Given the large number of competitors, the fact that barbers routinely tailor services to meet customer needs, and the lack of entry barriers, it is reasonable to assume that the market is perfectly competitive and that the average \$15 price equals marginal revenue, P = MR = \$15. Furthermore, assume that the barbershop's monthly operating expenses are typical of the 50 barbershops in the local market and can be expressed by the following total and marginal cost functions:

TC = \$7,812.50 + \$2.5Q + \$0.005Q2

MC =dTC/dQ = \$2.5 + \$0.01Q

where TC is total cost per month including capital costs, MC is marginal cost, and Q is the number of hair cuts provided. Total costs include a normal profit.

A. Calculate Calvin's profit-maximizing output level.
B. Calculate the Calvin's economic profits at this activity level. Is this activity level sustainable in the long run?

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#### Solution Preview

A. Calculate Calvin's profit-maximizing output level.

MC = 2.5 + 0.01Q
Average Price=P=\$15
Put MC=P for profit maximization
2.5+0.01Q=15
0.01Q=12.5
Q=1250

B. Calculate ...

#### Solution Summary

Solution describes the steps to calculate optimal output level and economic profit of a perfectly competitive firm.

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## Managerial Economics

1. Assume that a competitive firm has the total cost function:

TC = 1q3 - 40q2 + 890q + 1800

Suppose the price of the firm's output (sold in integer units) is \$600 per unit.

Using tables (but not calculus) to find a solution, what is the total profit at the optimal output level?

Please specify your answer as an integer.

2. Assume that a competitive firm has the total cost function:

TC = 1q3 - 40q2 + 880q + 2000

Suppose the price of the firm's output (sold in integer units) is \$550 per unit.

Using calculus and formulas (but no tables and restricting your use of spreadsheets to implementing the quadratic formula) to find a solution, how many units should the firm produce to maximize profit?

Please specify your answer as an integer.

3. Suppose a competitive firm has as its total cost function:

TC = 17 + 2q2

Suppose the firm's output can be sold (in integer units) at \$57 per unit.

Using calculus and formulas (but no tables or spreadsheets) to find a solution, how many units should the firm produce to maximize profit?

Please specify your answer as an integer. In the case of equal profit from rounding up and down for a non-integer initial solution quantity, enter the higher quantity.

4. Assume that the demand curve D(p) given below is the market demand for apples:

Q = D(p) = 280 - 13p, p > 0

Let the market supply of apples by given by:

Q = S(p) = 44 + 5p, p > 0

where p is the price (in dollars) and Q is the quantity. The functions D(p) and S(p) give the number of bushels (in thousands) demanded and supplied.

What is the equilibrium quantity in this market?

Round the equilibrium price to the nearest cent and round the equilibrium quantity DOWN to its integer part.

5. The demand curve for tickets at an amusement park is:

Q = D(p) = 1200 - 49p, p > 0

The marginal cost of serving a customer is \$18.

Using calculus and formulas (but no tables or spreadsheets) to find a solution, how many tickets will be sold at the profit-maximizing price?

Round the equilibrium quantity DOWN to its integer part and round the equilibrium price to the nearest cent.

6. Assume that the demand curve D(p) given below is the market demand for apples:

Q = D(p) = 280 - 13p, p > 0

Let the market supply of apples by given by:

Q = S(p) = 44 + 5p, p > 0

where p is the price (in dollars) and Q is the quantity. The functions D(p) and S(p) give the number of bushels (in thousands) demanded and supplied.

What is the equilibrium quantity in this market?

Round the equilibrium price to the nearest cent and round the equilibrium quantity DOWN to its integer part.

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