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# Calculating the profit maximizing price and output levels

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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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#### Solution Preview

Please refer attached file/s for complete solutions. Expressions typed with the help of equation writer are missing here.

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?

q TC MC*
0 1800
1 2651 851
2 3428 777
3 4137 709
4 4784 647
5 5375 591
6 5916 541
7 6413 497
8 6872 459
9 7299 427
10 7700 401
11 8081 381
12 8448 367
13 8807 359
14 9164 357
15 9525 361
16 9896 371
17 10283 387
18 10692 409
19 11129 437
20 11600 471
21 12111 511
22 12668 557
23 13277 609
24 13944 667

* MC=change in TC/change in output
Please refer attached Excel file for calculations.

A firm will increase its output level as long as MC is less than Marginal Revenue (Price, in this case).
We find that for a ...

#### Solution Summary

There are 6 problems. Solutions to these problems explain the methodology to find profit maximizing output and price levels.

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