# Calculating the profit maximizing price and output levels

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.

Monopoly

A high-end lamp monopolist operates in the Mid-West where the demand for lamps is given by Q1=200-P1. Producing one lamp costs 10 per unit.

(a) Derive the profit maximizing price and the profits at this price.

(b)What is the demand elasticity at this price?

Suppose now that the monopolist has the opportunity to expand to the East Coast.This would entail launching an advertising campaign at a cost of 1000,a one time expense. The demand on the East Coast is given by Q2 =160-3P2.The per-unit cost of selling lamps on the East Coast is identical to the cost of selling them in the Midwest.Suppose first that,because he is thinking of selling from his website,the monopolist must charge the same price in both markets .

(c) What is the total demand when the monopolist charges a price P?

(d) Derive the profit maximizing price and the profits at this price in the case where

the monopolist must charge the same price in both markets.Would you recommend the monopolist to expand to this market?

Now suppose that the monopolist will sell through a network of distributors,and can charge different prices on the East Coast and in the Mid West .

(e)What price would the monopolist charge in the Mid-West?What price would the

monopolist charge in the East?What are the total profits?Would yo recommend the monopolist to expand in this case?