1. Assume that a competitive firm has the total cost function: TC = 1q3 - 40q2 + 840q + 1800 Suppose the price of the firm's output (sold in integer units) is $750 per unit. Using calculus and formulas (but no tables and restricting your use of spreadsheets to implementing the quadratic formula) to find a solution, what is the total profit at the optimal output level? Please specify your answer as an integer. Hint: The first derivative of the total cost function is the marginal cost function. Set the marginal cost equal to the marginal revenue (price in this case) to define an equation for the optimal quantity q. Rearrange the equation to the quadratic form aq2 + bq + c = 0. Use the quadratic formula to solve for q: For non-integer quantity, round up and down to find the optimal value. 2. Suppose a competitive firm has as its total cost function: TC = 23 + 3q2 Suppose the firm's output can be sold (in integer units) at $72 per unit. Using calculus and formulas (but no tables or spreadsheets) to find a solution, what is the total profit at the optimal output level? Please specify your answer as an integer. 3. Assume that a monopolist faces a demand curve for its product given by: p = 120 - 3q Further assume that the firm's cost function is: TC = 420 + 11q Using calculus and formulas (but no tables or spreadsheets) to find a solution, what is the profit (rounded to the nearest integer) for the firm at the optimal price and quantity? Round the optimal quantity to the nearest hundredth before computing the optimal price, which you should then round to the nearest cent. Note: Non-integer quantities may make sense when each unit of q represents a bundle of many individual items. Hint: Define a formula for Total Revenue using the demand curve equation. Then take the derivative of the Total Revenue and Total Cost formulas. Use these derivative equations to perform a marginal analysis. 4. Assume that the demand curve D(p) given below is the market demand for apples: Q = D(p) = 260 - 10p, p > 0 Let the market supply of apples by given by: Q = S(p) = 46 + 7p, 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 price in this market? Round the equilibrium price to the nearest cent. 5. The demand curve for tickets at an amusement park is: Q = D(p) = 1000 - 42p, p > 0 The marginal cost of serving a customer is $12. Using calculus and formulas (but no tables or spreadsheets) to find a solution, what is the profit-maximizing price? Round the equilibrium quantity DOWN to its integer part and round the equilibrium price to the nearest cent. 6. Global Corp. sells its output at the market price of $7 per unit. Each plant has the costs shown below: Units of Output Total Cost ($) 0 10 1 12 2 16 3 22 4 30 5 40 6 52 How much output should each plant produce? Please specify your answer as an integer. 7. Suppose that you can sell as much of a product (in integer units) as you like at $60 per unit. Your marginal cost (MC) for producing the qth unit is given by: MC=7q This means that each unit costs more to produce than the previous one (e.g., the first unit costs 7*1, the second unit (by itself) costs 7*2, etc.). If fixed costs are $100, what is the profit at the optimal output level? Please specify your answer as an integer. 8. Assume that a competitive firm has the total cost function: TC = 1q3 - 40q2 + 820q + 1900 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, how many units should the firm produce to maximize profit? Please specify your answer as an integer.

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

TC = 1q3 - 40q2 + 840q + 1800

Suppose the price of the firm's output (sold in integer units) is $750 per unit.

Using calculus and formulas (but no tables and restricting your use of spreadsheets to implementing the quadratic formula) to find a solution, what is the total profit at the optimal output level?

Please specify your answer as an integer.

Hint: The first derivative of the total cost function is the marginal cost function.

Set the marginal cost equal to the marginal revenue (price in this case) to define an equation for the optimal quantity q.

Rearrange the equation to the quadratic form aq2 + bq + c = 0.

Use the quadratic formula to solve for q:



For non-integer quantity, round up and down to find the optimal value.




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

TC = 23 + 3q2

Suppose the firm's output can be sold (in integer units) at $72 per unit.

Using calculus and formulas (but no tables or spreadsheets) to find a solution, what is the total profit at the optimal output level?

Please specify your answer as an integer.




3. Assume that a monopolist faces a demand curve for its product given by:

p = 120 - 3q

Further assume that the firm's cost function is:

TC = 420 + 11q

Using calculus and formulas (but no tables or spreadsheets) to find a solution, what is the profit (rounded to the nearest integer) for the firm at the optimal price and quantity?

Round the optimal quantity to the nearest hundredth before computing the optimal price, which you should then round to the nearest cent. Note: Non-integer quantities may make sense when each unit of q represents a bundle of many individual items.

Hint: Define a formula for Total Revenue using the demand curve equation. Then take the derivative of the Total Revenue and Total Cost formulas. Use these derivative equations to perform a marginal analysis.




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

Q = D(p) = 260 - 10p, p > 0

Let the market supply of apples by given by:

Q = S(p) = 46 + 7p, 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 price in this market?

Round the equilibrium price to the nearest cent.



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

Q = D(p) = 1000 - 42p, p > 0

The marginal cost of serving a customer is $12.

Using calculus and formulas (but no tables or spreadsheets) to find a solution, what is the profit-maximizing price?

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




6. Global Corp. sells its output at the market price of $7 per unit. Each plant has the costs shown below:
Units of Output Total Cost ($)
0 10
1 12
2 16
3 22
4 30
5 40
6 52
How much output should each plant produce?

Please specify your answer as an integer.

7. Suppose that you can sell as much of a product (in integer units) as you like at $60 per unit. Your marginal cost (MC) for producing the qth unit is given by:

MC=7q

This means that each unit costs more to produce than the previous one (e.g., the first unit costs 7*1, the second unit (by itself) costs 7*2, etc.).

If fixed costs are $100, what is the profit at the optimal output level?

Please specify your answer as an integer.

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

TC = 1q3 - 40q2 + 820q + 1900

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, how many units should the firm produce to maximize profit?

Please specify your answer as an integer.