1. Suppose that a monopoly has a demand curve given by QD(p) = 2400−3p. The cost function for the monopoly firm is c(Q) = 5Q2.

(a) Find the monopolist’s profit-maximizing quantity.

(b) Find the monopolist’s profit-maximizing price.

(c) For practice, answer the questions above three ways:

i. Use marginal revenue. (This is what we’ll do most often.)

ii. Use a function for profit based on price, π(p).

iii. Use a function for profit based on quantity, π(Q).

(d) For part (ii), use second-order conditions to verify you are finding the maximum profit (rather than the minimum profit).

(e) For part (iii) above, use second-order conditions to verify you are finding the maximum profit (rather than the minimum profit).

(f) Show this solution graphically. Answer the following questions assuming the monopolist is profit-maximizing.

(If your graph gets too messy, you may want to show the different parts on more than one graph.)

i. On the graph, show the area for consumer surplus (CS).

ii. What is CS? (You’ll probably want a calculator for this one.)

iii. On the graph, show the area for the monopolist’s revenue. iv. On the graph, show the area for producer surplus (CS).

2. Thinking about equilibrium:

(a) If a system is in equilibrium then . . . (check all that apply)

⃝ . . . the system will always try to come back to that state. ⃝ . . . the system won’t change without an exogenous force. ⃝ . . . nobody is making a profit.

⃝ . . . somebody is making a profit.

⃝ . . . the system will tend to run in cycles.

⃝ . . . all the forces on the system cancel each other out.

(b) Suppose that a monopolist has MR = $25 and MC = $15. Is this business in equilibrium? ⃝ Yes ⃝ No

Explain, using the ideas above.

(c) What if MR = $25 and MC = $15 for a firm in a perfectly competitive market?

3. A monopoly faces the demand function QD(p) = (3 − p)2. The firm has a constant marginal cost MC = $1

in the short run.

(a) Find the function for marginal revenue. (b) Suppose the firm sets a price p = $2.

i. How much will the firm sell at that price?

ii. What is the marginal revenue at that price?

iii. Based on your answer to part (ii), how much would the firm like to sell? ⃝ a greater quantity ⃝ the same quantity ⃝ a smaller quantity Explain your answer

4. Suppose that a monopoly has a demand curve given by QD(p) = a − bp, and cost function c(Q) = F + gQ. (a) Find the function for marginal revenue.

i. Use the marginal revenue function you found to say what the marginal revenue function would be if the demand function was QD (p) = 1000 − 1 p.

2

ii. Use the marginal revenue function you found to say what the marginal revenue function would be if the demand function was QD (p) = 900 − 3p.

(b) Find the function for marginal cost.

(c) Find the monopolist’s profit-maximizing quantity.

(d) Find the monopolist’s profit-maximizing price.

(e) Consider changes in b. (And remember, as economists we normally put price on the vertical axis and quantity on the horizontal axis.)

i. A change in b is (choose one): ⃝ an equilibrium change

⃝ a first-order change ⃝ a second-order change ⃝ an exogenous change ⃝ an endogenous change ⃝ a normative change ⃝ a positive change

ii. If b gets larger, then (check all that apply): ⃝ the demand curve gets steeper

⃝ the demand curve gets less steep

⃝ the market becomes more competitive ⃝ the market becomes less competitive ⃝ demand becomes more elastic

⃝ demand becomes less elastic

⃝ QM increases

⃝ QM decreases

(f) Consider changes in the cost function c(Q) (Look out here for questions some might consider “trick questions”.)

i. If g gets larger, then (check all that apply):

⃝ the marginal cost curve gets steeper ⃝ the marginal cost curve gets less steep ⃝ pM increases

⃝ pM decreases

⃝ QM increases

⃝ QM decreases

ii. If F gets larger, then (check all that apply):

⃝ the marginal cost curve gets steeper ⃝ the marginal cost curve gets less steep ⃝ pM increases

⃝ pM decreases

⃝ QM increases

⃝ QM decreases

￼Page 2

(a) Find the monopolist’s profit-maximizing quantity.

(b) Find the monopolist’s profit-maximizing price.

(c) For practice, answer the questions above three ways:

i. Use marginal revenue. (This is what we’ll do most often.)

ii. Use a function for profit based on price, π(p).

iii. Use a function for profit based on quantity, π(Q).

(d) For part (ii), use second-order conditions to verify you are finding the maximum profit (rather than the minimum profit).

(e) For part (iii) above, use second-order conditions to verify you are finding the maximum profit (rather than the minimum profit).

(f) Show this solution graphically. Answer the following questions assuming the monopolist is profit-maximizing.

(If your graph gets too messy, you may want to show the different parts on more than one graph.)

i. On the graph, show the area for consumer surplus (CS).

ii. What is CS? (You’ll probably want a calculator for this one.)

iii. On the graph, show the area for the monopolist’s revenue. iv. On the graph, show the area for producer surplus (CS).

2. Thinking about equilibrium:

(a) If a system is in equilibrium then . . . (check all that apply)

⃝ . . . the system will always try to come back to that state. ⃝ . . . the system won’t change without an exogenous force. ⃝ . . . nobody is making a profit.

⃝ . . . somebody is making a profit.

⃝ . . . the system will tend to run in cycles.

⃝ . . . all the forces on the system cancel each other out.

(b) Suppose that a monopolist has MR = $25 and MC = $15. Is this business in equilibrium? ⃝ Yes ⃝ No

Explain, using the ideas above.

(c) What if MR = $25 and MC = $15 for a firm in a perfectly competitive market?

3. A monopoly faces the demand function QD(p) = (3 − p)2. The firm has a constant marginal cost MC = $1

in the short run.

(a) Find the function for marginal revenue. (b) Suppose the firm sets a price p = $2.

i. How much will the firm sell at that price?

ii. What is the marginal revenue at that price?

iii. Based on your answer to part (ii), how much would the firm like to sell? ⃝ a greater quantity ⃝ the same quantity ⃝ a smaller quantity Explain your answer

4. Suppose that a monopoly has a demand curve given by QD(p) = a − bp, and cost function c(Q) = F + gQ. (a) Find the function for marginal revenue.

i. Use the marginal revenue function you found to say what the marginal revenue function would be if the demand function was QD (p) = 1000 − 1 p.

2

ii. Use the marginal revenue function you found to say what the marginal revenue function would be if the demand function was QD (p) = 900 − 3p.

(b) Find the function for marginal cost.

(c) Find the monopolist’s profit-maximizing quantity.

(d) Find the monopolist’s profit-maximizing price.

(e) Consider changes in b. (And remember, as economists we normally put price on the vertical axis and quantity on the horizontal axis.)

i. A change in b is (choose one): ⃝ an equilibrium change

⃝ a first-order change ⃝ a second-order change ⃝ an exogenous change ⃝ an endogenous change ⃝ a normative change ⃝ a positive change

ii. If b gets larger, then (check all that apply): ⃝ the demand curve gets steeper

⃝ the demand curve gets less steep

⃝ the market becomes more competitive ⃝ the market becomes less competitive ⃝ demand becomes more elastic

⃝ demand becomes less elastic

⃝ QM increases

⃝ QM decreases

(f) Consider changes in the cost function c(Q) (Look out here for questions some might consider “trick questions”.)

i. If g gets larger, then (check all that apply):

⃝ the marginal cost curve gets steeper ⃝ the marginal cost curve gets less steep ⃝ pM increases

⃝ pM decreases

⃝ QM increases

⃝ QM decreases

ii. If F gets larger, then (check all that apply):

⃝ the marginal cost curve gets steeper ⃝ the marginal cost curve gets less steep ⃝ pM increases

⃝ pM decreases

⃝ QM increases

⃝ QM decreases

￼Page 2

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