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### Course: AP®︎/College Microeconomics>Unit 4

Lesson 2: Monopoly

# Monopolist optimizing price: Marginal revenue

Learn about marginal revenue for a monopolist. We find the slope of the total revenue curve to determine marginal revenue at different quantities, and discover that the marginal revenue curve is a downward-sloping line with twice the slope of the demand curve. This helps us understand how monopolists can optimize their profit by comparing marginal revenue to marginal costs. Created by Sal Khan.

## Want to join the conversation?

• isn't MR just like elasticity??
• In a way, yes. You could say that the elasticity of demand determines the slope of the MR-curve. The MR-curve is the expected revenue, so the quantity demanded times the price paid for it summed up and given per extra unit.

The elasticity curve determines the quantity demanded for every price change, whilst the MR-curve visualizes it per quantity change (extra unit).
• There are many calculus videos on this site, which one specifically should I watch to do through calculus what Sal did algebraically?
• Exactly, as Jan said.

Make use of the FOC (First-order-condition) which is the derivative equals 0.
You know the demand function is P = 6 - Q and the TR = P * Q.

Substitute P in the TR-formula, gives you TR = (6 - Q) * Q
TR = -Q^2 + 6Q

Now use the FOC... TR' = -2Q + 6 = 0
-2Q = -6
2Q = 6
Q = 3
• what is the relationship between shortrun and monopoly
• There is no distinction between short run and long run with monopolies. So long as the firm remains the only producer in the market and the market does not experience any shifts, the supply, demand, and optimum production quantity will remain the same.
• @ Why the MR has twice slope of demand in the case of monopolist alone ? Why it is not valid for competitive market ? If it is due to some equilibrium price, surely in case of monopoly too, we can have supply curve (cost curve) & can have some equilibrium price.
• I understand how sal gets the MR but it makes no sense to me about the result
when Q=1 MR=4 TR=5 then the next incremental unit should add TR by 4 than why the TR when Q=2 is 8 , It's so hard to believe for me
• When Q=1 and MR=4 the TR increases by 4 times as much as a very, very small change in quantity. For example: an increase in Q from 1 to 1.001 will increase the total revenue by approximately 4 * 0.001 = ~0.004, making the TR 5 + ~0.004 = ~5.004.

However, and here it goes wrong with your thinking, when I want to add a little bit more quantity the marginal revenue is no longer 4. It's now slightly lower (at Q=1.001 the MR is 3.998). That decrease in marginal revenue will continue. By the time Q=2, MR dropped to 2.

Because the MR-curve is a straight line it's safe to say the average MR in between Q=1 and Q=2 is (4 + 2) / 2 = 3, which makes the new TR 5 + 3 * 1 (the change in quantity) = 8.
• So that the total revenue = the area under the demand curve = integral of the demand curve,
and marginal revenue = the inst. slope of total revenue curve = derivative of the TR curve
does it mean that the MR curve = the demand curve?
• I have checked several sources and I am just very stuck: if you have a firm that is a monopoly what would cause it to stop production? I know that it stops production when marginal revenue is less than marginal cost, but if you are neglecting cost, what else would cause a monopoly to stop production?
• A firm doesn't stop production when MR < MC (it will just produce less). The condition you're referring to—the shutdown condition—is when price is less than average variable cost.

Since a monopoly is the only producer of a good in a market, it's difficult to think of non-cost reasons why they would stop producing. I keep thinking of examples, but they're all cost-related. Maybe barriers to entry (such as a patent) go away, and new entrants drive the original firm out of business. Alternatively, this good might be replaced by another, decreasing demand so much that the firm drops out of production altogether.
• I am studying in China and not great at Mandarin yet so I'm struggling through some classes.
I have a question "A producer of oil lamps estimates the following demand function for its product:
Q=120,000 - 10,000P
where Q is the quantity demanded per year and P is the price per lamp. Fixed costs are \$12,000 and variable costs are \$1.50 per lamp."
I need to write an equation for the total revenue function in terms of Q
Specify the marginal revenue function.
Write an equation for the total cost function in terms of Q
Specify the marginal cost function.
Write an equation for total profits in terms of Q. At what level are total profits maximized? What price will be charged? what are total profits at this output level?
Check answers by equating the MR and MC functions, and solve for Q

Can someone help me out?
• I was wondering this about the algebra:

If the area function (integral) underneath the demand (AR) curve is the total revenue TR curve and the derivative (gradient function) is MR curve then why are they different? Surely integrating the AR -> TR then differentiating it should produce the AR again?

Very confused, would appreciate an explanation perhaps algebraically?
(1 vote)
• Total revenue is not the area under the demand (AR) curve. Total revenue is the price times the quantity. TR would be the integral of the MR function (in which case, I'm guessing the rest of your question will now make more sense—integrating MR -> TR and then differentiating it will produce MR again).