Calculating Marginal Revenue: Qx-y Formula

In summary, Marginal revenue is the additional revenue that will be generated by increasing product sales by one unit. The cost to make one unit of a product is a function of the number of units that are sold, so marginal revenue is the amount of money that is earned by selling an additional unit of the product.
  • #1
Karol
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Homework Statement


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Homework Equations


marginal revenue[/B] (R') is the additional revenue that will be generated by increasing product sales by one unit

The Attempt at a Solution


I don't know how to start. Q is the number of items sold at price x. y is the marginal cost, the cost of producing one item. N is the net profit, the revenue (my english isn't perfect so i explain each variable) ##~N=Q(x-y)~##.
The derivative N' is the marginal revenue, no?
Q, x and y are variables. x and y can change according to Q, because if i produce more i can lower the price i charge (x) and also the cost y.
How do i express marginal revenue?
 

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  • #2
Quantity sold/quantity produced ## Q=Q(P) ##, meaning ## Q ## is a function of ## P ##, where ## Q## is the quantity sold at price ##P ##. ## \\ ## Revenue ## R=Q \cdot P ##. ## \\ ## The cost to make ## Q ## items is a function of ## Q ##, so that ## C=C(Q) ##. ## \\ ## Net profit ## N=R-C ##. ## \\ ## Since ## Q=Q(P) ##, we can also write it in the form ## P=P(Q) ##, i.e. ## P ## is a function of ## Q ##. Thereby, revenue ## R=Q \cdot P=Q \cdot P(Q) ## is a function of ## Q ##, so we can write ## R=R(Q) ##. ## \\ ## We see that ## N=N(Q)=R(Q)-C(Q) ##. ## \\ ## How do we maximize ## N ##, using calculus? Also, what is the definition of marginal revenue, using calculus? And what is the definition of marginal cost, using calculus? ## \\ ## Once we find the ## Q ## where marginal revenue =marginal cost, we can then go to the function ## P=P(Q) ##, the inverse function of ## Q=Q(P) ##, to determine what price we should charge. The net profit ## N ## will be maximized at that selling price.
 
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  • #3
Marginal revenue=R':
$$R=Q \cdot P=Q \cdot P(Q)~\rightarrow~R'=P(Q)+Q\cdot P'(Q)$$
Marginal cost=C': ##~C=C(Q)~\rightarrow~C'=C'(Q)##
Marginal revenue=marginal cost: ##~P(Q)+Q\cdot P'(Q)=C'(Q)##
It doesn't help. C=C(Q) needn't be ##~C=Q\cdot k~##, the more items you make it's cheaper for each item.
And also, i have to prove that when Marginal revenue=marginal cost N is biggest
 
  • #4
The maximum or minimum in ## N ## occurs when ## \frac{dN}{dQ}=0 ##. Buy the above, this clearly is when ## \frac{dR}{dQ}-\frac{dC}{dQ}= 0 ## so that ## \frac{dR}{dQ}=\frac{dC}{dQ} ##. The marginal revenue calculus definition is ## \frac{dR}{dQ} ##. Similarly, the marginal cost calculus definition is ## \frac{dC}{dQ} ##. I basically gave you the complete proof... ## \\ ## I have not proven that it is indeed a maximum. If you can show ## \frac{d^2 N}{dQ^2} <0 ##, then it is a maximum.
 
  • #5
Thank you Charles, this is the proof.
I am not familiar with economics and i felt uncomfortable with this question
 
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FAQ: Calculating Marginal Revenue: Qx-y Formula

1. What is marginal revenue?

Marginal revenue is the additional revenue a company earns by selling one more unit of a product or service.

2. How is marginal revenue calculated?

Marginal revenue is calculated using the formula Qx-y, where Q represents the change in quantity and y represents the change in price.

3. Why is it important to calculate marginal revenue?

Calculating marginal revenue is important because it helps a company determine the optimal price and quantity to maximize profits. It also provides insights into consumer demand and market trends.

4. What is the relationship between marginal revenue and total revenue?

Marginal revenue is the change in total revenue divided by the change in quantity. This means that marginal revenue is a part of total revenue and is influenced by it.

5. Can marginal revenue ever be negative?

Yes, marginal revenue can be negative if the change in quantity results in a decrease in total revenue. This can happen when the price elasticity of demand is low, and increasing the price leads to a decrease in demand for the product.

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