**why is marginal revenue less than price** This is a topic that many people are looking for. **khurak.net** is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, ** khurak.net ** would like to introduce to you **Proof: Marginal Revenue less than Price**. Following along are instructions in the video below:

“In this video. We re gonna do a mathematical proof. Showing that marginal revenue is is less than price. When demand slopes downwards.

So we start with a general form total revenue equals the price times. Quantity. The change in total revenue is approximated as a following and this here is just a total differential. So we need to get the partial derivative of total revenue with respect to price and plug it in here and we need to get the partial derivative of total revenue with respect to quantity and plug it in over here.

So here s our partial derivatives..

Partial derivative total revenue with respect to price is q partial derivative of total revenue with respect to quantity is p. And plugging those results into the delv equation. We have our total differential. Showing the change in total revenue is approximated as the quantity times.

A change in price plus. The price times a change in quantity. So we want to show that when demand slopes downwards and the demand curve slopes downwards marshal revenue is less than price. So i m going to divide now both sides by dq so dividing everything through by dq.

So we ll just get p right here..

Do you q. Divided by dq is p. And then we ll get dp divided by dq right over there. And that s being all multiplied by q.

And then the left hand side the same thing and will recognize the left hand side as marginal revenue. The derivative of total revenue with respect to quantity is marginal revenue. So making that substitution we got an expression for marginal revenue. Now let s start with the case.

Where we have a horizontal and perfectly elastic demand curve..

When we have a perfectly elastic or horizontal demand curve. The slope of that demand curve. Which is given by this right here derivative of the price with respect to quantity just zero derivative of the horizontal line is zero. So plugging zero into this equation means marginal revenue equals price.

So we just proved that for a perfectly competitive firm that marginal revenue will indeed equal price. Now let s go to the downward sloping demand part with a downward sloping demand curve. The slope of a downward sloping demand. Curve is negative.

So the derivative of the inverse demand or the price equation with respect to quantity is less than zero..

It s negative. So what is marginal revenue marginal revenue is price minus something over here so since dp dq over here is a negative number. We re gonna be taking away from price so marginal revenue will be less than price. And that s our proof.

Okay i hope you ” ..

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