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9. Revenue
Essay:
1. Define marginal revenue. Describe the relationship between marginal revenue and changes in total revenue as
output changes. What is true of marginal revenue when total revenue is at its highest level?

Problems and Diagrams

2.     Given a demand function, find the revenue function and marginal revenue function. Given an output, find total
       revenue and marginal revenue.
3.     Find the level of output that maximizes total revenue.
4.     Draw a total revenue curve. Show total revenue, marginal revenue, and price for a given output.
5.     Draw a demand and marginal revenue curve. Show marginal revenue, price, and total revenue for a given
       output.

Revenue
   Revenue is the amount of money a firm earns from selling a product. It is the
total amount buyers pay. It is equal to the price times the quantity.

Revenue = price * quantity

     The revenue function relates total revenue earned by a firm to the firm sells.

Revenue function = Q*price function.

R = Q* (3253 - .05 Q)

R = 3253Q-.05Q2

   If you are given a quantity, you can substitute into this revenue function and find the most revenue that can be
earned selling this quantity.

     Suppose the quantity (or output) is 10000. Then the revenue the firm earns is:

R = 3,253 * (10,000) - .05 * (10000)^2
 = 32,530,000 - .05* 100,000,000
 = 32,530,000 – 5,000,000
 = 27,530,000

The Revenue Curve
    The revenue curve shows the relationship between total revenue and quantity.

                             R


              R 27,530,000




                                     Q                       Q
                                     10,000

   You can see the revenue curve at first rise and then fall. The highest point represents the highest revenue
possible. The quantity providing that revenue is on the horizontal axis.
                                                                                                                            2
     Why does revenue at first increase and then decrease? Well, when nothing is sold, revenue is zero. As output
and sales rise, revenue rises, but because a lower price must be charged to sell more output, this slows the rate of
increase in revenue. Eventually, this need to lower price offsets the consequences of the increased amount sold so
that on net, revenue actually drops. If one tries to sell such a large amount that no one will purchase another unit at
any price above zero, then increasing output requires that the product be given away so that revenue is again zero.
Marginal Revenue
    Marginal revenue is the change in revenue as the firm sells another unit of output. Because the firm has to cut
its price to sell another unit, it makes that price on the additional unit, but earns less on all the units it could have
sold at the higher price. Because of this effect on the amount earned on other units, the marginal revenue will
always be less than the price. (Well, except in the extreme situation where a firm can unlimited amounts at the
  going market price.)
     The marginal revenue function is the first derivative of the revenue function.

R = 3253Q - .05Q2

MR= 3253-.1Q

Some simple rules--

Q = Q1

Derivative of sum is sum of derivatives

Derivative of a difference is equal to the difference of the derivatives.

Power rule    Derivative of aXb = b*a*Xb-1

Q0 = 1

   So using these rules--

R = 3253Q1 - .05Q2

MR = 1*3253Q1-1 - 2*.05Q2-1
   = 3253Q0 - .1Q1
   = 3253*1 - .1Q
   = 3253 -.1Q


   If the simple demand function is linear (as it will be in this course,) the marginal revenue function is easy to find
from the price function. Just multiply the coefficient on quantity by 2.

P = 3253 - .05Q

MR = 3253 – 2*.05Q

MR = 3253 - .1Q


This function will tell you the marginal revenue for any given quantity. For example, if
Q= 10000--

MR = 3253 - .1Q
    = 3253 - .1(10000)
   = 3253 – 1000
                                                                                                                  3
      = 2253
Marginal Revenue Curve
    The marginal revenue curve illustrates the relationship between marginal revenue and quantity. It is
conventional to show marginal revenue with the demand curve. If the demand curve is linear, then marginal
revenue and demand start at the same point but then the marginal revenue curve falls off faster than the demand
curve. The point where the marginal revenue curve crosses the quantity axis is ½ of the distance to where the
demand curve crosses the quantity axis.




       P 3253
      MR 2253


                              MR        D

                     Q                          Q
                     10,000



       The total revenue can be shown on the diagram above, but it is an area. Since revenue is price times
quantity, it is equal to the area of the rectangle that has a length of price and width of quantity.




         P 3253
       MR 2253


                                MR          D

                       Q                            Q
                       10,000

  The shaded area represents total revenue.

   The price and marginal revenue can be shown on the total revenue diagram.

                                   R
                                                        MR
                   R 27,530,000


                                              P


                                                    Q          Q
                                                    10000
                                                                                                                       4
   The slope of a line tangent to the revenue curve at the given quantity is the marginal revenue of that quantity.
And the slope of a line from the origin (zero) to the total revenue curve at the given quantity is the price of that
quantity


                      R

        R 52,910,045




                                    Q                  Q
                                    32,530
                                                                                                                  5


Revenue-Maximizing Output
    As long as revenue is increasing with quantity, then the marginal revenue must be positive. That is, the
change in revenue from a one unit increase in output is a positive number because revenue is going up. If
revenue is decreasing with quantity, then the marginal revenue must be negative. Revenue is decreasing,
so the change in revenue is negative--a decrease.
     Since revenue increases and then decreases, marginal revenue must start out positive and become
negative. The point where revenue is at its highest will be the point where marginal revenue is zero. This
can be discovered by setting the marginal revenue function equal to zero.

MR = 0
3253-.1Q = 0
3253 = .1Q
Q = 3253/.1
 Q = 32530


  To find the highest price that can be charged and still sell that output, simply substitute it into the price
function.

P = 3253 - .05*(32530)

=   1626.50

 To find the maximum revenue, just multiply the price times the quantity—

R = 32530 * 1626.50
 = 52,910,045

				
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