VIEWS: 16 PAGES: 5 POSTED ON: 5/26/2011
1 9. Revenue Essay: 1. Define marg inal revenue. Describe the relationship between marg inal 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, marg inal 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 fro m selling a product. It is the total amount buyers pay. It is equal to the price t imes 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, reve nue 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 ext reme situation where a firm can unlimited amounts at the going market price.) The marg inal 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 o f derivatives Derivative of a d ifference 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 simp le demand function is linear (as it will be in this course,) the marginal revenue function is easy to find fro m the price function. Just mu ltiply 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 quant ity. 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 marg inal revenue with the demand curve. If the demand curve is linear, then marginal revenue and demand start at the same point but then the marg inal 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 fro m the orig in (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 marg inal revenue must be negative. Revenue is decreasing, so the change in revenue is negative--a decrease. Since revenue increases and then decreases, marg inal 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 functi on 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 maximu m revenue, just mult iply the price times the quantity — R = 32530 * 1626.50 = 52,910,045