Price elasticity of demand and revenue implications

1. Price elasticity of demand and revenue implications Often in economics we look at how the value of one variable changes when another variable changes. The concept called elasticity is a summary statement about those changes.

2. Percentage change examples Say you put $1 in the bank and at the end of the year you have $2 (this is a special bank!). The percentage change in the amount of money in your account is found by the basic formula (end value – start value) divided by start value and we have (2 – 1)/1 = 1 (and you could multiply by 100 to say 100 percentage increase). As another example say you buy a share of stock for $10 and at the end of 1 year the stock has a price of $5. The percentage change in the price of the stock (5 – 10)/10 = -0.5. The minus sign here is important because it is an indication that the stock price went down (and here the fall is 50%).

3. Demand in a table of numbers P A 11 0 10 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1 10 0 11 Sometimes as a student you are faced with demand in a table of numbers. From the table other concepts may be developed. One such concept is the price elasticity of demand. Note the idea of price elasticity is calculated as the ratio of 2 separate percentage change calculations.

4. Elasticity The law of demand or the law of supply is a statement about the direction of change of the quantity demanded, or supplied, respectively, when there is a price change. The concept of elasticity adds to these concepts by indicating the magnitude of the change in quantity, given the price change. The magnitude of the change is reported in percentage terms.

5. price elasticity of demand Many textbooks define the price elasticity of demand as ε= (% change in Q)/(% change in P) by definition. Remember in our text the Q or quantity in this context is referred as A for attendance. Let’s look at the demand values (in P, A order) from our table (10, 1) and (9, 2). If (10, 1) is the start value The percentage change in A is (2 – 1)/1 = 1/1 (or just 1) and the percentage change in P is (9 – 10)/10 = -1/10. Putting the 2 together means (1/1)/(-1/10) = (1/1) times (-10/1) = -10.

6. Elasticity example P As the price falls from 10 to 9, the quantity rises from 1 to 2. The elasticity of demand is (2-1)/1 (9-10)/10 = - 10/1 10 9 D Q 1 2 I show the same data points here in a graph, but we really do not know the exact math form of the demand. The graph is really just a visual representation of our table.

7. Please note the following As you look at the example in the table you see that whenever the price falls the quantity demanded rises. Or, whenever the price rises the quantity demanded falls. This is a reflection of the law of demand. Because of the law of demand the 2 percentage changes we calculate to get the elasticity will always have opposite numerical signs and thus the ratio will always be negative. For this reason I choose to define the price elasticity of demand asε = -(% change in Q)/(% change in P). Note the minus sign put out front. This means the elasticity value will now be a positive number. This way I don’t have to type a minus sign all the time!

8. Elasticity can have three basic values If ε> 1 we say demand is elastic. This means the % change in the A is greater than the % change in price. If ε= 1 we say demand is unit elastic. This means the % change in the A is equal to the % change in price. If ε< 1 we say demand is inelastic. This means the % change in the A is less than the % change in price.

9. Demand in a table of numbers P A ε 11 0 10 1 9 2 10 8 3 4.5 7 4 2.67 6 5 1.75 5 6 1.20 4 7 .83 3 8 .57 2 9 .38 1 10 .22 0 11 .10 Here I have added the elasticity values from our example in the table. Please check the numbers to make sure you can get them and that I am correct. When I have the elasticity of 10, for example, I started at (10, 1) and ended at (9, 2) and put the elasticity in the row where I ended.

10. Elasticity again P In the upper left of the demand curve the % change in the A is greater than the % change in the P and thus the ε> 1 . P1 P2 A A1 A2 Without a real formal proof of the above statement, we can see the % change in A is about 100 % and the % change in P is less than 100 %. Demand is elastic here.

11. Elasticity has several ranges of values P In the lower right of the demand curve the % change in the A is less than the % change in the P and thus the ε< 1. P1 P2 A A1 A2 Without a real formal proof of the above statement, we can see the % change in A is less than 100 % and the % change in P is about 100 %. Demand is inelastic here.

12. Elasticity has several ranges of values P In the middle of the demand curve the % change in the A is equal to the % change in the P and thus the ε= 1. P1 P2 A A1 A2 Without a real formal proof of the above statement, we can see the % change in A is about equal to the % change in P. Demand is unit elastic here.

13. Demand in a table of numbers P A ε TR 11 0 0 10 1 10 9 2 10 18 8 3 4.5 24 7 4 2.67 28 6 5 1.75 30 5 6 1.20 30 4 7 .83 28 3 8 .57 24 2 9 .38 18 1 10 .22 10 0 11 .10 0 Note that when the price starts at 11 and falls to 10 the quantity rises from 0 to 1 and the % change in A is (1 – 0)/0 and this is undefined. Here I have added TR or total revenue which is just the price times the quantity A. Here once the price is any amount (like 9) then everyone can buy at that price.

14. Example continued From the table on the previous screen, as we look through the table and see the price fall, 1) The quantity demanded for attendance A always rises, 2) The elasticity value continues to fall, 3) The total revenue starts out rising, levels off, and then falls, and 4) The total revenue begins to fall when the elasticity value becomes less than 1 or what we call inelastic.

15. Demand in a table of numbers P A ε TR MR 11 0 0 10 1 10 10 9 2 10 18 8 8 3 4.5 24 6 7 4 2.67 28 4 6 5 1.75 30 2 5 6 1.20 30 0 4 7 .83 28 -2 3 8 .57 24 -4 2 9 .38 18 -6 1 10 .22 10 -8 0 11 .10 0 -10 Here I have added MR or marginal revenue. MR is the change in TR divided by the change in A. With a table like this we just go from row to row. Note the MR starts out positive and shrinks as we move down the table. The MR actually becomes negative when the TR is falling and the elasticity is inelastic.

16. Own price elasticity and total revenue changes Total revenue (TR) is price times quantity. Along the demand curve P and A move in opposite directions. Knowledge of εassists in knowing how TR will change.

17. Elasticity and total revenue relationship P TR in the market is equal to the price in the market multiplied by the quantity traded in the market. In this diagram TR equals the area of the rectangle made by P1, Q1 and P1 Q Q1 the horizontal and vertical axes. We know from math that the area of a rectangle is base times height and thus here that means P times Q. The area is TR!

18. Elasticity and total revenue relationship P Now in this graph when the price is P1 the TR = a + b(adding areas) and if the price is P2 the TR = b + c. The change in TR if the price should fall P1 P2 a b c Q Q1 Q2 from P1 to P2 is (b + c) - (a + b) = c - a. Similarly, if the price should rise from P2 to P1 the change in TR is a - c. I will focus on price declines next.

19. Elasticity and total revenue relationship P Since the change in TR is c - a, the value of the change will depend on whether c is bigger or smaller, or even equal to, a. In this diagram we see c > a and thus the change in TR > 0. P1 P2 a b c Q Q1 Q2 This means that as the price falls, TR rises. I think you will recall that in the upper left of the demand the demand is price elastic. Thus if the price falls in the elastic range of demand TR rises.

20. Elasticity and TR You will note on the previous screen that I had c - a. In the graph c is indicating the change in TR because we are selling more units. The area a is indicating the change in TR when there is a price change. We have to bring the two together to get the change in TR. Thus a lower price has a good and a bad. Good - sell more units. Bad - sell at lower price.

21. Elasticity and total revenue relationship P Now in this graph when the price is P1 the TR = a + b(adding areas) and if the price is P2 the TR = b + c. In this diagram we see c < a and thus the change in TR < 0. P1 P2 a b c Q Q1 Q2 I think you will recall that in the lower right of the demand the demand is price inelastic. Thus if the price falls in the inelastic range of demand TR falls.

22. Elasticity and total revenue relationship P Now in this graph when the price is P1 the TR = a + b(adding areas) and if the price is P2 the TR = b + c. In this diagram we see c = a and thus the change in TR = 0. P1 P2 a b c Q Q1 Q2 I think you will recall that in the middle of the demand the demand is unit elastic. Thus if the price falls in the unit elastic range of demand TR does not change.

23. Summary If price TR if demand is Falls rises elastic Falls does not change unit elastic Falls falls inelastic Rises falls elastic Rises does not change unit elastic Rises rises inelastic

24. As you look at the summary, here is a little memory device to help you put this all together. If demand is elastic, price and TR move in opposite directions. If demand is inelastic, price and TR move in the same direction. If demand is unit elastic, TR does not change as the price changes. NOTE: in the table example I had before when the price went from 6 to 5 the total revenue did not change but the elasticity was not exactly 1. Why? The numbers used did not give us the exact story we want to tell, but it is close enough for now.

25. Demand and other concepts in math form Say in a general form demand is of the form P = a – bA (A specific form might be P = 140 -4A). Then by definition, TR has the form TR = aA – bA2. And, also by definition, MR has the form MR = a – 2bA. Remember we had ε = -(% change in A)/(% change in P). This can be rewritten in more explicit form.

26. ε = -(% change in A)/(% change in P), or -[change in A/start A]/[change in P/start P], or -[change in A/start A] times [start P/change in P], or -[change in A/change in P] times [start P/start A]. Now, with demand of the form P = a – bA, the change in A over the change in P is [1/-b] and we can rewrite [start P/Start A] as [P/A]. Then elasticity can be rewritten ε = -[1/-b][P/A] = [1/b][P/A] = [P/bA]. When we see demand is P = a – bA and substitute this in ε = [a – bA]/bA = (a/bA) – 1 This is what I want you to remember!

27. Something to note If marginal revenue MR = a -2bA, then when MR = 0 we have a – 2bA = 0 or A = a/2b. If we put this into the elasticity we have ε = [a – bA]/bA = (a/bA) – 1= (a/b(a/2b)) – 1 = 2 – 1 = 1. So, the elasticity value is 1 when the MR is zero.

28. Some graphs P ε> 1 So, as price falls 1) Quantity demanded A rises 2) Elasticity falls 3) Total revenue rises and MR is positive when demand is elastic, TR stays same and MR = 0 when elasticity is unit elastic, TR falls and MR is negative when elasticity is inelastic. ε= 1 ε< 1 A TR A What happens as the price rises? These two graphs have us move toward more A as the Price P falls

29. Focus on when MR = 0 What we see on the previous screen is many things occur when MR = 0. At the level of A where MR = 0, 1) On the demand curve the elasticity value is 1 (unit elastic)-if you look at the A where MR = 0 and plug that A into elasticity formula you see elasticity is 1, 2) On the TR curve total revenue is at its highest or maximum value-if you take the same A and put into TR the TR is at its highest. Plus, the MR hits 0 at the A that is halfway between the origin and where the demand curve crosses the horizontal axis.

30. Application Once a stadium has been built it can be expected that the cost of operating the stadium is higher the more who attend the game because of things such as clean up cost and cost of personnel to assist fans. The author notes that ticket prices are often found to be in the inelastic range of demand. This means a price increase and the resultant decrease in attendance would both raise attendance revenue and lower operating cost. This means profit would rise. So, why don’t prices for attendance rise? Fans know when they go to a game they might also pay for parking, concessions, and merchandise. Higher ticket prices might raise attendance revenue but could cut back on the overall profit of the bundle of the game, concessions, parking and merchandise.

31. Another application If the demand curve could be shifted to the right (maybe owners of teams convince public there is a better team), then a given price structure generates more revenue. Note, the demand shifting out means level of A where MR = 0 and thus elasticity = 1 and TR is a max is higher.