Percentages and Elasticity

1. Percentages and Elasticity

2. percentage: “for each hundred” one per cent: one for each hundred ex: "I spend ten percent of my income on movies and other forms of entertainment" means that "I spend ten dollars of each hundred dollars of my income on entertainment."

3. Converting from decimal to percentage notation: Just move the decimal to the right two places. .65 65%

4. What percentages are equivalent to the following numbers? (a) .01 1% (b) .94 94% (c) 1.705 170.5% (d) .2386 23.86% (e) .8 80%

5. Converting from fractions to percentages: Convert from fraction to decimal by dividing, and then move the decimal to the right two places. 1/4 .25 25%

6. Questions What percent of 50 is 10? 20% 4 is what percent of 8? 50% What percent of 60 is 15? 25%

7. Percentage Change: absolute change = after - before before before

8. If spending increased from $50 to $60, by what percent did spending increase? percentage change = after - before before = 60 - 50 50 = 10 / 50 = .20 = 20 %

9. Suppose your financial aid increased by $50. You used to spend $10 per week on entertainment. You now spend $11. By what percentage did your entertainment spending increase? percentage change = after - before before = 11 - 10 10 = 1/10 = .10 = 10 %

10. Your financial aid is cut back by $50 to its original level. You reduce your entertainment spending from $11.00 back to $10.00. By what percent did your spending decrease? percentage change = after - before before = 10 - 11 11 = - 1 / 11 = - .0909 = - 9.09 %

11. A negative change indicates a decrease. A positive change indicates an increase.

12. It would be nice to be able to say the following: In response to a $50 increase, you increased your spending by some percent x. In response to a $50 decrease, you decreased your spending by that same x percent. When we get to the concept of elasticity, in particular, we will want to be able to do that.

13. So we will define percentage changes a little differently. Instead of using our "before" value as our denominator, we will use the average (or midpoint) of our "before" and "after" values as the denominator. average = (before + after) / 2

14. Our percentage change formula becomes percentage change = after - before average

15. Suppose your financial aid increased by $50. You increase your spending on entertainment from $10 per week to $11. By what percentage did your entertainment spending increase? (Use the midpoint formula.) First we need to determine the average of the before and after spending. average = (before + after) / 2 = (10 + 11) / 2 = (21) / 2 = 10.5

16. percentage change = after - before average = 11 - 10 10.5 = 1 / 10.5 = .0952 = 9.52 %

17. Financial aid is cut back by $50. You reduce your spending on entertainment from $11 per week to $10. Using the midpoint formula, determine by what percentage your spending decreases. percentage change = after - before average = 10 - 11 10.5 = - 1 / 10.5 = - .0952 = - 9.52 %

18. Suppose that financial aid for the semester increased from $995 to $1005. Using the midpoint formula, determine by what percent financial aid increased. percentage change = after - before average = 1005 - 995 1000 = 10 / 1000 = .01 = 1 %

19. Suppose the 1% increase in financial aid led you to increase your entertainment budget from $297 to $303. By what percent did you increase your entertainment budget? percentage change = after - before average = 303 - 297 300 = 6 / 300 = .02 = 2 %

20. fin. aid 1 % ent. budget 2 % In other words, the percentage increase in your entertainment budget was 2 times as large as the percentage increase in financial aid. This value (2) is the elasticity of your entertainment budget with respect to your financial aid.

21. The elasticity of X with respect to Y tells you by what percent X changes when Y changes by 1%.

22. The elasticity of X with respect to Y can be calculated asthe percentage change in X the percentage change in Y.

23. If in response to a 20% increase in financial aid, you increased your entertainment budget by 40%, what is the elasticity of your entertainment budget with respect to your financial aid? 2 By how much would you expect your entertain- ment budget to increase in response to a 1% increase in financial aid? 2%

24. Suppose financial aid increased from $1050 to $1100. In response, you increased your entertainment budget from $300 to $320. Calculate the following: • the percentage change in financial aid, • the percentage change in your entertainment budget, and • the elasticity of your entertainment budget with respect to financial aid.

25. percentage change in financial aid fin. aid: 1050, 1100 change in fin. aid = 1100 - 1050 = 50 avg. fin. aid = (1050 + 1100)/2 = 2150/2 = 1075 percentage change in financial aid = (change in fin. aid) / (avg fin. aid) = 50 / 1075 = .0465 = 4.65%

26. percentage change in ent. budget ent. budget: 300, 320 change in ent. budget = 320 - 300 = 20 avg ent. budget = (300 + 320) / 2 = 620/2 = 310 percentage change in entertainment budget = (change in ent. budget) / (avg ent. budget) = 20 / 310 = .0645 = 6.45%

27. elasticity of entertainment budget with respect to financial aid percentage change in entertainment budget percentage change in financial aid = 6.45/4.65 = 1.387. So when financial aid increases by1 percent, your entertainment budget increases by 1.387 percent.

28. In the elasticity formula, how do you remember which variable goes on top and which goes underneath? • The causegoes under the line. CAUSE and UNDER both have a U. • The effect goes on top. EFFECT and TOP both have a T.

29. elasticity = % change in effect % change in cause

30. for our financial aid & entertainment example: elasticity = % change in . % change in

31. Unit Elastic: |elasticity| = 1 % change in effect % change in cause = 1 % change in the effect = % change in the cause

32. fin. aid 5 % ent. budget 5 % .Your entertainment budget is unit elastic with respect to financial aid.

33. Elastic: |elasticity| > 1 % change in effect % change in cause > 1 % change in the effect > % change in the cause

34. fin. aid 5 % ent. budget 6 % Your entertainment budget is elastic with respect to financial aid.

35. Inelastic: |elasticity| < 1 % change in effect % change in cause < 1 % change in the effect < % change in the cause

36. fin. aid 5 % ent. budget 4 % Your entertainment budget is inelastic with respect to financial aid.

37. Price Elasticity of Demand (or Elasticity of Quantity Demanded with Respect to Price) measures the responsiveness of consumers' purchases to a change in the price of a commodity.

38. Notation means “change” and % means “percentage change” examples: Price or  P means “change in price” %  P means “percentage change in price”

39. Suppose the price of personal computers increased from $1600 to $1700. As a result, the number of PCs • the percentage change in the quantity demanded of PCs, • the percentage change in the price of PCs, and • the elasticity of demand for PCs with respect to the price of PCs. purchased per week by area consumers dropped from 500 to 400. Calculate the following:

40. percentage change in quantity demanded of PCsquantity: 500, 400  qty demanded of PCs = 400 - 500 = -100 avg qty demanded of PCs = (500 + 400) / 2 = 900 / 2 = 450 %  qty demanded of PCs = ( qty demanded of PCs) / (avg qty demanded) = -100/450 = -.2222 = - 22.22 % [The negative indicates a decrease in PCs.]

41. percentage change in price of PCsprice: 1600, 1700  Price of PCs = 1700 - 1600 = 100 avg price = (1600 + 1700) / 2 = 3300 / 2 = $1650 %  price of PCs = ( price of PCs) / (avg price) = 100 / 1650 = .0606 = 6.06 %

42. price elasticity of demand for PCs %  qty demanded of PCs %  price of PCs = - 22.22 / 6.06 = - 3.667 The negative indicates that there is an inverse relation between the qty demanded of PCs and the price of PCs. When the price of PCs increases by one percent, the quantity demanded of PCs decreases by 3.667 percent.

43. |-3.667| = 3.667 > 1 So, the demand for PCs is elastic with respect to the price of PCs.

44. Because it is almost always the case that the qty demanded of a good is inversely related to its price, the negative sign is frequently dropped. ex: The price elasticity of demand for PCs would be reported as 3.667 instead of -3.667. The negative is understood.

45. Suppose you have an ailment, for which you must take a particular medication. (Call it Medex.) Every thirty days you purchase one thirty-capsule bottle of Medex. The price of Medex increases from $4 to $5 Medex per bottle. You still purchase one bottle every thirty days. Calculate the elasticity of your quantity demanded of Medex with respect to the price of Medex.

46. percentage change in price of Medexprice: 4, 5  Price of Medex = 5 - 4 = 1 avg price = (4 + 5)/2 = 9/2 = 4.5 %  price of Medex = ( price) / (avg price) = 1 / 4.5 = .2222 So the price of Medex changed by 22.22 %.

47. percentage change in qty demanded of Medexquantity: 1, 1  qty demanded of Medex = 1 - 1 = 0 avg qty demanded of Medex = (1 + 1) / 2 = 2 / 2 = 1 %  qty demanded of Medex = ( qty demanded) / (avg qty demanded) = 0 / 1 = 0 So the quantity demanded of Medex changed by 0 %. (It didn’t change at all.)

48. price elasticity of demand for Medex %  qty demanded of Medex%  price of Medex = 0 / 22.22 = 0

49. Perfectly Inelastic elasticity = 0 It is a special case of inelastic.

50. Suppose the price of personal computers increases from $1500 to $1700. As a result, the number of PCs purchased per week by consumers in the area dropped from 850 to 750. Calculate the elasticity of demand for PCs with respect to the price of PCs.