1. Elasticity Chapter 9

2. 9.1 Introduction Consider a demand function q=q(p). The law of demand says that if price p goes up, the quantity demanded q goes down. Consider the following question: If the price increases by 1%, by how many percent will the quantity demanded decrease? Absolute versus proportionate changes: compare “wages went up by $5” to “wages went up by 1%”. Arc versus point elasticities: for continuous and discrete variables Marginal revenue versus price elasticity of demand. General definition of elasticity.

3. 9.2 Absolute, Proportionate, and Percentage Changes Definition. A change in the level of variable Y is called an absolute change. Notation: Definition. The ratio of an absolute change in variable Y to its initial value is called the proportionate change in variable Y. Notation: Example. Let variable Y represent income. Let : your income increases from the level of $200 per week to $240 per week. The absolute increase in your income is The proportionate increase in your income is

4. Percentage and Proportionate Change Mathematically,. For that reason, there are two ways to represent proportionate changes: 1)Decimal form: 2)Percentage form:

5. Units and Proportionate Changes Consider a change in income from $200 to $240. The absolute change is. The same absolute change is also (4000 US cents). The proportionate change will be the same in both cases: The measurement of proportionate change is independent of the units of measurement.

6. Arbitrary Zero Point Key point: when the zero level of a variable is chosen in an arbitrary fashion, the proportionate change will be different depending on the choice of the zero point. Example: Suppose your appointment is at 4PM, but in fact you arrived at 6PM. Formally, the proportionate measure of your being late should be computed as Remembering that 4PM=16.00 hours, the same change can be written as Elapsed time: this problem does not arise when applying the concept of proportionate change to the elapsed time: for instance, if it took the train 3 instead of 2 hours, the proportionate change is 50%, and it is meaningful.

7. Graphical Representation Consider a movement from A to B. Absolute change: Proportionate change:

8. 9.3 Arc Elasticity of Supply Consider a supply function. Definition. The ratio of a proportionate change in quantity supplied to proportionate change in price is called the arc elasticity of supply. Notation: The law of supply says that higher prices are associated with higher quantities supplied so that the arc elasticity of supply is a positive number: Interpretation: Let the price increase by 1%: In this case the arc elasticity of supply becomes, which implies that The arc elasticity of supply is a percentage change in the quantity supplied caused by a one-percent change in the price.

9. 9.4 Elastic and Inelastic Supply Suppose that a 1% increase in the price causes a more than 1% increase in quantity supplied, that is. In this case Alternatively, if it follows that Definition. Supply is called elastic at a point if. Definition. Supply is called inelastic at a point if. Interpretation. Elastic supply is sensitive to changes in the price: a small price increase causes a relatively large increase in quantity supplied. Inelastic supply is not sensitive to the price changes.

10. 9.5 Elasticity as a Rate of Proportionate Change Definition. A difference quotient measured at some point is called the rate of change of y. Definition. The ratio of two proportionate changes is called a rate of proportionate change. Consider a supply function q=f(p). The rate of proportionate change of the quantity supplied with respect to the price is given by: However, this expression is the arc elasticity of supply. Elasticity is the rate of proportionate change!

11. 9.6 Diagrammatic Treatment

12. Elasticities and Slopes Let us rewrite the expression for arc elasticity of supply as follows: The nominator is the difference quotient of the supply function at, which is approximating the slope of supply at that point. arc elasticity of supply is thus equal to the ratio of the slope of the supply function and the price ratio. Elasticities are not the same with slopes!

13. 9.7 Shortcomings of Arc Elasticity Shortcoming 1. The value of depends on the size of the price change. Shortcoming 2. Asymmetry: Computing a price change from J to K is based on initial price, while computing the same price change from K to J is based on initial price. As a result, we have two elasticities: and Which one do we choose? Shortcoming 3. Arc elasticity is difficult to calculate.

14. 9.8 Point Elasticity of Supply Definition. The limit of arc elasticity of supply at a point is called the point price elasticity of supply. Computation: Point elasticity of supply depends solely on the point at which it is measured: no ambiguity!

15. Diagrammatic Representation of Point Elasticity Point elasticity of supply is the ratio of two slopes: the slope of tangent line to the supply function at point J, and the slope of the ray OJ. Arc elasticity of supply is given by the ratio of chord JK to ray OJ.

16. 9.9 Reconciling Point and Arc Elasticities The arc and point elasticities are computed in a similar fashion: and The denominators are the same, while the nominators are different. If supply is linear, the two definitions result in the same number. When the price changes are small, the two elasticities are very close to each other. For larger price changes, the arc elasticity is preferable.

17. Example 9.1a Consider a supply function. a) Find arc elasticity of supply when p increases from 10 to 11. Let. Plug those price in the supply function to receive Compute absolute changes in prices and quantities: Compute proportionate changes prices and quantities: Arc elasticity of supply is the ratio of the proportionate change in quantities to the proportionate change in prices: Check to see that, if the starting point is the value of arc elasticity will be close, but different:

18. Example 9.1b Find point elasticity of supply at p=10.

19. 9.11 Arc Elasticity of Demand Consider a demand function q=g(p). Definition. The ratio of proportionate change in quantity demanded to the proportionate change in price is called arc elasticity of demand. Notation. Elasticity of demand is negative since the demand curve is downward sloping, so prices and quantities move in opposite directions.

20. 9.12 Elastic and Inelastic Demand Note. Since the demand elasticity is negative, it makes sense talking about the absolute value of the elasticity of demand. For instance, demand elasticity of -3 is mathematically less than demand elasticity of -2, but -3 corresponds to a more responsive demand. Definition. If at some point, demand is called elastic. Definition. If at some point, demand is called inelastic. Definition. If at some point, demand is called unit elastic.

21. Elastic versus Inelastic

22. 9.14 Point Elasticity of Demand Definition. The limit of the ratio of the proportionate change in quantity demanded to the proportionate change in the price at some point is called point elasticity of demand. Notation. What’s the mistake in the yellow rectangle?

23. Example 9.3b Consider a demand function. b) Find the point elasticity when p=5 and p=13. Observation. As price increases, demand becomes more elastic (i.e. more sensitive to the change in price). How does that relate to your personal consumption behavior?

24. Elasticity Changes Key features Demand is inelastic when prices are high. Demand is elastic when prices are low. Demand is unit elastic at one intermediate price. The slope of demand function is the same everywhere, but demand elasticity varies with changes in prices: elasticities are not slopes!

25. 9.17 Two Simplifications 1)From now on, when we say “elasticity” we will mean “point elasticity.” 2)We will omit zero subscripts for prices and quantities in the definition of elasticities: for instance, we will write

26. 9.18 Marginal Revenue and Demand Elasticity Key point: the sign of the marginal revenue at a point depends on the magnitude of elasticity of demand. Consider an inverse demand function. Total revenue is. By definition of the marginal revenue, Since p=f(q), we can rewrite the above as Taking p outside the brackets results in

27. Example 9.5 Consider a demand function 1)Compute elasticity of demand 2)Compute 3)Compute marginal revenue independently and see that it is equal to the result in (2)

28. Graphical Representation Key features 1)MR>0 if demand is elastic, TR increases 2)MR<0 if demand is inelastic, TR decreases 3)When demand is unit elastic, MR=0, and TR reaches its maximum

29. 9.19 Demand Elasticity under Perfect Competition is the market price. If the demand is zero since no one will buy above the market price. If the demand is the size of the whole market. Key point: under perfect competition, is not well-defined since is not well-defined.

30. Thinking in the Limit Consider the case of a very flat demand function: the firm’s product is just slightly different from the other firms’ products. As a result, an increase above the market price will not result in losing all customers, and a decrease below the market price will not attract the whole market. Let the inverse demand be. Demand 1 P Q is getting bigger as demand gets flatter: Demand 2

31. Example 9.7 Consider a demand function. a)Find the price at which TR is at its maximum. b)Show that demand has unitary elasticity at this price. c)Show that demand is elastic at higher prices and inelastic at lower prices.

32. Example 9.8 Consider a demand function. a)Find the price p* at which TR is at its maximum. b)Show that demand is elastic when p>p* and inelastic when p<p*.

33. 9.21 Other Elasticities in Economics Definition. The ratio of the proportionate change in total costs to the proportionate change in output is called elasticity of total cost with respect to output. Notation. The elasticity of total cost with respect to output is equal to the ratio of marginal to average costs.

34. 9.23 Aggregate Consumption Function Consider a consumption function C=f(Y) where C is aggregate consumption, and Y is income (e.g. GDP). Definition. The ratio of aggregate consumption to income is called average propensity to consume, computed as. Definition. The derivative of the consumption function at some point is called marginal propensity to consume, Note. Marginal propensity to consume can be defined in terms of the difference quotient, too. Interpretation: Marginal propensity to consume is equal to the increase in consumption as a result of an increase in income. It is normally less than 1.

35. Income Elasticity of Consumption Definition. The ratio of the proportionate change in consumption to the proportionate change in income is called income elasticity of consumption and is computed as. Note. Since and, the income elasticity of consumption can be re-written as Graphically, the income elasticity of consumption is the ratio of the slope of tangent DE to the slope of the ray OA.

36. 9.24 General Concept of Elasticity Definition. For an arbitrary function y=f(x), the ratio at some point is called the elasticity of the function at that point. Definition. The first derivative of function y=f(x) at some point is called the marginal function of the primitive function f(x), denoted as M. Definition. Given a primitive function f(x), the ratio f(x)/x is called average function of the primitive function f(x), denoted as A. In general, the elasticity of function y=f(x) at some point can be written at this point as: Exercise. Show that the average function is rising if M>A, and vice versa.