1. Lecture 3:Elasticities

2. 2 This lecture covers: Why elasticities are useful Estimating elasticities Arc formula Deriving elasticities from demand functions Constructing a linear demand function Price, cross, and income elasticities Retail versus farm-level demand elasticities Price flexibility

3. 3 Why elasticities are useful : Elasticities are very useful when analyzing the impacts of price changes on quantity demanded (supplied). Elasticities are independent of the units of measurement; they always have an interpretation in terms of percentage change.

4. 4 Why elasticities are useful : We will use E YZ to denote the elasticity of variable Y with respect to variable Z. In symbols, In words, E YZ is the percentage change in Y for a 1 % change in Z.

5. 5 Why elasticities are useful : In microeconomics we usually make use of price and income elasticities. However, the concept is more general. Suppose E YX = -1.5, and X is expected to increase by 20 percent. What do we expect will happen to Y ?.

6. 6 Different points, different elasticities 4 a 2 b

7. 7 The Arc Formula Q is consumption, and P is price of a commodity. Suppose we have two observations of these variables: (Q 1, P 1 ) and (Q 2, P 2 ). These could represent two points on a demand curve.

8. 8 The Arc Formula The elasticity of Q with respect to P can be calculated from the Arc formula: Note that we are measuring elasticity at an average between 2 points.

9. 9 The Arc Formula Why use the average? An alternative would be to choose a starting point, say (Q 1, P 1 ), and measure proportionate change relative to it. However, our elasticity estimate would then depend on an arbitrary choice of starting point. Choosing (Q 2,P 2 ) as starting point yields a different answer.

10. 10 Example (Arc formula) For example, suppose we want to calculate the income elasticity of demand for Z, given the following information: QIncome (I) Observation 1Q 1 = 200I 1 = 30 Observation 2Q 2 = 250I 2 = 50 What kind of good is this? (What does the elasticity tell you?).

11. 11 Deriving Elasticities from Demand Function The arc formula measures elasticity between two observations. But what if we can express the relationship between variables in equation form? Then we can use calculus to derive elasticity. If Y is a function of X, the elasticity of Y with respect to X is computed

12. 12 Linear Demand Function Because linear demand functions are commonly specified in economics, it is useful to examine what they imply about price elasticity. Consider the demand function Q = a – bP The formula for price elasticity is

13. 13 Linear Demand Function Construct a graph of the demand function with P on the vertical axis and Q on the horizontal. The horizontal intercept is ‘a’. What is the vertical intercept? It turns out that the midpoint of the linear demand schedule has a price elasticity of –1. (Why?)

14. 14 Constructing a Linear Demand Schedule If we are given the price elasticity and values of P and Q, it is possible to construct a linear demand function. Suppose E QP = -2, P=10, and Q=25. We seek a demand function of the form: Q = a – bP. Recall that price elasticity for such a function is

15. 15 Constructing a Linear Demand Schedule Begin by solving for b, the slope parameter. E QP = b = Now use the demand equation Q = a – bP, substituting values of b, Q, and P to solve for the intercept ‘a’. So the linear demand function is

16. 16 Elasticity Estimates As we have seen, elasticities can vary with levels of variables contained in the demand function. For this reason, it is common to use the mean values of variables when calculating the elasticities

17. 17 Elasticity Estimates Example, Suppose the following demand function has been estimated by linear regression: Q = 50 – 3P Assume that the mean values in the sample are and. Then the price elasticity of demand (evaluated at sample means) is

18. 18 Deriving Price, Cross, and Income Elasticities If Y is a function of Z, the elasticity of Y with respect to Z is computed

19. 19 Deriving Price, Cross, and Income Elasticities If we have a function of more than one variable, we use the notation of partial derivatives to signify that an elasticity holds everything else constant. Suppose Y = f(X, Z). Then we have 2 partial derivatives.

20. 20 Example Calculate the price elasticity of demand for the following functions. Variables are defined as Q K : consumption of kiwifruit, lbs per capita P K : price of kiwifruit, $/lb P C : price of cantaloupe, $/lb DI:disposable income, $ thousand per capita 1) Linear demand schedule Q K = 100 - 2P K + 1.5P C + 3I

21. 21 Example 2) Multiplicative (Cobb-Douglas) demand schedule

22. 22 Example 2) Another example For this function, what is the formula for direct price elasticity of demand? What is the cross price elasticity (with respect to Pc) formula? What is the income elasticity?

23. 23 In example 2, we saw a demand function with constant elasticity: In fact, all the exponents in this function can be interpreted as elasticities. After taking logarithms: ln(Q K ) = ln(10) – 0.32ln(P K ) +0.2ln(P C ) +1.3ln(I)

24. 24 Demand Elasticity and Total Revenue If own price elasticity (E QP ) is -0.4, how do you expect wheat farmer’s total revenue (TR) as wheat price declines due to extra production in the market?

25. 25 Retail vs. Farm-level Demand Elasticities Retail demand and farm-level demand for a commodity can be represented on a single graph.

26. 26 Retail vs. Farm-level Demand Elasticities The vertical gap between the two lines is the farm- retail marketing margin. In drawing parallel lines, we are assuming that the margin is not influenced by the quantity sold. Now pick a quantity and draw a vertical line. Suppose this is the quantity sold in a given marketing period. The retail price differs from the farm-level price by the marketing margin.

27. 27 Retail vs. Farm-level Demand Elasticities At this (arbitrary) quantity, which demand schedule is more price elastic? How can you be sure?

28. 28 Price Flexibility This concept is related to price elasticity. However, it answers a different question: if _______were to change by 1%, how would the ______ change in percentage terms? If demand is specified as a simple function of price, then the price flexibility is just the inverse of the price elasticity _____________________________.

29. 29 Price Flexibility The flexibility concept is useful in many agricultural markets where quantities are determined (because of production lags and limited possibilities for storage) before price. In such situations, it is important to know how much the price will change, in proportionate terms, for a given change in production.

30. 30 Price Flexibility Suppose E QP = -0.6, and the quantity produced (and consumed) is expected to rise by 10 percent. How will prices change?

31. 31 Class Talk A hypothetical market demand is given as follows: Q = 10 – 5P + 3I If P = 5 and Q = 50, what is price elasticity and price flexibility?