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Lecture 8 Some Arithmetic of Elasticities

Lecture 8 Some Arithmetic of Elasticities. How do we calculate elasticities from data? Point vs. arc methods No fundamental contradiction— each has its uses. Two ways to measure any elasticity. Point elasticity Measures elasticity at an exact point on a function

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Lecture 8 Some Arithmetic of Elasticities

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  1. Lecture 8Some Arithmetic of Elasticities • How do we calculate elasticities from data? • Point vs. arc methods • No fundamental contradiction— each has its uses

  2. Two ways to measure any elasticity • Point elasticity • Measures elasticity at an exact point on a function • Useful when there are small changes in neighborhood of point • Arc elasticity • Measures elasticity over a range of points • Useful when there are large changes along a function

  3. Point elasticity • Recall for elasticity of demand ε = (%∆Q)/(%∆P) (NOTE: I am suppressing subscript on Q to reduce clutter) • By definition, • %∆Q = dQ/Q • %∆P = dP/P • Thus, • (dQ/Q)/(dP/P) = (dQ/dP)*(P/Q)

  4. Point elasticity (con’t) • From before: • (dQ/Q)/(dP/P) = (dQ/dP)*(P/Q) • Note: • First term on right is inverse of slope • (This is why intuitive shapes aren’t strictly correct) • We now see why elasticity varies along linear demand curve • Slope constant • Rate of price to quantity changes

  5. The “problem” with point elasticity • Along linear demand curve, elasticity varies • Thus, for anything other than small price changes, beginning and ending points will have much different elasticities • We solve by using the average or “arc” measures of elasticity

  6. Arc elasticity • To avoid sensitivity to beginning and ending points, we use their average • The formula; [dQ/(Qb +Qe)] /[dP/ (Pb+Pe)] =[(dQ/dP)*(Pb+Pe)/(Qb+Qe)] (Note: when averaging start and end and Q and P, the “2s” cancel out)

  7. An example • Amazon.com occasionally engages in “dynamic pricing,” altering prices to evaluate the response • When Amazon.com prices my book at $36, it sells 20 copies per month • When it prices it at $30, it sells 30 copies per month • What is the elasticity of demand for my book?

  8. The solution • Recall arc elasticity: • (dQ/dP)*[(Pb+Pe)/(Qb+Qe)] • In this case, Pb = 36, Pe = 30, Qb = 20, Qe = 30 dQ = 30-20 = +10, dP = 30-36 = -6 (Pb+Pe) = 66, (Qb+Qe) = 50 • Thus, (10/-6)*(66/50) = -1.67*1.32 = -2.20 • The demand for my book is elastic: |ε| > 1

  9. Criteria for Selection • No hard and fast rules • When price change is “small” use point • When price change is “large” use arc • Use arc when extrapolating outside range of data (but extrapolating is not recommended using any method!)

  10. Some useful approximations • For any a = b*c, or f = g/h, it is approximately true that %∆a = %∆b + %∆c And %∆f = %∆g - %∆h • Revenue or expenditure is given by R = P*Q And thus: %∆R = %∆P + %∆Q But because: ε = %∆Q/%∆P We get: %∆R = %∆P(1+ ε) = %∆P(1- |ε|)

  11. An example • Given • %∆R = %∆P (1+ ε) • Suppose •  = -1.5, %∆P = +20% • How much will revenue (or expenditure) change? • %∆R = 20% (1- 1.5) = 20% (-.5) = - 10% • Expenditure (revenue) fall 10% when price increases 20% • This happens because the decline in quantity is 30%, more than offsetting the effect of the price increase

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