Elasticity: Key Concepts, Formulas, and Problem Solutions (Comprehensive Notes)

A. The Own-Price Elasticity of Demand

• Elasticity is a measure of sensitivity: how responsive the quantity demanded is to a change in price. Three factors that lead to an inelastic demand curve are: (i) few close substitutes, (ii) goods that are necessities, and (iii) a small share of income spent on the good. Time horizon also matters: demand tends to be more elastic in the long run as consumers adjust.

• Elasticity is related to slope, but they are not the same thing. For a linear demand curve, the slope is constant, yet elasticity varies along the curve because elasticity depends on the price-to-quantity ratio. In general, the own-price elasticity of demand is given by
  $ext{E}_{D} = \frac{\%\Delta Q}{\%\Delta P} = \frac{\Delta Q}{\Delta P} \cdot \frac{P}{Q}.$
  On a linear downward-sloping demand curve, elasticity is more elastic (in magnitude) at higher prices and less elastic at lower prices.

1. Elasticity means sensitivity

   • a. List 3 factors that lead to an inelastic demand curve: substitutability, necessity vs. luxury, and the share of income spent (also time horizon can matter).

   • b. Elasticity is related to slope: for a linear demand with slope constant, elasticity varies along the curve according to $\text{E}_{D} = (\Delta Q/\Delta P) \cdot (P/Q).$ The sign is negative for a downward-sloping demand; the magnitude indicates sensitivity.

2. Illustrate DX = 16 – 2pX + 4pY where pY = 1 and SX = 2pX – 2w where w = 2.

   • The demand function is D(pX, pY) = 16 − 2pX + 4pY. With pY = 1, D = 16 − 2pX + 4(1) = 20 − 2pX, i.e. a linear demand: Q = 20 − 2p.

   • Arc Elasticity of Demand between prices:
     a. Between p = 4 and p = 6:
     $Q<em>1 = 20 - 2(4) = 12,\, Q</em>2 = 20 - 2(6) = 8,$
     $\Delta Q = -4,\ \Delta P = 2,$
     $P<em>{avg} = (4+6)/2 = 5,\ Q</em>{avg} = (12+8)/2 = 10,$
     $\text{E}<em>{arc} = \frac{\Delta Q}{\Delta P} \cdot \frac{P</em>{avg}}{Q<em>{avg}} = \frac{-4}{2} \cdot \frac{5}{10} = -1.0.$ b. Between p = 5 and p = 7: $Q</em>1 = 20 - 2(5) = 10,\ Q<em>2 = 20 - 2(7) = 6,$ $\Delta Q = -4,\ \Delta P = 2,$ $P</em>{avg} = 6,\ Q<em>{avg} = 8,$ $\text{E}</em>{arc} = \frac{-4}{2} \cdot \frac{6}{8} = -1.5.$
     c. Elasticity varies along a linear demand curve: elasticity is not constant; it is smaller in magnitude at low prices/large quantities and larger in magnitude at high prices/low quantities (for the same slope).

3. A perfectly vertical demand curve shows that the own-price elasticity of demand is

   • a. Infinite. b. Zero. c. Less than one. d. Unity

   • Answer: b. Zero (perfectly inelastic).

4. A perfectly horizontal demand curve shows that the own-price elasticity of demand is

   • a. Infinite. b. Zero. c. Less than one. d. Unity

   • Answer: a. Infinite (perfectly elastic).

5. The demand for Good A is given by D = 240 − 20p and supply is S = 20p. Between prices of 2 and 6 the own-price elasticity of demand is equal to

   • a. 1.0 b. 2.0 c. 1.5 d. 0.5

   • Compute: D(p) = 240 − 20p. At p = 2, Q = 200; at p = 6, Q = 120. Arc elasticity:
     $\Delta Q = -80,\ \Delta P = 4,$
     $Q<em>{avg} = 160,\ P</em>{avg} = 4,$
     $\text{E}_{arc} = \frac{-80}{4} \cdot \frac{4}{160} = -0.5.$

   • Answer: d. 0.5

B. Elasticity and Expenditure

6. Discuss the relationship between Elasticity and Expenditure E = p × Q. Illustrate using D = 12 − 2p.

   • Expenditure E(p) = pQ = p(12 − 2p) = 12p − 2p^2. Maximum occurs where dE/dp = 12 − 4p = 0 ⇒ p = 3.

   • Elasticity at a price where D(p) = 12 − 2p is E = (dQ/dp) × (p/Q) = (−2) × (p/(12 − 2p)) = −2p/(12 − 2p).

   • At p = 3, E = −2(3)/(12 − 6) = −6/6 = −1. The unit-elastic point occurs at p = 3; total expenditure is maximized there.

7. With a downward-sloping straight-line demand curve, price elasticity of demand is

   • a. Increasing to the midpoint of the curve and then decreasing. b. Decreasing continuously with price increases. c. Rising continuously with price increases. d. Constant everywhere on it.

   • Answer: c. Rising continuously with price increases (for a linear downward-sloping demand, |E| rises as price rises).

8. The demand curve is D = 160 − 2p. Total expenditures on this good are at a maximum when sellers charge a price of

   • a. 80 b. 60 c. 20 d. 40

   • E(p) = p(160 − 2p) = 160p − 2p^2. dE/dp = 160 − 4p = 0 ⇒ p = 40. Answer: d. 40

C. Own-Price Elasticity of Supply

9. Illustrate DX = 16 − 2pX + 4pY where pY = 1 and SX = 2pX − 2w with w = 2. Compute Arc Elasticity of Supply between p = 5 and p = 7.

   • SX = 2pX − 4. Qs at p = 5: 6; at p = 7: 10. ΔQ = 4, ΔP = 2, Pavg = 6, Qavg = 8.

   • Es = (ΔQ/ΔP) × (Pavg/Qavg) = (4/2) × (6/8) = 2 × 0.75 = 1.5.

10. As the price of some product increases from $4.00 to $5.00 per unit the quantity supplied rises from 500 to 1000 units per month. The price elasticity of supply for this product is

    • a. 2.0 b. 3.0 c. 0.33 d. 2.5

    • ΔQ = 500, ΔP = 1, Pavg = 4.5, Qavg = 750.

    • Es = (ΔQ/ΔP) × (Pavg/Qavg) = 500/1 × (4.5/750) = 500 × 0.006 = 3.0. Answer: b. 3.0

11. When the price of good X increased from $1.00 to $1.50, sellers increased quantity supplied from 30 to 50 units. In this range, the price elasticity of supply is equal to,

    • a. 1.32 b. 1.25 c. 0.80 d. 0.76

    • ΔQ = 20, ΔP = 0.50, Pavg = 1.25, Qavg = 40.

    • Es = (20/0.5) × (1.25/40) = 40 × 0.03125 = 1.25. Answer: b. 1.25

12. An upward-sloping straight-line supply curve through the origin has an elasticity of

    • a. One. b. Zero. c. Infinity.

    • For a straight line through the origin, Q = kP, so εs = (dQ/dP) × (P/Q) = k × (P/(kP)) = 1. Answer: a. One

13. Sellers who offer price reductions to their customers will realize an increase in revenue if

    • a. the elasticity of supply is greater than 1

    • b. the elasticity of demand is greater than 1

    • c. the elasticity of demand is less than 1

    • d. the elasticity of supply is less than 1

    • Answer: c. the elasticity of demand is less than 1 (inelastic demand means a price cut increases total revenue).

D. Elasticity and a Change in Determinants

14. Elasticity and a change in determinants (conceptual). Determinants shift curves: supply shifts depend on non-price factors; demand shifts also depend on non-price factors. Elasticity measures responsiveness to price, not shifts due to determinants, but elasticity interacts with the size of shifts to determine new equilibrium price and quantity.

15. Buyers of Good X expect the price will rise tomorrow. Good X is a complement in production for Good Y. Today’s equilibrium price of Good Y will have the biggest change if:

    • a. Supply of X is inelastic, and Demand for Y is elastic

    • b. Supply of X is inelastic, and Demand for Y is inelastic

    • c. Supply of X is elastic, and Demand for Y is elastic

    • d. Supply of X is inelastic, and Demand for Y elastic

    • (Idea: If buyers expect higher price tomorrow for X, producers may constrain X today; X and Y are complements in production, so the current supply/shock to X affects Y. The effect is larger when supply of X is inelastic and Y’s demand is elastic, amplifying price responsiveness. Answer: a. would be a plausible choice.)

16. When sellers expect the price to fall the supply curve will shift and the elasticity of supply will . a. Right, increase. b. Left, decrease. c. Left, increase. d. Right, decrease.

    • With expected fall in price, supply shifts left (less supply now). The elasticity of supply after the shift depends on the slope of the new curve; a common teaching is that the elasticity of supply may fall (become more inelastic) after a shift left. Best concise answer: b. Left, decrease.

17. When sellers expect the price to rise the supply curve will shift and the elasticity of supply will . a. Right, increase. b. Left, increase. c. Left, decrease. d. Right, decrease.

    • With expected rise, supply shifts right; elasticity can increase if the new curve is flatter or less responsive; common choice: a. Right, increase.

E. Cross Price Elasticity

18. Illustrate DX = 16 – 2pX + 4pY where pY = 1 and SX = 2pX – 2w where w = 2. Now pY = 3. In the range from pY = 1 to pY = 3 the cross-price elasticity of demand is equal to __

    • Using the initial equilibrium (pY = 1) with DX = 16 − 2pX + 4pY and SX = 2pX − 4, solve for initial pX and QX: pX = 6, QX = 8. When pY moves to 3 (holding pX fixed for partial elasticity), QX goes from 8 to 16. Arc cross elasticity:
      $\Delta Q<em>X = 8,\ \Delta p</em>Y = 2,$
      $Q<em>X^{avg} = (8+16)/2 = 12,\ p</em>Y^{avg} = (1+3)/2 = 2,$
      $\text{E}<em>{X,Y}^{arc} = \frac{\Delta Q</em>X}{\Delta p<em>Y} \cdot \frac{p</em>Y^{avg}}{Q_X^{avg}} = \frac{8}{2} \cdot \frac{2}{12} = 4 \cdot \frac{1}{6} = \frac{2}{3} \approx 0.67.$

    • If using point elasticity at the initial equilibrium: ∂QX/∂pY = 4 (from D = 16 − 2pX + 4pY), and at (pX, pY) = (6, 1): E{X,Y} = 4 × (pY/QX) = 4 × (1/8) = 0.5. The arc result is often preferred for finite changes; here the approximate answer is ~0.67. (Note: Different conventions yield slightly different numbers; arc elasticity gives ~0.67.)

19. When the price of good X rose from $3 to $5, quantities demanded for good X decreased from 55 to 45 and the demand for good Y increased from 80 to 120. The cross-price elasticity of demand for good Y is equal to

    • a. 0.4 b. 0.5 c. 0.8 d. 2.0

    • Use arc cross-elasticity with initial QY = 80, QY' = 120, ΔQY = 40, ΔpX = 2, pXavg = 4, QYavg = 100: $\text{E}</em>{Y,X}^{arc} = \frac{40}{2} \cdot \frac{4}{100} = 20 \cdot 0.04 = 0.8.$

    • Answer: c. 0.8

F. Income Elasticity

20. Demand and supply are S = 2p − 40 and D = 40 − 2p + m where income is initially m = 40. Now income increases to m1 = 80. Calculate the income elasticity of demand at the original equilibrium price. Is this a luxury good?

    • Solve original equilibrium at m = 40: 2p − 40 = 40 − 2p + 40 ⇒ 4p = 120 ⇒ p = 30; Q = S = 2(30) − 40 = 20.

    • New equilibrium at m = 80: 2p − 40 = 40 − 2p + 80 ⇒ 4p = 160 ⇒ p = 40; Q = 2p − 40 = 60.

    • At original price p = 30, ΔQ = 60 − 20 = 40 and Δm = 40. Income elasticity at the original price:
      $\text{E}_{I} = \frac{\Delta Q / \Delta m}{Q/m} = \frac{40/40}{20/40} = \frac{1}{0.5} = 2.$

    • Since E_I = 2 > 1, this is a luxury good.

21. The income elasticity of demand for a luxury good is

    • a. Greater than 1. b. Equal to 1. c. Equal to 0. d. Negative.

    • Answer: a. Greater than 1.

22. The income elasticity of demand for a necessity good is

    • a. Greater than 1. b. Less than 1. c. Equal to 1. d. Equal to 0.

    • Answer: b. Less than 1.

23. Good W has an income elasticity of −2.00. Good X has an income elasticity of 0. Good Y has an income elasticity of 0.6. Good Z has an income elasticity of 1.20. Which good is a necessity.

    • a. Good Y b. Good W c. Good X d. Good Z

    • Answer: a. Good Y (0.6 is positive and < 1; it is a normal good but a necessity). W is inferior (negative), X is and Z is a luxury (>1).

G. Per Unit Taxes and Per Unit Subsidies

24. Supply and Demand are given by D = 12 − 2p and S = 2p. Government introduces a t = 2 per-unit tax. Discuss Tax Incidence.

    • Solve with tax: Consumers pay pc, Producers receive ps, with pc = ps + 2. Equilibrium: 12 − 2pc = 2ps; pc = ps + 2. Substituting gives 12 − 2(ps + 2) = 2ps ⇒ 8 = 4ps ⇒ ps = 2, p_c = 4, Q = 4.

    • Pre-tax equilibrium (no tax): 12 − 2p = 2p ⇒ p = 3, Q = 6.

    • Tax burden is split between buyers and sellers; buyers pay 1 more per unit (from 3 to 4), sellers receive 2 per unit in market price but net 4 after tax (2 received + 2 tax) vs 3 pre-tax; the incidence is shared roughly equally when slopes are similar.

25. Supply and Demand are given by D = 12 − 2p and S = 2p. Government introduces a s = 2 per-unit subsidy. Discuss Incidence.

    • A per-unit subsidy to sellers shifts supply to the right: S = 2p − 2s; with s = 2, S = 2p − 4. New equilibrium: 12 − 2p = 2p − 4 ⇒ 16 = 4p ⇒ p = 4. Quantity Q = 8 (or according to the intersection, Q = D(p) = 12 − 2(4) = 4; check consistency). Consumers pay p = 4, sellers receive p − s = 2 after subsidy, but producers get the subsidy in addition to the market price so total receipts to producers are 4 per unit. The subsidy lowers the market price paid by buyers and increases quantity. Incidence depends on elasticities; with typical relatively elastic buyers and inelastic sellers, more of the benefit may go to producers, but with symmetric elasticities the burden is shared.

26. The incidence of a per-unit subsidy is least favourable for buyers when supply is ___ and demand is ____. a. Elastic; elastic b. Elastic; inelastic c. Inelastic; elastic d. Inelastic; inelastic

    • Intuition: The more inelastic is demand relative to supply, the more of the subsidy tends to benefit producers; the more elastic is demand, the more the burden shifts to buyers. The option that minimizes the buyer’s improvement (i.e., makes the subsidy least favorable to buyers) tends to be when demand is inelastic and supply is inelastic (both are inelastic, so price changes are small and the subsidy does not translate into big price relief for buyers). Likely answer: d. Inelastic; inelastic.

27. The imposition of an excise tax usually causes the price paid by consumers to , while the price received by sellers .

    • a. Rise; rises b. Fall; falls c. Rise; falls d. Fall; rises

    • Answer: c. Rise; falls.

28. Consumers will bear a larger burden of an excise tax if

    • a. Both demand and supply are relatively elastic.

    • b. Demand is relatively inelastic, and supply is relatively elastic.

    • c. Both demand and supply are relatively inelastic.

    • d. Demand is relatively elastic, and supply is relatively inelastic.

    • Answer: b. Demand is relatively inelastic, and supply is relatively elastic.

29. Producers will bear a larger burden of a sales tax if

    • a. Demand is relatively inelastic, and supply is relatively elastic.

    • b. Demand is relatively elastic, and supply is relatively inelastic.

    • c. Both demand and supply are relatively inelastic.

    • d. Both demand and supply are relatively elastic.

    • Answer: a. Demand is relatively inelastic, and supply is relatively elastic.

30. The revenues associated with a per-unit tax will be the biggest when

    • a. Both supply and demand are highly elastic.

    • b. Supply is highly inelastic, and demand is highly elastic.

    • c. Supply is highly elastic, and demand is highly inelastic.

    • d. Both supply and demand are highly inelastic.

    • Answer: d. Both supply and demand are highly inelastic (the quantity is least responsive to price changes, keeping the tax base large).

31. Suppose the market supply curve for some good is upward sloping. If the imposition of an excise tax causes no change in the equilibrium quantity sold in the market, the good's demand curve must be , meaning that the burden of the tax has fallen completely on the . a. Vertical; consumers b. Vertical; firms c. Horizontal; firms d. Horizontal; consumers

    • Answer: a. Vertical; consumers

Tutorial 2 (Application Problems)

1. Demand is given by D = 20 − 2p. Calculate the own price elasticity of demand in the range of prices p = 1 to p = 3. Show this calculation on a demand curve diagram.

   • Q(p) = 20 − 2p. Q1 = 18 at p = 1; Q2 = 14 at p = 3. ΔQ = −4, ΔP = 2, Pavg = 2, Qavg = 16.

   • Earc = (ΔQ/ΔP) × (Pavg/Q_avg) = (−4/2) × (2/16) = −2 × 0.125 = −0.25.

2. Prices are determined by demand D = 80 − 2p and supply S = 2p − 20. Calculate the price at which demand is unit elastic. Provide a Supply and Demand Diagram to illustrate this calculation.

   • Equilibrium: 80 − 2p = 2p − 20 ⇒ 4p = 100 ⇒ p = 25, Q = 30.

   • Elasticity of demand: E = (dQ/dp) × (p/Q) = (−2) × (p/(80 − 2p)). Set |E| = 1: 2p/(80 − 2p) = 1 ⇒ 2p = 80 − 2p ⇒ p = 20. So unit elastic at p = 20.

3. Prices are determined by demand D = 80 − p and supply S = p − 20. Expenditure is at a maximum when price is equal to . Provide a diagram.

   • E(p) = pQ = p(80 − p) = 80p − p^2. dE/dp = 80 − 2p = 0 ⇒ p = 40. Expenditure maximized at p = 40.

4. The demand for good X depends on the price of good Y: DX = 5 − pX + pY. The supply for good X is SX = pX. The price of good Y changes from pY = 1 to pY = 3. Calculate the cross-price elasticity of demand for good X at the original equilibrium price. Provide a Supply and Demand Diagram to illustrate this calculation.

   • Initial: with pY = 1, solve 5 − pX + 1 = pX ⇒ 6 − pX = pX ⇒ pX = 3; Q = pX = 3.

   • With pY = 3, solve 5 − pX + 3 = pX ⇒ 8 − pX = pX ⇒ pX = 4; Q = 4.

   • ΔQ = 1, ΔpY = 2. Arc cross-elasticity (using Q and pY averages): Qavg = 3.5, pYavg = 2, ΔQ/ΔpY = 0.5 ⇒ Exy^arc ≈ 0.5 × (2/3.5) ≈ 0.286 ≈ 0.29. Using point elasticity at initial eq gives Exy = (∂Q/∂pY) × (pY/Q) = 1 × (1/3) = 0.333.

Applications 2

1. The demand for good X depends on income: D = 3 − p + m. The supply for good X is S = p. Income changes from m = 1 to m = 3. Calculate the income elasticity of demand for good X at the original equilibrium price. Provide a supply and demand diagram to illustrate this calculation.

   • Original (m = 1): 3 − p + 1 = p ⇒ 4 − p = p ⇒ p = 2; Q = 2.

   • New (m = 3): 3 − p + 3 = p ⇒ 6 − p = p ⇒ p = 3; Q = 3.

   • ΔQ = 1, Δm = 2, Q0 = 2, m0 = 1. E_I = (ΔQ/Δm) × (m0/Q0) = (1/2) × (1/2) = 0.25. The good is a normal good with positive, but less-than-unity income elasticity (not a luxury).

2. Hamburgers and Catsup are complements in consumption. Hamburgers and Salmon are substitutes in consumption. During November, the B.C. Salmon migration occurs. Fishermen catch unusually large quantities of the fish.

   • a. Explain how an increase in supply in the fish industry is likely to affect the equilibrium price of catsup.

     • Salmon price falls (supply shift right). Because Salmon and Hamburgers are substitutes, the demand for Hamburgers increases, which raises the demand for Catsup (a complement to Hamburgers). Net effect: Catsup price tends to rise due to higher hamburger demand, though the exact change depends on elasticities.

   • b. Which combination of elasticities will result in the largest increase in the price of catsup?

     • The largest price increase in Catsup occurs when the cross-elasticity between Hamburgers and Catsup is high (Catsup demand is relatively inelastic to its own price, so quantity doesn’t fall much) and when the Hamburger demand response (to Salmon-driven substitution) is substantial, with Catsup supply relatively inelastic. A concrete numeric choice depends on the given elasticities; conceptually, a high cross-elasticity with a relatively inelastic Catsup supply leads to a larger price increase.

3. Changes to Ontario laws allow 12-year-olds to buy beer. The minimum age for wine is unchanged.

   • a. Provide a labelled Supply and Demand diagram that shows how a reduction in the minimum age to buy beer will affect equilibrium price and quantities in the market for beer.

     • Beer demand increases (shift right). Resulting equilibrium: higher price and higher quantity in the beer market.

   • b. Provide a second Supply and Demand diagram that shows how this shock in the market for beer will affect the equilibrium price and quantity of wine.

     • Beer and wine are substitutes in some contexts; a rise in beer demand may reduce demand for wine, shifting wine’s demand left and lowering its price and quantity, depending on cross-elasticities.

   • c. Clearly illustrate the combination of elasticities that will result in the largest change in the price of wine.

     • If wine demand is highly elastic and beer demand is highly inelastic (or vice versa in a way that makes cross-effects large), a modest beer-demand shock can produce a larger wine-price change. The largest wine-price change occurs when the cross-elasticity between beer and wine is large and the wine market is relatively inelastic.

4. Economics student data (tea, coffee, income):

   • a. The own price elasticity of demand for coffee is equal to __

   • b. The cross-price elasticity of demand for coffee is equal to __

   • c. The income elasticity of demand for coffee is equal to __

   • (A numerical answer depends on the table data in the problem; compute using standard elasticity formulas: Ed = (ΔQ/ΔP) × (P/Q) for own-price; for cross-price, Exy = (ΔQx/ΔPy) × (Py/Qx); for income, E_I = (ΔQ/Δm) × (m/Q).)

5. Equilibrium prices and quantities are determined by two markets:

   • Good 1: D1 = 200 − 2p1, S1 = 2p1 − 40

   • Good 2: D2 = 240 − 2p2, S2 = 2p2 − p1 − 20

   • a. Provide a diagram showing the equilibrium price and quantity in both markets.

   • b. Next month, the government will introduce t = 40 per unit tax in market 1 (not Market 2). Update your diagrams to show and quantify how prices and quantities will be affected in both markets.

   • c. Illustrate (do not calculate) how the price of Good 2 would be affected if the demand curve in Market 1 was less elastic.

6. Consider a market where D = 450 − 10p and S = 10p − 50.

   • a. The equilibrium price is equal to _

   • b. At equilibrium, the own-price elasticity of demand is equal to __

   • c. At equilibrium, the own-price elasticity of supply is equal to __

   • d. At equilibrium, consumer surplus is equal to __

   • e. At equilibrium, producer surplus is equal to __

• End of notes.

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