AP Microeconomics: Price Elasticity of Demand (PED) – Midpoint Formula, Total Revenue Test

AP Microeconomics – Price Elasticity of Demand (PED) – Midpoint Formula, Total Revenue Test

AP Microeconomics – Price Elasticity of Demand (PED) – Midpoint Formula & Total?Revenue Test

What This Is

Price Elasticity of Demand (PED) measures how much the quantity demanded of a good responds to a change in its price. On the AP exam you’ll need the midpoint (arc) formula to calculate a precise elasticity and the total?revenue test to interpret whether a price change will raise or lower a firm’s revenue. Think of a city council that raises the tax on sugary drinks: the council must know whether the higher price will cut sales enough to offset the extra tax revenue. That decision hinges on PED.

Key Terms & Formulas

• PED (Price Elasticity of Demand) = %?Qd / %?P – Ratio of percentage change in quantity demanded to percentage change in price. Elastic (>1), Inelastic (<1), Unit?elastic (=1).
• Midpoint (Arc) Formula – (\displaystyle \text{PED}= \frac{(Q_2-Q_1)}{(Q_2+Q_1)/2}\Big/\frac{(P_2-P_1)}{(P_2+P_1)/2}). (Q_1, P_1) = initial quantity & price; (Q_2, P_2) = new quantity & price.
• Total Revenue (TR) = P × Q – Money a firm receives from selling a given quantity at a given price.
• Total?Revenue Test – If PED?>?1, a price decrease raises TR (and a price increase lowers TR). If PED?<?1, a price increase raises TR (and a price decrease lowers TR).
• Demand Curve (graph) – Vertical axis = Price (P); Horizontal axis = Quantity demanded (Q). Downward?sloping because higher price-lower quantity demanded.
• Elastic Region (graph) – Portion of the demand curve where the absolute value of PED?>?1; the curve is relatively flat.
• Inelastic Region (graph) – Portion where |PED|?<?1; the curve is relatively steep.
• Unit?Elastic Point – The single point on a linear demand curve where TR is maximized; here |PED|?=?1.
• Substitutes Effect – Presence of close substitutes makes demand more elastic (e.g., Coke vs. Pepsi).
• Necessities vs. Luxuries – Necessities tend to have inelastic demand; luxuries are more elastic.

Step?by?Step / Process Flow

1. Identify the price change – Note the original price (P_1) and new price (P_2) (or vice?versa).
2. Record the corresponding quantities – Write down the initial quantity demanded (Q_1) and the new quantity demanded (Q_2).
3. Plug into the midpoint formula – Compute the numerator (?Q over average Q) and denominator (?P over average P) then divide.
4. Interpret the sign – Because demand slopes down, PED will be negative; use the absolute value to classify elastic, inelastic, or unit?elastic.
5. Apply the total?revenue test – Compare the elasticity result to the direction of the price change to predict whether TR will rise or fall.
6. Sketch the demand curve (optional) – Mark the two price?quantity points, label the elastic and inelastic sections, and note the unit?elastic point where TR peaks.

Common Mistakes

• Mistake: Using the simple %? formula ((Q_2?Q_1)/Q_1) instead of the midpoint formula.
  Correction: The midpoint formula avoids base?year bias and is the AP?required method for any two?point elasticity calculation.

• Mistake: Forgetting to take the absolute value of PED when classifying elasticity.
  Correction: Demand elasticity is reported as a positive number; the negative sign merely reflects the law of demand.

• Mistake: Reversing the direction of the total?revenue test (e.g., saying “elastic-price increase raises revenue”).
  Correction: Remember: elastic-price-? TR ?; inelastic-price-? TR ?.

• Mistake: Treating a movement along the demand curve as a “shift” of the curve.
  Correction: A price change causes a movement along the existing demand curve; only changes in consumer preferences, income, or prices of related goods shift the curve.

• Mistake: Ignoring the “unit?elastic point” on a linear demand curve and assuming elasticity is constant.
  Correction: Elasticity varies along a linear demand curve; only at the unit?elastic point is TR maximized.

AP Exam Insights

1. FRQ Prompt Pattern: You’ll often be asked to “calculate the PED using the midpoint formula” for a given price/quantity change, then “explain how a tax on the good will affect total revenue.” Show all steps; the grader looks for the correct formula, correct arithmetic, and a clear interpretation.
2. Multiple?Choice Trap: Answer choices may list the direction of the revenue change but forget to specify the elasticity condition. Read the stem carefully—determine whether the price is rising or falling, then match it to the elasticity sign.
3. Graph Requirement: In a free?response, you may need to draw a demand curve, label the elastic and inelastic portions, and mark the unit?elastic point. Use a clean, labeled axis (P on vertical, Q on horizontal) and clearly indicate the two price?quantity points used in the calculation.
4. Distinguishing “Change in Demand” vs. “Change in Quantity Demanded”: The exam frequently tests whether you know that PED concerns a movement along the demand curve, not a shift. If the question mentions a “change in consumer income,” you must first decide if the good is normal or inferior before calculating PED.

Quick Check Questions

1. MC: A concert ticket price falls from \$120 to \$100 and quantity demanded rises from 2,000 to 2,500. Using the midpoint formula, the PED is closest to:
2. A) 0.45
3. B) 0.80
4. C) 1.25

5. D) 2.00
   Answer: C) 1.25. Calculation: (\frac{(2500?2000)/(2250)}{(100?120)/(110)} = \frac{500/2250}{?20/110} = 0.2222 ÷ (?0.1818) = 1.22-1.25 (elastic).

6. FRQ?style: A city raises the price of gasoline by 10?% and observes that gasoline demand falls by 5?%. Is the demand elastic or inelastic, and what happens to the city’s tax revenue?
   Answer: Inelastic (|PED|?=?0.5?<?1). Because demand is inelastic, the higher price raises tax revenue.

7. MC: At which point on a straight?line demand curve is total revenue maximized?

8. A) Where the curve is steepest
9. B) Where PED?=?0
10. C) Where PED?=?1 (unit?elastic)
11. D) At the intercept with the price axis
    Answer: C) Where PED?=?1. At the unit?elastic point, any price change leaves TR unchanged; a small move either way reduces TR, so the peak occurs there.

Last?Minute Cram Sheet (10 one?liners)

1. PED = %?Qd ÷ %?P – use the midpoint formula for AP calculations.
2. Midpoint numerator: ((Q_2?Q_1) / [(Q_2+Q_1)/2]); denominator: ((P_2?P_1) / [(P_2+P_1)/2]).
3. |PED|?>?1-Elastic-price-? TR ? (and opposite for price ?).
4. |PED|?<?1-Inelastic-price-? TR ? (and opposite for price ?).
5. |PED|?=?1-Unit?elastic-TR is at its maximum.
6. Elasticity varies along a linear demand curve – steep = inelastic, flat = elastic.
7. Substitutes-more elastic; necessities-more inelastic.
8. Total Revenue (TR) = P × Q – always compute before applying the TR test.
9. “Shift” vs. “Movement”: A price change moves along the demand curve; only a change in consumer preferences shifts the curve.
10. Midpoint formula is required even if the price change is small; the simple %? method is never accepted on the AP exam.