Consider the demand equation Q = 25 – 3P, where Q represents quantity demanded and P the selling price.
a. Calculate the arc-price elasticity of demand when P1= $4 and P2 = $3.
b. Calculate the point-price elasticity of demand at these prices. Is the demand for this good elastic or inelastic at these prices?
c. What, if anything, can you say about the relationship between the price elasticity of demand and total revenue at these prices?
d. What is the price elasticity of demand at the price that maximizes total revenue?
The Correct Answer and Explanation is:
Let’s solve this problem step-by-step using the demand equation:
Demand Equation:
Q=25−3PQ = 25 – 3P
a. Arc-Price Elasticity of Demand
Formula: Earc=ΔQΔP×P1+P2Q1+Q2E_{arc} = \frac{\Delta Q}{\Delta P} \times \frac{P_1 + P_2}{Q_1 + Q_2}
First, calculate Q1Q_1 and Q2Q_2:
- When P1=4P_1 = 4:
Q1=25−3(4)=13Q_1 = 25 – 3(4) = 13 - When P2=3P_2 = 3:
Q2=25−3(3)=16Q_2 = 25 – 3(3) = 16
Now plug into the formula: Earc=16−133−4×4+316+13=3−1×729=−3×729=−2129≈−0.72E_{arc} = \frac{16 – 13}{3 – 4} \times \frac{4 + 3}{16 + 13} = \frac{3}{-1} \times \frac{7}{29} = -3 \times \frac{7}{29} = -\frac{21}{29} \approx -0.72
b. Point-Price Elasticity of Demand
Formula: Ep=dQdP×PQE_p = \frac{dQ}{dP} \times \frac{P}{Q}
From the demand function Q=25−3PQ = 25 – 3P, the derivative is: dQdP=−3\frac{dQ}{dP} = -3
- At P=4P = 4, Q=13Q = 13: Ep=−3×413=−1213≈−0.92E_p = -3 \times \frac{4}{13} = -\frac{12}{13} \approx -0.92
- At P=3P = 3, Q=16Q = 16: Ep=−3×316=−916≈−0.56E_p = -3 \times \frac{3}{16} = -\frac{9}{16} \approx -0.56
Elastic or Inelastic?
- Since the absolute values are less than 1 at both prices, demand is inelastic at both.
c. Elasticity and Total Revenue Relationship
Total revenue (TR) is P×QP \times Q. When demand is inelastic (∣Ep∣<1|E_p| < 1), a price increase leads to higher revenue, and a price decrease lowers revenue.
- At P=4P = 4:
TR = 4×13=524 \times 13 = 52 - At P=3P = 3:
TR = 3×16=483 \times 16 = 48
Since TR decreases as price decreases (from $4 to $3), and elasticity is inelastic in this range, the inverse relationship holds.
d. Price Elasticity at Revenue-Maximizing Price
To find the price that maximizes TR, set elasticity equal to -1 (unit elastic). Use: Ep=−3×PQ⇒−3×P25−3P=−1E_p = -3 \times \frac{P}{Q} \Rightarrow -3 \times \frac{P}{25 – 3P} = -1
Solving: 3P25−3P=1⇒3P=25−3P⇒6P=25⇒P=256≈4.17\frac{3P}{25 – 3P} = 1 \Rightarrow 3P = 25 – 3P \Rightarrow 6P = 25 \Rightarrow P = \frac{25}{6} \approx 4.17
At this price, demand is unit elastic, and total revenue is maximized.
Summary (Explanation)
The price elasticity of demand measures how sensitive quantity demanded is to a change in price. The arc-price elasticity over the range from $4 to $3 is approximately -0.72, indicating inelastic demand. Point-price elasticity confirms this: at both $4 and $3, demand remains inelastic. When demand is inelastic, total revenue moves in the same direction as price—raising prices increases revenue. This explains why total revenue fell when price dropped from $4 to $3.
The price that maximizes total revenue occurs when the elasticity is exactly -1 (unit elastic). Solving the elasticity equation gives us a revenue-maximizing price of approximately $4.17. At this point, any further price changes (up or down) would reduce revenue, showing the balance between price and demand response. Understanding this relationship helps businesses and economists make strategic pricing decisions to optimize revenue.