Given a demand equation

where Q represents the quantity demanded and P the selling price

a) Calculate the arc-price elasticity of demand

when P1 = 4 GHS and P2= 3 GHS (5 marks)

b) Calculate the point-price elasticity of demand at these prices; (4 marks)

i.Is the demand for this good elastic or inelastic at these prices? (2 marks)

c) What else can you say about the relationship between the price elasticity of demand and total revenue at these prices. (4 marks)

d) What is the price elasticity of demand at the price that maximises total revenue? (5 marks)

The Correct Answer and Explanation is:

Sure! To answer your question fully, I need the demand equation you mentioned but didn’t provide. Since it’s missing, I’ll assume a common linear demand function for illustration: Q=a−bPQ = a – bP

Let’s assume a specific example to work through your question. Suppose the demand function is: Q=100−10PQ = 100 – 10P

where QQ is quantity demanded and PP is the price in GHS.

a) Calculate the arc-price elasticity of demand between P1=4P_1 = 4 GHS and P2=3P_2 = 3 GHS

Arc elasticity formula is: Ed=ΔQΔP×P1+P2Q1+Q2E_d = \frac{\Delta Q}{\Delta P} \times \frac{P_1 + P_2}{Q_1 + Q_2}

where:

ΔQ=Q2−Q1\Delta Q = Q_2 – Q_1
ΔP=P2−P1\Delta P = P_2 – P_1
Q1=100−10×4=100−40=60Q_1 = 100 – 10 \times 4 = 100 – 40 = 60
Q2=100−10×3=100−30=70Q_2 = 100 – 10 \times 3 = 100 – 30 = 70

Calculate: ΔQ=70−60=10\Delta Q = 70 – 60 = 10 ΔP=3−4=−1\Delta P = 3 – 4 = -1

Plug into formula: Ed=10−1×4+360+70=−10×7130=−10×0.0538=−0.538E_d = \frac{10}{-1} \times \frac{4 + 3}{60 + 70} = -10 \times \frac{7}{130} = -10 \times 0.0538 = -0.538

Arc elasticity = -0.538

b) Calculate the point-price elasticity of demand at P=4P = 4 and P=3P = 3

The point elasticity formula is: Ed=dQdP×PQE_d = \frac{dQ}{dP} \times \frac{P}{Q}

For a linear demand Q=100−10PQ = 100 – 10P, derivative is: dQdP=−10\frac{dQ}{dP} = -10

At P=4P = 4: Q=60Q = 60 Ed=−10×460=−4060=−0.67E_d = -10 \times \frac{4}{60} = -\frac{40}{60} = -0.67

At P=3P = 3: Q=70Q = 70 Ed=−10×370=−3070=−0.43E_d = -10 \times \frac{3}{70} = -\frac{30}{70} = -0.43

b(i) Is demand elastic or inelastic?

Elasticity absolute values:

At P=4P=4, ∣Ed∣=0.67<1|E_d| = 0.67 < 1 → Inelastic demand
At P=3P=3, ∣Ed∣=0.43<1|E_d| = 0.43 < 1 → Inelastic demand

So, demand is inelastic at both prices.

c) Relationship between price elasticity of demand and total revenue

Total revenue TR=P×QTR = P \times Q.

When demand is inelastic (∣Ed∣<1|E_d| < 1), a decrease in price causes total revenue to decrease because the percentage increase in quantity demanded is less than the percentage decrease in price.
At P=4P=4 and P=3P=3, demand is inelastic, so lowering the price from 4 to 3 reduces total revenue.

Calculate:

TR1=4×60=240TR_1 = 4 \times 60 = 240
TR2=3×70=210TR_2 = 3 \times 70 = 210

Total revenue falls from 240 to 210 when price drops from 4 to 3, consistent with inelastic demand.

d) Price elasticity of demand at the price that maximizes total revenue

Total revenue maximizes where demand elasticity ∣Ed∣=1|E_d| = 1 (unit elastic).

For a linear demand Q=a−bPQ = a – bP, the price maximizing total revenue is: P∗=a2bP^* = \frac{a}{2b}

Using a=100a=100, b=10b=10: P∗=1002×10=10020=5P^* = \frac{100}{2 \times 10} = \frac{100}{20} = 5

At P=5P = 5: Q=100−10×5=50Q = 100 – 10 \times 5 = 50 Ed=−10×550=−1E_d = -10 \times \frac{5}{50} = -1

Price elasticity at P=5P=5 is -1 (unit elastic).

Explanation (300 words)

Price elasticity of demand measures how sensitive the quantity demanded of a good is to changes in its price. The arc-price elasticity calculates this sensitivity between two price points by considering the average percentage change in both price and quantity, providing an overall estimate over the range. The point-price elasticity calculates this at a specific price and quantity, giving a more precise local measure.

In our example, the arc elasticity between prices 4 GHS and 3 GHS is approximately -0.538, indicating inelastic demand over this price range. The point elasticities calculated at the individual prices confirm this, with values of -0.67 and -0.43, both less than one in absolute value, thus inelastic.

When demand is inelastic, consumers do not respond strongly to price changes. Hence, reducing prices in this inelastic range decreases total revenue because the percentage increase in quantity demanded does not compensate for the price drop. This is shown by total revenue decreasing from 240 GHS to 210 GHS when price falls from 4 to 3 GHS.

Total revenue is maximized at the point where demand is unit elastic (∣Ed∣=1|E_d|=1), balancing the effect of price and quantity changes. For the linear demand assumed, this occurs at a price of 5 GHS, where elasticity is exactly -1. At this point, any further price changes will reduce total revenue, with demand becoming elastic if price falls below 5 or inelastic if price rises above 5.

Understanding elasticity and its relationship with total revenue is crucial for firms when making pricing decisions to maximize revenue and profits. If demand is inelastic, increasing prices can increase total revenue; if elastic, lowering prices might increase revenue. The unit elastic point is the critical balance where total revenue peaks.