Calculate price elasticity of demand and supply for the following functions when (a) P=8 and (b) Q=6.
i. P= 40 – 0.5Q
ii. Q= -40 + 0.75P
iii. Q – P + 2 = 0
iv. 2P + 0.2Q = 40
v. P = 20 – 2Q
vi. 4P + 4Q = 64
The Correct Answer and Explanation is:
To calculate price elasticity of demand (Ed) and price elasticity of supply (Es) at specific points, we use: Price Elasticity of Demand (Ed)=(dQdP)⋅(PQ)andPrice Elasticity of Supply (Es)=(dQdP)⋅(PQ)\text{Price Elasticity of Demand (Ed)} = \left(\frac{dQ}{dP}\right) \cdot \left(\frac{P}{Q}\right) \quad \text{and} \quad \text{Price Elasticity of Supply (Es)} = \left(\frac{dQ}{dP}\right) \cdot \left(\frac{P}{Q}\right)
i. P=40−0.5QP = 40 – 0.5Q
Rewriting: Q=80−2PQ = 80 – 2P, so dQdP=−2\frac{dQ}{dP} = -2
- At P = 8:
Q=80−2(8)=64Q = 80 – 2(8) = 64
Ed=−2⋅864=−0.25Ed = -2 \cdot \frac{8}{64} = -0.25 - At Q = 6:
P=40−0.5(6)=37P = 40 – 0.5(6) = 37
Ed=−2⋅376=−12.33Ed = -2 \cdot \frac{37}{6} = -12.33
ii. Q=−40+0.75PQ = -40 + 0.75P, dQdP=0.75\frac{dQ}{dP} = 0.75
- At P = 8:
Q=−40+0.75(8)=−34Q = -40 + 0.75(8) = -34
Es=0.75⋅8−34=−0.176Es = 0.75 \cdot \frac{8}{-34} = -0.176 - At Q = 6:
6=−40+0.75P⇒P=61.336 = -40 + 0.75P \Rightarrow P = 61.33
Es=0.75⋅61.336=7.67Es = 0.75 \cdot \frac{61.33}{6} = 7.67
iii. Q−P+2=0⇒Q=P−2Q – P + 2 = 0 \Rightarrow Q = P – 2, dQdP=1\frac{dQ}{dP} = 1
- At P = 8: Q=6Q = 6, so Ed=1⋅86=1.33Ed = 1 \cdot \frac{8}{6} = 1.33
- At Q = 6: P=8P = 8, so Ed=1.33Ed = 1.33
iv. 2P+0.2Q=40⇒Q=200−10P2P + 0.2Q = 40 \Rightarrow Q = 200 – 10P, dQdP=−10\frac{dQ}{dP} = -10
- At P = 8: Q=200−10(8)=120Q = 200 – 10(8) = 120
Ed=−10⋅8120=−0.667Ed = -10 \cdot \frac{8}{120} = -0.667 - At Q = 6: P=19.4P = 19.4
Ed=−10⋅19.46=−32.33Ed = -10 \cdot \frac{19.4}{6} = -32.33
v. P=20−2Q⇒Q=10−0.5PP = 20 – 2Q \Rightarrow Q = 10 – 0.5P, dQdP=−0.5\frac{dQ}{dP} = -0.5
- At P = 8: Q=10−0.5(8)=6Q = 10 – 0.5(8) = 6
Ed=−0.5⋅86=−0.667Ed = -0.5 \cdot \frac{8}{6} = -0.667 - At Q = 6: Same result, so Ed=−0.667Ed = -0.667
vi. 4P+4Q=64⇒Q=16−P4P + 4Q = 64 \Rightarrow Q = 16 – P, dQdP=−1\frac{dQ}{dP} = -1
- At P = 8: Q=8Q = 8
Ed=−1⋅88=−1Ed = -1 \cdot \frac{8}{8} = -1 - At Q = 6: P=10P = 10, so Ed=−1⋅106=−1.67Ed = -1 \cdot \frac{10}{6} = -1.67
Explanation
Price elasticity of demand (Ed) and supply (Es) measure how responsive quantity demanded or supplied is to changes in price. The general formula is: E=(dQdP)⋅(PQ)E = \left( \frac{dQ}{dP} \right) \cdot \left( \frac{P}{Q} \right)
This combines the slope of the demand/supply function and the price-to-quantity ratio at a specific point.
In linear demand curves (e.g., P=a−bQP = a – bQ), demand elasticity becomes more inelastic as we move down the curve. For example, in function (i), at a high quantity (Q = 64), Ed is only –0.25 (inelastic), but at low quantity (Q = 6), Ed becomes highly elastic (–12.33).
Supply functions, such as in (ii), may produce unrealistic outputs (like negative quantity at low prices), highlighting the importance of evaluating economic meaning. A negative quantity supplied is not feasible, suggesting a need to limit the domain or re-specify the model.
Functions (iii) and (vi) are linear and symmetric: as price rises, quantity rises or falls by a constant rate. Elasticity values are relatively consistent and depend more directly on the price and quantity values used.
Some functions like (iv) and (v) demonstrate that elasticity can change significantly with quantity or price, even for linear relationships. In particular, (iv) shows a steep change in elasticity from –0.67 to –32.33 as quantity falls from 120 to 6, reflecting how steep curves have more volatile elasticities.
Understanding elasticity is crucial in real-world economics: inelastic demand means price changes won’t affect quantity much (e.g., for necessities), while elastic demand means consumers are sensitive to price changes (e.g., for luxury goods).
