Question 2.19: Given the demand function for computers as P = 2400 − 0.5Q. ......
Given the demand function for computers as P = 2400 − 0.5Q
(a) Determine the coefficient of point elasticity of demand when (i) P = 1800, (ii) P = 1200, and (iii) P = 600. Give a verbal description of each result.
(b) If the price of computers increases by 12%, calculate the percentage change in the quantity demanded at P = 1800, P = 1200 and P = 600.
(i) First use the definition of elasticity:
ε_d = \frac{\text {percentage change in quantity demanded}}{\text {percentage change in price}} = \frac{ΔQ_d(\%)}{ΔP(\%)}
(ii) Then calculate the exact percentage changes and compare them with the answers in (i).
(c) Graph the demand function, indicating where demand is elastic, unit elastic and inelastic.
(a) (i) At P = 1800 the quantity of computers demanded, Q, is calculated by substituting P = 1800 into the demand function:
P = 2400 − 0.5Q → 1800 = 2400 − 0.5Q → 0.5Q = 600 → Q = 1200
The value of point elasticity of demand at (P = 1800, Q = 1200) is calculated by substituting these values along with b = 0.5 into formula (2.12):
ε_d = – \frac{1}{b} \frac{P_0}{Q_0}
=-\frac{1}{0.5}\frac{1800}{1200}=-\frac{1800}{600}=-\frac{3}{1}=-3
The coefficient of point elasticity of demand is ε_d =−3, which indicates that at the price P = 1800 a 1% increase (decrease) in price will cause a 3% decrease (increase) in the quantity of computers demanded. Demand is elastic, |ε_d | > 1.
(ii) and (iii) The calculations for price elasticity of demand for P = 1200 and P = 600 are carried out in the same way as (i) and are summarised below.
Notes:
ε_d = −1 indicates that at P = 1200, a 1% increase (decrease) in price will cause a 1% decrease (increase) in the quantity of computers demanded.
ε_d =−0.33 (correct to 2 decimal places) indicates that at P=600, a 1% increase (decrease) in price will cause a 0.33% decrease (increase) in the quantity of computers demanded.
(b) (i) The definition
ε_d = \frac{\text {percentage change in quantity demanded}}{\text {percentage change in price}} = \frac{ΔQ_d(\%)}{ΔP(\%)}
may be rearranged as
ΔQ(%) = ΔP(%) × ε_d (2.13)
- At P = 1800 we have calculated that ε_d =−3. To calculate the percentage change in Q, substitute ε_d =−3 and ΔP = 12% into equation (2.13):
ΔQ = 12%×−3 = −36%
The quantity demanded decreases by 36%, a larger percentage decrease than the 12% price increase. Demand is strongly responsive to price change and is described as elastic demand.
- P = 1200: ε_d =−1, ΔP = 12%. Substitute into equation (2.13):
ΔQ = 12%×−1 = −12%
The quantity demanded decreases by 12%; the percentage increase in price is 12%. This is described as unit elastic demand.
- P = 600: εd =−0.33, ΔP = 12%. Substitute into equation (2.13):
ΔQ = 12%×−0.33 = −3.96%
The quantity demanded decreases by 3.96%, a smaller percentage decrease than the 12% price increase. So demand is weakly responsive to price, hence it is described as inelastic demand.
(ii) The calculation of the exact percentage change in Q requires basic arithmetic. Start by calculating the price which results from an increase of 12% on the initial price.
Increase P = 1800 by 12% → P_{new} = \frac{112} {100} P = 1.12(1800) = 2016
Then calculate the corresponding values of Q from the equation of the demand function. When P = 1800, Q = 1200 and when P = 2016, Q = 768.
Comment: The exact value of elasticity, −1/3, not the rounded value, −0.33, was used for accurate results when P = 600. The rounded value of elasticity, −0.33, would give approximately the same answer, −3.96%, as obtained by direct calculation. At prices P = 1800 and P = 1200, no rounding of ε_d was required; the percentage change in demand was exactly the same, whether calculated directly or calculated by equation (2.13). In general, the computer firm is not only interested in the percentage change in the price of computers, but also in the percentage change in demand which results from the price changes.
(c) The graph of the demand function is given in Figure 2.35.
| P_0 | 1800 | 1200 | 600 |
| Calculate Q_0 | P_0 = 2400 − 0.5Q_0
1800 = 2400 − 0.5 Q_0 0.5 Q_0 = 600 Q_0 = 1200 ε_d = – \frac{1}{b} \frac{P_0}{Q_0} |
P_0 = 2400 − 0.5Q_0
1200 = 2400 − 0.5 Q_0 0.5 Q_0 = 1200 Q_0 = 2400 ε_d = – \frac{1}{b} \frac{P_0}{Q_0} |
P_0 = 2400 − 0.5Q_0
600 = 2400 − 0.5 Q_0 0.5 Q_0 = 1800 Q_0 = 3600 ε_d = – \frac{1}{b} \frac{P_0}{Q_0} |
| ε_d for P = 2400 − 0.5Q |
=-\frac{1}{0.5}\frac{1800}{1200} = -3 | =-\frac{1}{0.5}\frac{1800}{2400}
=-\frac{1200}{1200}
=−1 (see notes) |
=-\frac{1}{0.5}\frac{600}{3600}
=-\frac{1}{3}
≈−0.33 (notes) |
| Demand | Demand is elastic |ε_d| > 1 | Demand is unit elastic |ε_d| > 1 | Demand is inelastic |ε_d| > 1 |
| nitial P, Q | New P, Q | Percentage change in Q by direct calculation | ε_d | ΔQ(%) = ΔP(%) × ε_d by formula (2.13) |
| 1800, 1200 | 2016, 768 | \frac {768 − 1200}{1200} × 100 = −36% | -3 | ΔQ=−3 × 12=−36% |
| 1200, 2400 | 1344, 2122 | \frac {2112 − 2400}{2400} × 100 = −12% | -1 | ΔQ=−1 × 12=−12% |
| 600, 3600 | 672, 3456 | \frac {3456 − 3600}{3600} × 100 = −4% | \frac {1}{3} ≈ −0.33 | ΔQ = −\frac {1}{3} × 12 ≈ −4% |
| ≈ means approximately equal to | ||||
