When P0 and P1 and Q0 and Q1 denote before and after change in the price and quantity respectively and in both the situations, total outlay remains the same, which of the following formulae give the similar value of the arc price - elasticity of demand ? (a) \(\dfrac{Q_{0}-Q_{1}}{P_{0}-P_{1}}\times\dfrac{P_{0}+P_{1}}{Q_{0}+Q_{1}}\) (b) \(\dfrac{Q_{0}-Q_{1}}{P_{0}-P_{1}}\times\dfrac{P_{0}}{Q_{1}}\) (c) \(\dfrac{Q_{0}-Q_{1}}{P_{0}-P_{1}}\times\dfrac{P_{0}}{Q_{0}}\) (d) \(\dfrac{Q_{0}-Q_{1}}{P_{0}-P_{1}}\times\dfrac{P_{1}}{Q_{1}}\) (e) \(\dfrac{Q_{0}-Q_{1}}{P_{0}-P_{1}}\times\dfrac{P_{1}}{Q_{0}}\) Code :
(a), (b) and (e)
(a), (b) and (e) — option 2.
The condition that does all the work. “Total outlay remains the same” means the consumer’s expenditure is unchanged before and after :
\(P_{0}Q_{0}=P_{1}Q_{1}\)
This is exactly the case of unitary elasticity, and it forces a relationship between the prices and the quantities that makes several of the formulae collapse into one another.
Rewriting the condition. From \(P_{0}Q_{0}=P_{1}Q_{1}\) we get
\(\dfrac{P_{0}}{Q_{1}}=\dfrac{P_{1}}{Q_{0}}\)
Call that common value t. Then \(P_{0}=tQ_{1}\) and \(P_{1}=tQ_{0}\).
Now test each formula. Every one shares the factor \(k=\dfrac{Q_{0}-Q_{1}}{P_{0}-P_{1}}\), so only the second factor matters :
| Formula | Second factor | Value under the condition | Same as (a)? |
|---|---|---|---|
| (a) | \(\dfrac{P_{0}+P_{1}}{Q_{0}+Q_{1}}\) | \(\dfrac{tQ_{1}+tQ_{0}}{Q_{0}+Q_{1}}=t\) | — the benchmark |
| (b) | \(\dfrac{P_{0}}{Q_{1}}\) | \(\dfrac{tQ_{1}}{Q_{1}}=t\) | YES |
| (c) | \(\dfrac{P_{0}}{Q_{0}}\) | \(\dfrac{tQ_{1}}{Q_{0}}\ne t\) unless Q0 = Q1 | No |
| (d) | \(\dfrac{P_{1}}{Q_{1}}\) | \(\dfrac{tQ_{0}}{Q_{1}}\ne t\) | No |
| (e) | \(\dfrac{P_{1}}{Q_{0}}\) | \(\dfrac{tQ_{0}}{Q_{0}}=t\) | YES |
So (a), (b) and (e) all reduce to the same number, and (c) and (d) do not.
A numerical check. Take P0 = 2, Q0 = 30 and P1 = 3, Q1 = 20, so outlay is 60 in both cases. Here k = (30−20)÷(2−3) = −10. Then (a) gives −10 × 5÷50 = −1; (b) gives −10 × 2÷20 = −1; (e) gives −10 × 3÷30 = −1. But (c) gives −10 × 2÷30 = −0.67 and (d) gives −10 × 3÷20 = −1.5. The value −1 confirms unitary elasticity, exactly as constant outlay requires.
Why arc elasticity uses averages at all. Point elasticity is exact only for an infinitesimal change. Over a discrete move the answer differs depending on which end you treat as the base, so the arc formula — formula (a), which divides by the sums — takes the midpoint and gives one figure regardless of direction.
Hence, the answer is (a), (b) and (e).
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