When price elasticity of demand = -4, the optimal markup on cost is:

Question

BlackTom AI · Accepted Answer

To approach this question, I’ll lay out the relationship between price, cost, and elasticity, then apply it to the given elasticity value.
First, recall the Lerner index formula: L = (P − MC)/P, which links markup relative to price to the elasticity of demand. When ε (price elasticity of demand) is negative, the standard result is L = −1/ε. Here ε = −4, so L = −1/(−4) = 0.25. This tells us that the markup relative to price is 25% of the price.
Next, convert this to markup on cost, which is defined as (P − MC)/MC. Using the relationships P = MC/(1 − L) and (P − MC)/MC = (P/MC) − 1, we can compute:
- P/MC = 1/(1 − L) = 1/(1 − 0.25) = 1/0.75 ≈ 1.333...
- Therefore, (P − MC)/MC = (P/MC) − 1 ≈ 1.333... − 1 = 0.333... = 33.33%.
Now let’s evaluate the options:
- Option a: 25% — This would be the markup relative to price (the Lerner index), not the markup on cost. It misrepresents the conversion from L to the cost-based markup.
- Option b: 33% — This matches the calculation above for markup on cost when ε = −4, since 0.25/(1 − 0.25) = 0.333... ≈ 33.3%. This is the correct interpretation.
- Option c: 75% — This would occur if you erroneously treated the 25% as the markup on cost without the necessary division by (1 − L); it overstates the markup.
- Option d: 400% — This is far off from the economic relationship here and indicates a misapplication of the formulas or a misunderstanding of elasticity effects.
In summary, the correct derivation shows that with elasticity −4, the optimal markup on cost is about 33%, corresponding to option b, while the other choices reflect common misinterpretations of the Lerner index and cost-based markup relationships.

When price elasticity of demand = -4, the optimal markup on cost is:

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