Salt City Donuts is a local donut shop selling artisanal donuts such as a maple bacon donut with real bacon on top, a stuffed peaches and cream variety, and a full breakfast donut featuring egg and sausage on top. Salt City Donuts has marginal cost of $2 per donut. The donut shop knows that the price elasticity of demand for their donuts is -4 when they are charging a price of $8. The firm’s markup is:

Question

BlackTom AI · Accepted Answer

The problem asks for the firm's markup given marginal cost, the price, and the price elasticity of demand at that price. We start by identifying the known values and the typical ways to express markup.

- Known values: marginal cost (MC) = $2, price (P) = $8, elasticity of demand (in absolute value) |ε| = 4 at that price.
- A common way to think about markup relative to price is (P − MC) / P, which is the portion of the price that covers costs. Another related measure is the Lerner index, (P − MC) / P = 1/|ε| when price is set optimally for a monopolist with constant elasticity, though the data here gives explicit P and MC, so we can compute the markup directly from those.

Option by option analysis:

Option 1: 1.25
- If the markup were 1.25, that would imply (P − MC) / P = 1.25, which is impossible since a markup over price cannot exceed 1 (the numerator cannot be larger than the denominator in a ratio of positive quantities like (P − MC) to P). This value is inconsistent with the given P = 8 and MC = 2. Therefore this option is incorrect.

Option 2: 0.75
- Compute the markup as (P − MC) / P = (8 − 2) / 8 = 6 / 8 = 0.75. This yields the proportion of the price that covers costs, and matches the given numbers. This option is consistent with P = 8 and MC = 2, so it is correct.

Option 3: 0.5
- If the markup were 0.5, then (P − MC) / P = 0.5, which implies P − MC = 0.5P, so MC = 0.5P, and with P = 8 we would get MC = 4. But the marginal cost is given as 2, not 4, so this option does not fit the data. This option is incorrect.

Option 4: 0.33
- A markup of 0.33 would imply (P − MC) / P ≈ 1/3, so P − MC ≈ (1/3)P. With P = 8, that would give P − MC ≈ 2.666..., hence MC ≈ 5.333..., which again contradicts MC = 2. Therefore this option is incorrect.

Summary of reasoning: among the provided options, only 0.75 is consistent with the given price and marginal cost when evaluating the markup as the fraction of the price that covers costs, (P − MC) / P. The other options imply inconsistent relationships between P, MC, and the markup, given the specified values.

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