Suppose a profit maximizing monopolist faces the following demand curve: Q = 90 – 2P, and sets a uniform price where the elasticity of demand is - 2. What must be the MC of this firm at the profit-maximizing level of output?

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BlackTom AI · Accepted Answer

We start by organizing the given information and the standard monopoly relationships.
From the demand function Q = 90 − 2P, the inverse demand is P = 45 − Q/2. The total revenue is TR = P·Q = (45 − Q/2)Q = 45Q − Q^2/2, so marginal revenue is MR = d(TR)/dQ = 45 − Q.
For a profit-maximizing monopoly with MC constant, the condition is MR = MC.
Next, we use the elasticity of demand. The elasticity is ε = (dQ/dP)·(P/Q). Since Q = 90 − 2P, we have dQ/dP = −2. Substituting gives ε = (−2)·(P/Q) = −2P/Q.
We are told the elasticity is −2, so set −2P/Q = −2, which simplifies to P/Q = 1, i.e., Q = P.
Plugging Q = P into the demand equation Q = 90 − 2P yields P = 90 − 2P → 3P = 90 → P = 30 and thus Q = 30.
Now evaluate MR at this output: MR = 45 − Q = 45 − 30 = 15.
Under a constant MC firm, MR = MC at the optimum, so MC equals 15.
In summary, using the elasticity condition to locate the price and quantity where ε = −2, then applying the MR = MC rule, yields the marginal cost consistent with profit maximization and the given demand schedule.

COMM_V 295 105 106 2025W1 Participation/Math Quiz #1

Suppose a profit maximizing monopolist faces the following demand curve: Q = 90 – 2P, and sets a uniform price where the elasticity of demand is - 2. What must be the MC of this firm at the profit-maximizing level of output?

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