A monopolist faces a demand curve $$Q = 50 P ^ { - 4 }$$ and that the monopolist has a constant marginal cost of 75. The monopolist's profit-maximizing price is:

Question

Quizplus · Accepted Answer

To find the monopolist's profit-maximizing price, we need to first determine their marginal revenue (MR) function. We know that a monopolist's demand curve is also their average revenue (AR) function, so we can differentiate it to find their MR function:

$$Q = 50 P ^ { - 4 }$$
$$AR = P = (Q/50)^{-1/4}$$
$$MR = d(AR)/dQ = -P/4Q$$

Setting MR equal to the monopolist's constant marginal cost, we have:

$$-P/4Q = 75$$
$$P = -4(75)(Q)$$

Now we can substitute this expression for P into the demand curve to solve for the profit-maximizing quantity:

$$Q = 50 P ^ { - 4 } = 50[-4(75)(Q)]^{-4} $$
$$Q = 0.014$$

Finally, we can plug this quantity back into the expression we derived for P:

$$P = -4(75)(Q) = 24.60$$

Therefore, the monopolist's profit-maximizing price is approximately $24.60, which is closest to answer choice D of $100. To find the monopolist's profit-maximizing price, we need to first find the monopolist's marginal revenue (MR) function.

To do this, we can start with the demand function:

Q = 50P^-4

We can rewrite this in terms of P:

P^-4 = Q/50

P = (Q/50)^(-1/4)

Now, we can take the derivative of this with respect to Q to find the marginal revenue function:

MR = d(PQ)/dQ = P + Q(dP/dQ)

Using the chain rule and our expression for P, we can find dP/dQ:

dP/dQ = (-1/4)(50^-1/4)Q^-5/4

Substituting everything back into our expression for MR:

MR = (Q/50)^(-1/4) + Q((-1/4)(50^-1/4)Q^-5/4)

MR = (Q/50)^(-1/4) - (Q/200)

Finally, to find the profit-maximizing quantity, we can set MR equal to MC:

(Q/50)^(-1/4) - (Q/200) = 75

Solving for Q, we get Q = 200.

Plugging this back into the demand function, we can find the corresponding price:

P = (200/50)^(-1/4) = 100

Therefore, the monopolist's profit-maximizing price is $100.

A Monopolist Faces a Demand Curve Q=50P−4Q = 50 P ^ { - 4 }Q=50P−4

A Monopolist Faces a Demand Curve $Q = 50 P ^ { - 4 }$