# An engineering industry has the following market supply and demand equations: P=100-2Qd P=30+3QSSuppose the industry is taken over by a monopolist. What price would the monopolist charge to maximize its profits?A) 80B) 90C) 100D)110I worked out the equilibrium price and used it with P= 100-4Qd but not getting the right answer. Your intuition that the monopoly faces a marginal revenue curve with twice the slope of the demand curve is correct. I'll walk through the problem and see if I can find where you might have made a mistake.

From what you described, it sounds like you might have missed the fact that the equilibrium price will change when the market switches from competition to a monopoly.

Part of being a monopoly is that they get to choose both how much they make and how much they charge, subject to their costs and the market demand. They aren't constrained by the market price the way a competitive firm would be.

In general the monopolist maximizes this function (technically called a Lagrangian, though it's often taught without using that terminology):

profit = P*Q - C(Q) = max

The first-order condition for a local maximum is that the derivative must be zero:

0 = dP/dQ * Q + P(Q) - C'(Q)

The supply curve is assumed to be the same as the marginal cost function, so

P = 30 + Q_S
C'(Q) = 30 + 3Q

The demand curve is the function that gives us P(Q) and dP/dQ:

P = 100 - 2 Q_d
P(Q) = 100 - 2 Q
dP/dQ = -2

Substituting all that back in gives us:

0 = -2*Q + 100 - 2Q - (30 + 3Q)
100 - 4Q = 30 + 3Q

70 = 7Q
10 = Q

That's the quantity that the monopolist wishes to produce and sell. To see how much they can charge and still sell that many, we plug back into the demand curve:

P = 100 - 2(10) = 80

So the price that the monopoly wants to charge is \$80 per unit, choice A.

(As a quick test-taking strategy, you can eliminate C and D immediately based on this demand curve. For P = 100 - 2Q, P = 110 can't possibly be right because Q would have to be negative and a negative quantity sold makes no sense. P = 100 is also very unlikely because you can't be making profits at Q = 0. That is at least possible, if it turns out that this whole market is failing proposition and your best bet is to drop out completely; but that doesn't seem to fit the context of this problem. Choices A and B are the only plausible answers from the very start.)

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