Question

# Consider a market with demand given by P = a−bQ, where a,b > 0. Suppose the...

Consider a market with demand given by P = a−bQ, where a,b > 0. Suppose the only supplier in the market has costs given by C(Q) = mQ+F, where ﬁxed cost is F > 0 and sunk and m is positive but small enough that the ﬁrm does not shut down.

(1) Find the monopolist’s proﬁt-maximizing price and quantity and the monopolist’s proﬁts.

(2) Show that the proﬁt-maximizing monopolist produces on the elastic portion of its demand curve (in the short run).

1)

P = a - bQ (Market demand curve)

Total revenue = PQ = (a-bQ)Q

Marginal revenue

MR =

Cost function C(Q) = mQ + F where F > 0 are the fixed costs

For the profit maximizing monopolist,

(Equilibrium quantity)

Equilibrium price is found from the inverse demand curve:

And monopolist's profits

2)

Inverse demand curve: P = a - bQ

Price elasticity of demand

From our inverse demand curve, we can find that

But

Thus proﬁt-maximizing monopolist produces on the elastic portion of its demand curve.

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