Question

Demand: Qd=90-4P, where Qd is quantity demanded and P is price

Supply: Qs=-100+15P, where Qs is quantity supplied and P is price

Recall that equilibrium price was 19, while quantity was 50. At that price, the price elasticity of demand was -0.80.

1. Now I want you to rearrange each equation, putting P on the left-hand side, and solve again for equilibrium P and Q (you ought to get the same answer).
   1. Now we want to figure the monopoly price. Take the supply equation that you just developed (with P on the left-hand side), and make it your marginal cost equation. That is, just replace P with MC.

2. Total revenue is (P*Qd) and marginal revenue is the first derivative of total revenue with respect to Qd. Calculate total revenue and marginal revenue equations, derived from the demand equation developed above.

3. Now calculate price and quantity at the monopoly equilibrium (set MR=MC, solve for Q, the solve for P from the demand equation). How does monopoly P and Q compare to that calculated under competition?

4. What’s the price elasticity of demand at the monopoly price and quantity? Is it more or less elastic than the price elasticity at the competitive equilibrium?

1. A price elasticity of demand, estimated at the profit-maximizing monopoly price, will always be in the elastic range (that is, less than -1.0). Can you explain why?

Answer 1

A) Qs = -100 + 15P

15P = Qs + 100

P = Q/15 + 6.67

MC = Q/15 + 6.67

B) Qd = 90-4P

4P = 90-Qd

P = 22.5 - 0.25Qd

TR = PQd = 22.5Qd - 0.25Qd^2

MR = dTR/dQd = 22.5 - 0.5Qd

C)

At profit maximizing point

MR = MC

22.5-0.5Q = Q/15 + 6.67

15.83 = 0.567Q

Q* = 28 units

P* = $15.5

Both monopoly P and Q are lower than perfect competition

D)

For monopoly,

1/e = (P-MC)/P

1/e = (15.5-8.53)/15.5

1/e = 0.45

e = 2.22 (highly elastic)

E)

This is because a monopoly operates at profit maximizing point and increasing any price further would lead to a drop in sales and thus total revenues (which is a trait of elastic demand)

Hence, the firm operates at the highest possible range of elastic portion of its demand curve, where increasing price further would lead to fall in sales and revenues.