Consider a market with demand given by Q 20p2 A firm has a

Consider a market with demand given by Q = 20/p^2. A firm has a constant marginal cost of 2 with no other fixed costs. Use the Lerner Index rule to answer the following questions. Suppose the government grants this firm an exclusive monopoly in this market. Find the price-cost margin. Then use that to find the profit-maximizing price. Suppose that, in addition to the firm, the government decides to supply d/p62 units of the good, where p is the price charged by the firm. Show that price and total quantity do not change.

Solution

a) As we know the demand is Q= 20/P² . Clearly, demand is of the form Q=Ap
Elasticity = dQ/dP *P/Q = (20*-2 * P^-2-1 ) * P/Q
   = -40 * (P^-2/Q)
   = -2 * (20/P²Q) = -2 * Q/Q
So, elasticity = -2
Price cost margin using lerner index will be :-
(P-MC)/P = 1/2
=> 2P- 2*2 = P
=> P= 4$
Therefore, Q* = 20/16 = 1.25
b) Let Q_r be the residual demand and q* be the quantity which government produces.
So, Q_r (p) = Q_d (p) - q*(p)
= 20/P² - 4/P²  = 16/P²
Again the demand is of the form Q=Ap so, the elasticity is again -2. The price cost margin will again be 1/2 and as MC has not changed the Price will not change. Similarly, we can say quantity will also not change.