Monopolist Price and Quantity

Alzheimer's problem of the day.

The residents of Seltzer Springs, Michigan, consume bottles of mineral water according to the demand function D(p) = 1; 000 −p. Here D(p) is the demand per year for bottles of mineral water if the price per bottle is p. The sole distributor of mineral water in Seltzer Springs, Bubble Up,

purchases mineral water at c per bottle from their supplier Perry Air.  Perry Air is the only supplier of mineral water in the area and behaves as a profit-maximizing monopolist. For simplicity we suppose that it has zero costs of production.

(a)  What is the equilibrium price charged by the distributor Bubble Up?

(b)  What is the equilibrium quantity sold by Bubble Up?

This is an inverse demand function so rearrange the function to: P = 1000-Q.  Next, derive an expression for profit, which is total revenue minus total cost.  Profit = (1000 - Q)Q - cQ.  Next, distribute and take the first-order condition. 1000 - 2Q - c.  Next, set marginal revenue equal to marginal cost.  c = 1000 - 2Q.  Solve for Q.  Q = (1000 - c)/2.  Now, plug in this expression into the demand function. P = (1000 + c) /2