# Price elasticity and optimal pricing in a monopoly with zero marginal cost

I am having trouble understanding how to calculate the optimal price P for a good and understand the optimal price elasticity of demand in the following condition:

• The firm is a monopoly seeking to maximise profit.
• It also has a zero marginal cost (MR = 0).
• The firm can only produce/sell an upper limit/number of the good (I am not sure if this changes anything in the analysis)

I found a lot of documentation on the web regarding price elasticity, but I did not find (enough) details about the conditions described above. My understanding is that, even in the above conditions, the optimal price elasticity of demand is at unit elastic (E= -1) [is this correct?], but I have no clue how to calculate the optimal price of the good [is there any formula for it?]. Apologies if my questions is a bit broad, I am very new to this area of study and any help (or reference to a worked solution) is appreciated.

You are probably working with (or given) a linear demand function of the form $Q=\frac{a}{b}-\frac1bP$, or its equivalent inverse form $P=a-bQ$, where $a$ and $b$ are positive numbers.
Given that price elasticity of demand at the optimum is $-1$, which you were right to point out (assuming linear demand), you can use the elasticity formula $$-1=\frac{\mathrm dQ}{\mathrm dP}\cdot\frac{P}{Q}=-\frac1b\cdot\frac{P}{Q}$$ and the demand equation to solve for the two unknowns $P$ and $Q$.
• Is the formula for optimal pricing −1=dQ/dP⋅P/Q=−1/b⋅P/Q still applicable if there is an upper limit on the number of products that can be sold/produced? – NadTeX Sep 17 '18 at 11:25