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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.

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Hint

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 \begin{equation} -1=\frac{\mathrm dQ}{\mathrm dP}\cdot\frac{P}{Q}=-\frac1b\cdot\frac{P}{Q} \end{equation} and the demand equation to solve for the two unknowns $P$ and $Q$.

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  • $\begingroup$ 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? $\endgroup$
    – NadTeX
    Sep 17, 2018 at 11:25
  • $\begingroup$ @NadTeX you just need to compare the quantity obtained from the formula to the upper limit. If the upper limit is higher then the production constraint is not binding. If the upper limit is binding, output would have to be at the upper bound. $\endgroup$
    – Herr K.
    Sep 17, 2018 at 12:29
  • $\begingroup$ Thanks. Indeed I double checked the optimal output and prices where the upper limit is binding using Linear Programming and got the same answers. $\endgroup$
    – NadTeX
    Sep 24, 2018 at 6:17

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