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So this is very beginner level doubt.

I know $ MR = \frac{\partial TR}{\partial q} $ . So TR need to be in terms of q to differentiate. This helps us to find Optimal Output level in Monopoly by equating $MR=MC$

But can $ MR = \frac{\partial TR}{\partial p}$ when TR is in terms of p ? Will this help us in discovering Optimal Price level directly in Monopoly by equating $MR=MC$ without finding the Optimal Output ?

I had this doubt when I was working on some examples and getting the same Optimal Price but faster by doing it in the second method.'

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2 Answers 2

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You can take derivative with respect to price to derive optimum price and hence also quantity in monopoly case (to clarify you need to take derivative of whole profit function which yields $\partial TR/ \partial p= \partial TC/ \partial p$).

However, note $\partial TR/ \partial p$ is not marginal revenue, since marginal revenue is defined as increase in revenue when you sell more product not when you change price.

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  • $\begingroup$ Are you saying that you can obtain optimum price and quantity using $\partial TR/ \partial p$? Surely costs need to come into the calculation: shouldn't it be $\partial \Pi/\partial p$? $\endgroup$ Oct 10, 2023 at 8:02
  • $\begingroup$ @AdamBailey that’s what I meant in the first half of my comment $\endgroup$
    – 1muflon1
    Oct 10, 2023 at 9:09
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I think you are getting confused by looking at the math only. The only way a monopoly could "set" a price is by producing certain quantity. That is how OPEC "sets" prices for oil, they announce whether they gonna produce more or less and the market "prices in" these announcements. They do not just call prices of oil.

To bring it back to the math that you are looking at, price is not a choice variable here at all. The only choice variable is quantity which pins down the price via demand equation.

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  • $\begingroup$ What about Apple (which, unlike OPEC does have exclusive control over their product)? Do they set the price by announcing quantities? $\endgroup$
    – Giskard
    Oct 10, 2023 at 4:40
  • $\begingroup$ No monopoly can choose to either set quantity and let price adjust or set price and let demand adjust. Consider following problem: monopolist has cost function $TC=10q$, demand function is given by $Q=100-p$. You can solve this problem both with p and q as a choice variable and you will get the same result. OPEC works through quantity because they are actually not complete monopoly and in fact even they are cartel and hence option of setting price is not open to them but real monopoly can choose $\endgroup$
    – 1muflon1
    Oct 10, 2023 at 9:44
  • $\begingroup$ Monopoly takes the demand curve as given and the only way they can "choose" the price is by providing a certain quantity to the market. For any given demand curve there would be only one profit maximizing (p*,q*), where the demand is unit elastic. Any monopoly could then achieve a profit-maximizing point by producing the corresponding q*. Market conditions decide the price for the monopoly, unless we enter the world of non-profit maximizing monopolies. $\endgroup$
    – IPII
    Oct 10, 2023 at 15:25
  • $\begingroup$ Profit maximizing monopoly cannot choose arbitrary price, but that does not mean it cannot choose profit maximizing price. Again just try it yourself. Take any demand function and calculate $\max_q \Pi$ and $\max_p \Pi$ you will find that in both cases $\Pi^*(q)= \Pi^*(p)$. $\endgroup$
    – 1muflon1
    Oct 10, 2023 at 20:38

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