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Suppose that monopoly produces product $X$. Let $MC = 20$, Quantity demanded function is given by equation $Q_d = 600 - 2P$ (where $Q_d$ - demand curve, $P$ - price). Calculate the amount of production and the price in which the profit will be maximized.

I'd like to know why is the following solution wrong:

$MC = 20 ,\ Q_d = 600 - 2P$

I know that

$(Q_d = AR $ & $ TR = Q*AR) \implies TR = Q*Q_d$

$TR = Q*(600-2P)$

$TR = 600Q-2PQ + C$

$MR = \frac{\partial TR}{\partial Q} = 600 - 2P$

since the profit is maximsed when $MR = MC$:

$MR = MC$

$20 = 600 -2P \implies P = 290$

since $Qd = 600-2P,$ & $ P^* = 290 \implies Q^* = 20$

so the solution should be:

$ \left\{ \begin{array}{c} P^* = 290 \\ Q^* = 20 \\ \end{array} \right. $

But when we solved this task in classroom, the correct solution was:

$MC = 20 ,\ Q_d = 600 - 2P$

$Q_d = 600 - 2P \implies P=-\frac{Q_d}{2} + 300$

since $TR = P*Q \implies TR = (-\frac{Q_d}{2} + 300)*Q = -\frac{Q_d^2}{2} + 300Q_d$ (I don't understand why the $Q = Q_d$ here)

we know that the profit is maximsed when $MR = MC$, so:

$MR = MC$

$-Q+300 = 20 \implies Q = 280$

$P=\frac{Q_d}{2} + 300 \implies P = 160$

so the solution is:

$ \left\{ \begin{array}{c} P^* = 160 \\ Q^* = 280 \\ \end{array} \right. $

Why the answer is different when we solve this exercise in two different ways? What am I doing wrong in the first solution? Why is $Q_d = Q$?

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    $\begingroup$ average revenue is price, not quantity. your solution makes no sense. $\endgroup$ – Fato Sep 16 '17 at 16:36
  • $\begingroup$ Unfortunately it's not correct. Please check the following sources quora.com/… and acted.co.uk/forums/index.php?threads/demand-curve-ar-or-mr.9296 $\endgroup$ – user3130324 Sep 16 '17 at 17:03
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    $\begingroup$ Unfortunately you did not understand your own sources correctly. Try to grasp the concept instead of taking the literal meaning of AR is the same as demand curve. What they meant was in graphical representation, since they both plot price against quantity, although the more correct term they should have used is inverse demand curve. Reading your own sources would also have led you to conclude that they actually meant AR = P, not AR = Q $\endgroup$ – MH.Q Sep 19 '17 at 3:17
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The reason your solution is wrong is because this assumption is wrong Qd =/= AR.

The comments from your question has already stated that point. But lets put it into perspective.

The way to arrive at Average Revenue (AR) is by taking Total Revenue dividing by the quantity you sold.

Total revenue is derived from the price per unit multiplying by the number of units you sold (quantity, Q)

Essentially, this becomes AR = (P*Q)/Q = P

Therefore, AR = P

Now that we have established that, you can go on to use TR = Q*AR which will eventually bring you through the steps of the second solution which you went through in the classroom.

For your second question, I do not really get where the confusion is coming from where Qd = Q.

I assume you get confused because you wrote TR = P * Q

But read out the whole expression to yourself, the expression is saying Total revenue is equals to Price multiplied by the Quantity sold (AKA quantity demanded)

AND, the quantity demanded is a function of the price which is provided to you in the question, Qd = 600 - 2P

So at a particular price (P), you have a particular quantity demanded (Qd) which leads you to a particular Total Revenue. The Qd is equal to Q then, in your terms.

Hope this helps!

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The reason why the answers are different is because the way you solved the for marginal revenue on your own includes $P$ in your solution.

The reason why this is problematic is because $MR=P$ for a monopolist. You'll end up with some mathematically impossible statements when attempting to find the optimal quantity produced.

I.e.

$$MR = \frac{\partial TR}{\partial Q} = 600 - 2P$$ $$MR = 600 - 2MR$$ $$ MR = 200$$

the monopolist prodouces where $MC=MR$ therefore $$\color{red}{20=200} $$

As for the second point $Q_d$ is $Q$ because we can only have a single quantity supplied and demanded in equilibrium.

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