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In one market, the demand is $D_1=80-P_1$ and in the other $D_2=100-2P_2$. Suppose the markets are separated. What output level should be produced and what price will prevail. Since $D_1=80-P_1$, then $Profits=Revenue-Cost=Demand*Price-Demand*Cost=P_1(80-P_1)-10(80-P_1)$

For market 2, $D_2=100-2P_2$ so $Profit=P_2(100-2P_2)-10P_2$

To maximize profit just do calculus to work out the maximum of each of the resulting quadratics. Is this correct?

stackdsewew
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1 Answers1

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The inverse demand curve for Market 1 is given to be

$ P_1 = 80 -q_1$

Revenue function $= R_1(q_1) = p_1q_1 = (80-q_1)*q_1$

Marginal Revenue $MR_1 = 80-2q_1$

Similarly the inverse demand curve for Market 2 is given to be

$p_2 = 50-0.5q_2$

Revenue function $= R_2(q_2) = p_2q_2 = (50-0.5q_2)*q_2$

Marginal Revenue $MR_2 = p_2q_2 = (50 - q_2)$

Set $MR_1$ = Marginal Cost = 10

Set $MR_2$ = Marginal Cost = 10

At equilibrium, $80-2\hat q_1 = 10$

and

$50-\hat q_2 = 10$

Thus the output levels are $\hat q_1 = 35$ and $\hat q_2 = 40$

The equilibrium prices at the two markets are $\hat p_1 = 45$ and $\hat p_2 = 30$

Goodluck

Satish

Satish Ramanathan
  • 11,984
  • Why have you taken the inverse of the demand curve? Algebraically, the expression $(80-q_1)q_1$ is the same as $(80-p_1)p_1$ and how did you get $MR_1=80-2q_1$? I mean ultimately we want to maximize profits so shouldn't there be some calculus involved to see where this max occurs? – stackdsewew Aug 20 '16 at 03:54
  • If you take the demand curve and solved it, you will fist find the equilibrium price and then quantity. If you take the inverse, you will find q and then p. Either way is OK. To answer you second question, equilibrium occurs when marginal revenue is equal to the marginal cost. Revenue for each market is pq. With that in mind, $R(p_1) = (80-q_1)q_1$ which when you differentiate gives $(80-2q_1)$. Now set this equal to marginal cost which is 10 and find $q_1$ and subsequently $p_1$. Do a similar thing for market 2 and find $q_2,p_2$. There is very minimal calculus that is used – Satish Ramanathan Aug 20 '16 at 04:03
  • By the way, did you check the answers, were they correct if it is a book problem – Satish Ramanathan Aug 20 '16 at 04:04
  • Ok so I have verified doing it via the calculus method also works and produces same results. Thank you for showing me a different way of doing it though. Much appreciated! – stackdsewew Aug 20 '16 at 04:11
  • A monopolist's maximizing output occurs when MR and MC intersect. Calculus is used to find MR. You can be rest assured that the answers are indeed correct. If it is homework problem, you can confidently submit the solution that I have provided. – Satish Ramanathan Aug 20 '16 at 04:11
  • If you like the answer, please choose to accept my solution by clicking on the tick mark to the side of the solution – Satish Ramanathan Aug 20 '16 at 04:12