Monopoly in Math Terms

1. Monopoly in Math Terms

2. Say a competitive market is monopolized by having the monopoly buy up all the firms. The one firm now meets the demand from all the consumers in the market. Say the demand is Qd = 6000/9 – (50/9)P. We know that firms that maximize profit produce the level of output where MR = MC (as long as P>=AVC). We need to find the MR for the monopoly firm. It comes from the demand curve. In general, when the inverse demand is P = A – BQ, the MR is MR = A – 2BQ.

3. In our example, express the demand in inverse form P = 120 – (9/50)Q. The total revenue, TR, is PQ = 120Q – (9/50)Q2. Marginal revenue, MR, is just the first derivative, or MR = 120 –(18/50)Q. If the supply in the market before monopolization was Qs = 25MC – 250, then the MC for the monopoly is just a re-expression of the supply as MC = Q/25 + 250/25 = 10 + Q/25. (I am assuming we have a multi-plant monopoly that just operates each company it bought out as a separate plant.) Now the monopoly will make the Q where MR = MC and charge the price for that Q on the inverse demand curve.

4. So we have MR = MC, or 120 –(18/50)Q = 10 + Q/25, or 110 = Q/25 + (18/50)Q = (20/50)Q, or Q = 110 (50/20) =275. Plug this into the inverse demand to get the price, P = 120 –(9/50)Q = 120 – (9/50)275 = 70.50. Note the MR at the optimal level of output is MR = 120 – (18/50)275 = 21. In each plant the firm will want the MR = MC. Now, if each plant had total cost, TC = 100 + 10Q + Q2, then the MC = 10 +2Q.

5. Each plant will thus produce 21 = 10 +2Q, or Q = (21 – 10)/2 = 5.5 units. The profit in each plant is then TR – TC in the plant, or 70.50(5.5) – (100 + 10(5.5) + (5.5)2) = 387.75 – 185.25 = 202.50.

6. P D1 S1=MC summed across each plant Pm = 70.50 Pc=30 MR of monopoly Q Qc=500 Qm = 275 Market