Multiplant Monopoly Here we study the situation where a monopoly sells in one market but makes the output in two facilities.

multiplant monopolist The type of monopolist we will consider here is one that produces output in two facilities, but sells the output in one market. The questions we want to answer are: 1) How much output to sell in the market, 2) What price to charge in the market, 3) How much of the total output should come from each of the facilities?

firm level marginal cost Each facility will have a marginal cost schedule. The first unit of output out of each facility will have a MC value, of course. Say the first plant has a MC = 3 for the first unit and plant 2 has a MC = 4 for the first unit. The first unit the firm sells should come form plant 1. Now say the second unit from plant 1 has a MC = 3.5. The second unit the firm sells should come from plant 1 as well. If the third unit from plant 1 has a MC = 4.25, then the third unit the firm should sell should be the first unit from plant 2. In a graph this is accomplished by horizontally adding the plant level MC’s to get the firm level MC.

Firm level marginal cost $ MC firm MC1 MC2 equal

decision time $ MC firm Pm MC1 MC2 D MR Q1 Q2 Q = Q1 + Q2

analysis The firm will sell the level of output where firm level MC = MR, charge the price on the demand curve at this level of output, recognize the MR at the optimal Q is a firm level MR now and each plant should produce the amount where MR = MC. Since the firm level MR drives the revenue side of things, don’t sell additional units that have higher MC than MR. Thus plants have MC’s that are equal and also equal to MR.

In general, when the inverse demand is P = A – BQ, the MR is MR = A – 2BQ. Example: Say demand is P = 70 – 0.5Q, then MR = 70 – Q. Note in the example that B = 0.5 and 2B = 2(0.5) = 1. So, this is just a rule you will use in working with a monopoly. You have to put the demand in this inverse form to apply the rule.

Decision in Math Terms Say we have demand in inverse form of P = 40 – Q. Plus say the total cost in each plant is (with subscripts) TC1 = Q1 + Q12, TC2 = 4Q2 + 0.5Q22. The MR for the firm is: MR = 40 – 2Q. The MC at each plant is MC1 = 1 + 2Q1, MC2 = 4 + Q2.

Decision in Math Terms There are several ways we could proceed at this time to answer the questions we set out above. I have chosen the path that is consistent with using the graph we saw before. Next you will see a crucial step. Re-express each MC in Q form as MC1 = 1 + 2Q1, as Q1 = .5 MC1 - .5 MC2 = 4 + Q2, as Q2 = MC2 – 4. Now, the Q’s add up to Q and on the right side we add to get Q = 1.5MC – 4.5 (you ignore the subscripts on the MC terms and add them together.)

Output and Price for the Firm Re-express the MC for the firm as MC = (2/3)Q + 3. We have MR = 40 – 2Q. Thus the level of output to maximize profit occurs where MR = MC and we have 40 – 2Q = (2/3)Q + 3, or Q = 37(3/8) = 13.875. The price to charge is on the demand curve P = 40 – 13.875 = 26.125. Note at the optimal level of output the MR = 40 – 27.75 = 12.25.

Production in each plant The MC in each plan, again, is MC1 = 1 + 2Q1, MC2 = 4 + Q2. With MR = 12.25 at the optimal output, set each MC = 12.25 to see how much to make in each plant: Plant 1  1 + 2Q1 = 12.25, or Q1 = 5.625 and Plant 2  4 + Q2 = 12.25, or Q2 = 8.25. Note the two Q’s add up to what we saw before.

Profit The profit for the firm is 258.1875. Find this by looking at the TR and TC in each plant.

Demo problem 8-6 P = 70 - .5Q is demand, So MR = 70 – Q because of rule we saw before. Say MC1 = 3Q1 and MC2 = Q2. Then Q1 = [1/3]MC1 and Q2 = MC2. Q1+ Q2 = Q 4/3MC and MC = [3/4]Q Make the Q where MR = MC, so 70 – Q = [3/4]Q, or 70 = [7/4]Q, or Q = 40. Set this Q into the demand to get the price P = 70 -0.5[40] = 50. The value of MR at Q = 40 is 30 so in each plant MR = MC gives In plant 1 30 = 3Q1, or Q1 = 10 and in Plant 2 30 = Q2.