1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,910 Your support will help MIT OpenCourseWare continue to 4 00:00:06,910 --> 00:00:10,560 offer high-quality educational resources for free. 5 00:00:10,560 --> 00:00:13,460 To make a donation or view additional materials from 6 00:00:13,460 --> 00:00:18,340 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:18,340 --> 00:00:19,590 ocw.mit.edu. 8 00:00:22,780 --> 00:00:24,170 GREG HUTKO: Welcome back to the 14.01 9 00:00:24,170 --> 00:00:25,770 problem-solving videos. 10 00:00:25,770 --> 00:00:29,490 Today I'm going to be working on Fall 2010 PSET 8, 11 00:00:29,490 --> 00:00:31,170 Problem Number 2. 12 00:00:31,170 --> 00:00:33,610 And we've seen the case with the monopolist where we've 13 00:00:33,610 --> 00:00:36,290 only had one producer in a market, and we've seen how 14 00:00:36,290 --> 00:00:39,040 that affects consumer surplus, produce surplus, and 15 00:00:39,040 --> 00:00:40,520 deadweight loss. 16 00:00:40,520 --> 00:00:42,100 Now we're going to look at the case where we 17 00:00:42,100 --> 00:00:44,080 have two firms competing. 18 00:00:44,080 --> 00:00:46,010 And specifically, for the first part, we're going to 19 00:00:46,010 --> 00:00:48,360 have the Cournot equilibrium, where they're going to be 20 00:00:48,360 --> 00:00:51,090 competing not by setting the price of their product. 21 00:00:51,090 --> 00:00:53,170 Instead they're going to compete by setting how much 22 00:00:53,170 --> 00:00:55,520 they're going to produce. 23 00:00:55,520 --> 00:00:58,130 Let's go ahead and read part A of this problem. 24 00:00:58,130 --> 00:01:00,610 Consider a market in which two firms produce 25 00:01:00,610 --> 00:01:02,320 a homogeneous product. 26 00:01:02,320 --> 00:01:06,430 Market demand is given by quantity equals 200 minus p. 27 00:01:06,430 --> 00:01:11,710 The cost functions for Firm A and Firm B, the total cost for 28 00:01:11,710 --> 00:01:15,860 A equals 5qA and the total cost for b is going to equal 29 00:01:15,860 --> 00:01:19,680 1/2 qB squared, respectively. 30 00:01:19,680 --> 00:01:21,490 Find the Cournot equilibrium quantity 31 00:01:21,490 --> 00:01:22,940 supplied by each firm. 32 00:01:22,940 --> 00:01:24,990 We're going to graph our results using reaction 33 00:01:24,990 --> 00:01:27,620 functions and we're going to find the market price, and 34 00:01:27,620 --> 00:01:30,900 then calculate the profits for each firm. 35 00:01:30,900 --> 00:01:34,070 Now in the duopoly model, both firms in the Cournot 36 00:01:34,070 --> 00:01:35,330 equilibrium are going to set their 37 00:01:35,330 --> 00:01:37,490 quantities at the same time. 38 00:01:37,490 --> 00:01:39,970 But what they're going to do is they're going to know what 39 00:01:39,970 --> 00:01:42,530 each other's revenues and costs look like. 40 00:01:42,530 --> 00:01:45,750 So they can know, I know that if I make this move and set 41 00:01:45,750 --> 00:01:49,650 this quantity, I know that my competition is going to have 42 00:01:49,650 --> 00:01:51,100 this reaction. 43 00:01:51,100 --> 00:01:54,510 Since they can plan for each other's reaction, they can 44 00:01:54,510 --> 00:01:57,080 decide how much quantity they're going to produce at 45 00:01:57,080 --> 00:01:59,940 the same time and they're going to reach an equilibrium. 46 00:01:59,940 --> 00:02:04,740 What this looks like on our reaction curves, up here we're 47 00:02:04,740 --> 00:02:07,050 going to have qA. 48 00:02:07,050 --> 00:02:13,670 And down here is going to be qB. 49 00:02:13,670 --> 00:02:17,850 We're going to graph the reaction function for Firm A 50 00:02:17,850 --> 00:02:28,020 first. And we're now going to graph the reaction function 51 00:02:28,020 --> 00:02:36,970 for Firm B. 52 00:02:36,970 --> 00:02:40,420 Now all this tells us is if we're at this point, if we're 53 00:02:40,420 --> 00:02:43,480 Firm A and we decide to produce this much, then we 54 00:02:43,480 --> 00:02:47,040 know that Firm B's best reaction is going to produce 55 00:02:47,040 --> 00:02:50,410 this quantity here, the intersection of how much I'm 56 00:02:50,410 --> 00:02:53,060 producing with their reaction curve. 57 00:02:53,060 --> 00:02:56,830 Similarly, if Firm B is going to decide to produce this 58 00:02:56,830 --> 00:03:00,610 much, then my best reaction going over to Firm A's 59 00:03:00,610 --> 00:03:03,450 reaction function straight across, is going to be to 60 00:03:03,450 --> 00:03:06,110 produce this much. 61 00:03:06,110 --> 00:03:08,550 Now since they're both choosing at the same time, 62 00:03:08,550 --> 00:03:10,890 neither of the firms can decide to 63 00:03:10,890 --> 00:03:12,270 declare a larger amount. 64 00:03:12,270 --> 00:03:15,810 And they have to kind of use their intuition to determine 65 00:03:15,810 --> 00:03:17,380 what they're going to produce. 66 00:03:17,380 --> 00:03:22,280 So since they're choosing at the same time, they have to 67 00:03:22,280 --> 00:03:24,100 plan for each other's reactions. 68 00:03:24,100 --> 00:03:26,810 They're going to produce at this point where the two 69 00:03:26,810 --> 00:03:28,420 reaction curves meet. 70 00:03:28,420 --> 00:03:29,780 And that's what we're going to be calculating. 71 00:03:29,780 --> 00:03:32,470 We're going to be calculating the quantity that Firm A 72 00:03:32,470 --> 00:03:37,640 produces and the quantity that Firm B produces when the 73 00:03:37,640 --> 00:03:42,190 reaction curves are set equal to each other. 74 00:03:42,190 --> 00:03:44,230 Now to get the reaction curves, we're going to start 75 00:03:44,230 --> 00:03:48,490 off with our revenue function for both Firm A and Firm B. 76 00:03:48,490 --> 00:03:50,930 And revenue is just going to be the price times the 77 00:03:50,930 --> 00:03:55,100 quantity that either Firm A or Firm B is producing. 78 00:03:55,100 --> 00:03:59,170 Now instead of saying that we are just going to take the 79 00:03:59,170 --> 00:04:04,410 marginal revenue according to this P times Q, we're going to 80 00:04:04,410 --> 00:04:08,860 plug-in for P from the demand curve 200 minus a 81 00:04:08,860 --> 00:04:11,490 disaggregated quantity where we disaggregate into the 82 00:04:11,490 --> 00:04:13,800 amount Firm A is producing and the 83 00:04:13,800 --> 00:04:15,710 amount Firm B is producing. 84 00:04:15,710 --> 00:04:19,190 We're going to plug this in to the revenue function. 85 00:04:19,190 --> 00:04:21,640 And so just like the monopolist did where they're 86 00:04:21,640 --> 00:04:24,350 maximizing by setting marginal revenue equal to marginal 87 00:04:24,350 --> 00:04:27,800 cost, Firm A and Firm B we're going to do the same thing. 88 00:04:27,800 --> 00:04:29,840 We're going to set marginal revenue equal to marginal 89 00:04:29,840 --> 00:04:34,730 cost. And then we're going to solve through for qA in terms 90 00:04:34,730 --> 00:04:38,070 of the quantity that the other firm is producing. 91 00:04:38,070 --> 00:04:40,570 That's why it's considered a reaction curve because it's in 92 00:04:40,570 --> 00:04:43,340 terms of what the other firm is producing. 93 00:04:43,340 --> 00:04:50,010 So solving through for Firm A's marginal revenue, we're 94 00:04:50,010 --> 00:04:53,690 going to find that marginal revenue for Firm A is equal to 95 00:04:53,690 --> 00:05:02,690 200 minus 2qA minus qB. 96 00:05:02,690 --> 00:05:05,160 And it makes sense that the more that they're producing, 97 00:05:05,160 --> 00:05:08,190 Firm A and Firm B, the lower the revenue that they're going 98 00:05:08,190 --> 00:05:10,320 to be taking in. 99 00:05:10,320 --> 00:05:12,410 Now we're going to also calculate the marginal cost 100 00:05:12,410 --> 00:05:15,490 for Firm A by taking the derivative with respect to the 101 00:05:15,490 --> 00:05:17,630 total cost function. 102 00:05:17,630 --> 00:05:19,980 We're going to find that the marginal cost is going to be 103 00:05:19,980 --> 00:05:21,560 equal to 5. 104 00:05:21,560 --> 00:05:25,730 Now we're just going to set the marginal cost and the 105 00:05:25,730 --> 00:05:28,400 marginal revenue for Firm A equal. 106 00:05:28,400 --> 00:05:32,850 And we're going to solve through for qA. 107 00:05:32,850 --> 00:05:38,820 When we do that, we're going to find that qA is equal to 108 00:05:38,820 --> 00:05:46,270 97.5 minus 0.5 qB. 109 00:05:46,270 --> 00:05:50,410 And we're going to repeat this exact same process for Firm B. 110 00:05:50,410 --> 00:05:52,410 But when we do it for Firm B, instead of taking the 111 00:05:52,410 --> 00:05:54,570 derivative with respect to qA, we're going to take the 112 00:05:54,570 --> 00:05:56,940 derivative with respect to qB. 113 00:05:59,540 --> 00:06:04,810 So the marginal revenue for Firm B is going to be equal to 114 00:06:04,810 --> 00:06:12,520 200 minus qA minus 2 qB. 115 00:06:12,520 --> 00:06:19,695 And the marginal cost is just going to be equal to qB. 116 00:06:22,330 --> 00:06:26,540 Again, we're going to set marginal cost and marginal 117 00:06:26,540 --> 00:06:27,910 revenue equal to each other. 118 00:06:27,910 --> 00:06:31,630 And now we're going to solve through for qB. 119 00:06:31,630 --> 00:06:35,590 And when we do this, we're going to have Firm B's 120 00:06:35,590 --> 00:06:36,840 reaction curve. 121 00:06:52,240 --> 00:06:54,110 And now what we're going to do since we have these two 122 00:06:54,110 --> 00:06:57,220 reaction curves, we have a reaction for Firm A and a 123 00:06:57,220 --> 00:07:00,330 reaction for Firm B, all we're going to do is we're going to 124 00:07:00,330 --> 00:07:06,140 plug-in for this qB, qB's reaction curve. 125 00:07:06,140 --> 00:07:11,570 And when we do that, when we plug-in 66.67 minus 0.33qA, we 126 00:07:11,570 --> 00:07:14,840 can solve through for just qA. 127 00:07:14,840 --> 00:07:18,180 Doing this we're going to find that Firm A is going to 128 00:07:18,180 --> 00:07:23,450 produce approximately 77 units. 129 00:07:23,450 --> 00:07:27,080 And then taking this 77 and plugging it in to Firm B's 130 00:07:27,080 --> 00:07:29,920 reaction function, we can solve for Firm 131 00:07:29,920 --> 00:07:31,170 B's production amount. 132 00:07:34,200 --> 00:07:36,270 And we're going to find that Firm B is going to produce 133 00:07:36,270 --> 00:07:41,380 approximately 41 units. 134 00:07:41,380 --> 00:07:43,530 Now to find the equilibrium price, we're just going to 135 00:07:43,530 --> 00:07:47,470 come back up here to our disaggregated demand function, 136 00:07:47,470 --> 00:07:50,460 and we're going to plug-in for qA and qB. 137 00:07:50,460 --> 00:08:01,360 And we can solve through for the price being equal to 82. 138 00:08:01,360 --> 00:08:06,050 Now what we've just done for this problem is we solved on 139 00:08:06,050 --> 00:08:09,810 our graph for the intersection of the two reaction functions. 140 00:08:09,810 --> 00:08:14,920 We found that qB is going to be equal to 41 and we found 141 00:08:14,920 --> 00:08:18,790 that qA is going to be equal to 77. 142 00:08:18,790 --> 00:08:23,500 So we just calculated the intersection point for the 143 00:08:23,500 --> 00:08:25,500 Cournot equilibrium. 144 00:08:25,500 --> 00:08:27,940 Now the last part of this problem asks us to calculate 145 00:08:27,940 --> 00:08:35,330 the profits for both the firms. The profits for Firm A 146 00:08:35,330 --> 00:08:40,299 are just going to be price times the quantity that A is 147 00:08:40,299 --> 00:08:49,130 producing minus the cost as a function of qA. 148 00:08:49,130 --> 00:08:51,760 So we're just going to take the total revenues minus the 149 00:08:51,760 --> 00:08:53,610 total costs. 150 00:08:53,610 --> 00:09:02,990 For Firm A, we're going to find that the total profits 151 00:09:02,990 --> 00:09:06,350 are going to be about $5,929. 152 00:09:06,350 --> 00:09:10,530 And doing the same process for Firm B, we can find that the 153 00:09:10,530 --> 00:09:13,960 profits for Firm B are going to be equal to 154 00:09:13,960 --> 00:09:23,470 approximately $2,521. 155 00:09:23,470 --> 00:09:26,670 Now part B of this problem is going to ask us instead of 156 00:09:26,670 --> 00:09:29,670 having this Cournot equilibrium where neither firm 157 00:09:29,670 --> 00:09:32,670 can go ahead and produce a higher quantity or move first 158 00:09:32,670 --> 00:09:34,960 in the market, we're going to look at something different 159 00:09:34,960 --> 00:09:36,610 than the Cournot equilibrium. 160 00:09:36,610 --> 00:09:38,460 We're going to look at the case where one of the firms 161 00:09:38,460 --> 00:09:40,570 gets to decide how much they're going to produce 162 00:09:40,570 --> 00:09:42,080 before the other firm. 163 00:09:42,080 --> 00:09:45,010 And if you get to decide first, you get to produce a 164 00:09:45,010 --> 00:09:48,100 higher quantity and get more of the profits. 165 00:09:48,100 --> 00:09:51,450 Part B says, now suppose that Firm A chooses how much to 166 00:09:51,450 --> 00:09:54,640 produce before firm B does. 167 00:09:54,640 --> 00:09:56,580 In this case, Firm A is a Stackelberg 168 00:09:56,580 --> 00:09:58,470 leader and B a follower. 169 00:09:58,470 --> 00:10:01,220 We're going to calculate the quantities, the market price, 170 00:10:01,220 --> 00:10:04,090 and the profit for each firm. 171 00:10:04,090 --> 00:10:07,920 Now coming over to this side of the board, we see that I'm 172 00:10:07,920 --> 00:10:10,960 going to keep Firm B's reaction function the same. 173 00:10:10,960 --> 00:10:13,930 So Firm B is going to be reacting in the same way to 174 00:10:13,930 --> 00:10:15,730 Firm A's decision. 175 00:10:15,730 --> 00:10:18,770 Only now, the only difference is when we calculate marginal 176 00:10:18,770 --> 00:10:24,940 revenue equal the marginal cost for Firm A, instead of 177 00:10:24,940 --> 00:10:27,390 just saying the qB is going to be random, we 178 00:10:27,390 --> 00:10:29,760 can't account for it. 179 00:10:29,760 --> 00:10:32,320 We're going to plug-in, we're going to take into account 180 00:10:32,320 --> 00:10:36,350 Firm B's reaction when we're maximizing or taking the 181 00:10:36,350 --> 00:10:38,720 derivative with respect to qA. 182 00:10:38,720 --> 00:10:41,440 So instead of having qB in here, I'm going to plug-in 183 00:10:41,440 --> 00:10:43,350 this reaction function. 184 00:10:43,350 --> 00:10:45,020 So in this case, the equation that I'm going to be 185 00:10:45,020 --> 00:11:06,730 maximizing is going to be this one right here. 186 00:11:06,730 --> 00:11:11,480 I'm going to take the derivative with respect to qA 187 00:11:11,480 --> 00:11:28,550 to find the marginal revenue for A. 188 00:11:28,550 --> 00:11:30,890 And again, I'm just going to set this equal to the marginal 189 00:11:30,890 --> 00:11:34,070 cost, which we found earlier is equal to 5. 190 00:11:38,090 --> 00:11:40,460 And when we set these equal, we can solve through for the 191 00:11:40,460 --> 00:11:42,960 quantity that Firm A is going to produce. 192 00:11:42,960 --> 00:11:44,920 And we're going to find just like we predicted that the 193 00:11:44,920 --> 00:11:46,310 leader is going to produce more. 194 00:11:53,390 --> 00:11:55,160 In this case, Firm A has increased their 195 00:11:55,160 --> 00:11:58,270 production to 96.25. 196 00:11:58,270 --> 00:12:02,610 And then plugging in this quantity in to Firm B's 197 00:12:02,610 --> 00:12:09,410 reaction function, we can find that Firm B in this case, is 198 00:12:09,410 --> 00:12:17,330 going to produce 34.6. 199 00:12:17,330 --> 00:12:20,840 Now in this case, we can again calculate the price by taking 200 00:12:20,840 --> 00:12:27,410 the demand function that we have. We can take the demand 201 00:12:27,410 --> 00:12:28,680 function that we're given in the problem. 202 00:12:36,170 --> 00:12:40,240 Plugging in for qA and qB, we find that the new price in the 203 00:12:40,240 --> 00:12:51,460 Stackelberg problem is going to be 69.15. 204 00:12:51,460 --> 00:12:54,510 And again, we can calculate the profits going through the 205 00:12:54,510 --> 00:12:58,270 same process of doing total revenue minus total cost. And 206 00:12:58,270 --> 00:13:02,110 we're going to have that the profit for Firm A is going to 207 00:13:02,110 --> 00:13:09,450 be equal to about 6,174. 208 00:13:09,450 --> 00:13:12,840 And the profit for Firm B is going to be 209 00:13:12,840 --> 00:13:20,810 equal to about 1,794. 210 00:13:20,810 --> 00:13:23,480 And so what we can do here is we can compare the profits 211 00:13:23,480 --> 00:13:26,660 that we had in the Stackelberg case to the profits that we 212 00:13:26,660 --> 00:13:28,440 had at the start of our problem. 213 00:13:58,600 --> 00:14:02,490 So before we can see that Firm A was not as profitable when 214 00:14:02,490 --> 00:14:04,750 they had to choose their quantity at the same time as 215 00:14:04,750 --> 00:14:08,360 Firm B. We can see that their profits have increased. 216 00:14:08,360 --> 00:14:11,880 But we can see that Firm B, their profits have actually 217 00:14:11,880 --> 00:14:13,630 decreased because they're a follower in 218 00:14:13,630 --> 00:14:15,690 the Stackelberg model. 219 00:14:15,690 --> 00:14:17,560 Now the last thing, and the thought I want to leave you 220 00:14:17,560 --> 00:14:20,680 with is, how do we actually interpret this when we look at 221 00:14:20,680 --> 00:14:23,010 the reaction functions on our graph? 222 00:14:23,010 --> 00:14:25,420 We're no longer at the point where we're setting the two 223 00:14:25,420 --> 00:14:27,590 reaction functions equal. 224 00:14:27,590 --> 00:14:34,320 What's happening now is we're way up here and qA is choosing 225 00:14:34,320 --> 00:14:37,840 their production way up here. 226 00:14:37,840 --> 00:14:44,770 And qB is forced to react by choosing their production 227 00:14:44,770 --> 00:14:46,510 right here. 228 00:14:46,510 --> 00:14:49,040 And what happens is since they both increased their 229 00:14:49,040 --> 00:14:53,760 production or since qA has increased their production and 230 00:14:53,760 --> 00:14:56,110 qB has decreased their production, but since 231 00:14:56,110 --> 00:15:01,610 production has increased overall, the price has dropped 232 00:15:01,610 --> 00:15:03,720 compared to when they were at the Cournot equilibrium. 233 00:15:03,720 --> 00:15:07,250 So total profits have actually dropped as well. 234 00:15:07,250 --> 00:15:09,370 So really what the first two parts of these problems were 235 00:15:09,370 --> 00:15:11,240 having us look at, they were looking at two different 236 00:15:11,240 --> 00:15:13,520 situations of duopoly where we have two 237 00:15:13,520 --> 00:15:15,190 competitors in the market. 238 00:15:15,190 --> 00:15:16,600 The first one they were choosing their 239 00:15:16,600 --> 00:15:18,220 outputs at the same time. 240 00:15:18,220 --> 00:15:20,750 And in the second problem, one of the firms had the advantage 241 00:15:20,750 --> 00:15:23,990 of getting to choose a higher quantity and making a credible 242 00:15:23,990 --> 00:15:27,050 threat that they were going to make that 243 00:15:27,050 --> 00:15:29,050 quantity to begin with. 244 00:15:29,050 --> 00:15:31,200 For the last parts of these problems, you're going to go 245 00:15:31,200 --> 00:15:34,190 ahead and you can look at what the implications are when we 246 00:15:34,190 --> 00:15:36,920 think about what the total quantity is produced in a 247 00:15:36,920 --> 00:15:40,420 competitive market aggregating the supplies of these two 248 00:15:40,420 --> 00:15:43,940 firms. And then you can compare the output in the 249 00:15:43,940 --> 00:15:45,390 three different scenarios. 250 00:15:45,390 --> 00:15:47,150 But for now, I'm going to leave you here. 251 00:15:47,150 --> 00:15:49,040 Go ahead and finish the rest of the problem, and I hope you 252 00:15:49,040 --> 00:15:50,290 found this part helpful.