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 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,210 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,210 --> 00:00:17,865 at ocw.mit.edu. 8 00:00:22,170 --> 00:00:25,300 PROFESSOR: Hi, and welcome back to the 14.01 problem solving 9 00:00:25,300 --> 00:00:25,990 videos. 10 00:00:25,990 --> 00:00:29,620 Today, we're going to work on Fall 2010, problem set one, 11 00:00:29,620 --> 00:00:30,727 problem number four. 12 00:00:30,727 --> 00:00:32,310 And in this problem, we're going to be 13 00:00:32,310 --> 00:00:34,062 working with elasticities. 14 00:00:34,062 --> 00:00:36,020 But instead of starting with a demand function, 15 00:00:36,020 --> 00:00:37,980 and starting with a supply function, 16 00:00:37,980 --> 00:00:41,210 and calculating the elasticity given those functions, 17 00:00:41,210 --> 00:00:43,210 we're going to be given the elasticity of demand 18 00:00:43,210 --> 00:00:44,672 and the elasticity of supply. 19 00:00:44,672 --> 00:00:46,130 And we're going to have to back out 20 00:00:46,130 --> 00:00:48,741 what the demand functions and the supply functions 21 00:00:48,741 --> 00:00:49,740 should have looked like. 22 00:00:49,740 --> 00:00:51,620 So we're basically just working in reverse 23 00:00:51,620 --> 00:00:53,570 from what we did in lecture. 24 00:00:53,570 --> 00:00:57,210 Let's go ahead and read the full problem up through part A. 25 00:00:57,210 --> 00:01:00,040 You have been asked to analyze the market for steel. 26 00:01:00,040 --> 00:01:01,870 From public sources, you are able to find 27 00:01:01,870 --> 00:01:05,120 that last year's price for steel was $20 per ton. 28 00:01:05,120 --> 00:01:08,750 At this price, 100 million tons were sold on the world market. 29 00:01:08,750 --> 00:01:10,810 From trade association data, you are 30 00:01:10,810 --> 00:01:14,170 able to obtain estimates for their own price elasticities 31 00:01:14,170 --> 00:01:18,690 of demand and supply on the world markets as negative 0.25 32 00:01:18,690 --> 00:01:21,610 for demand and 0.5 for supply. 33 00:01:21,610 --> 00:01:24,440 Assume the steel has linear demand and supply 34 00:01:24,440 --> 00:01:25,960 curves throughout. 35 00:01:25,960 --> 00:01:29,850 Part A asks us to solve for the equations of demand and supply 36 00:01:29,850 --> 00:01:34,010 in this market, and to sketch the demand and supply curves. 37 00:01:34,010 --> 00:01:37,580 So looking at the formal definition of elasticity 38 00:01:37,580 --> 00:01:40,300 of demand and elasticity of supply, 39 00:01:40,300 --> 00:01:43,500 we basically are going to have three different parts to it. 40 00:01:43,500 --> 00:01:46,360 We have the derivative of either demand or supply 41 00:01:46,360 --> 00:01:50,180 function with respect to P, in this case the own price of P, 42 00:01:50,180 --> 00:01:52,300 or the price of steel. 43 00:01:52,300 --> 00:01:54,470 And we also have the equilibrium price, 44 00:01:54,470 --> 00:01:57,270 or any price at the point on the curve, and a quantity. 45 00:01:57,270 --> 00:02:00,420 In this case, it's going to be the equilibrium quantity. 46 00:02:00,420 --> 00:02:02,820 So basically, what we have now is 47 00:02:02,820 --> 00:02:05,740 we are given-- for the elasticity of demand, 48 00:02:05,740 --> 00:02:07,080 we're given three variables. 49 00:02:07,080 --> 00:02:09,509 We're given the price, the quantity, 50 00:02:09,509 --> 00:02:11,390 and the elasticity of demand. 51 00:02:11,390 --> 00:02:14,730 And that means the only thing that we don't know 52 00:02:14,730 --> 00:02:19,260 is the derivative of the demand curve with respect to P. 53 00:02:19,260 --> 00:02:21,940 So if we can isolate this derivative, 54 00:02:21,940 --> 00:02:24,750 then we can integrate the number that we're 55 00:02:24,750 --> 00:02:26,280 able to solve through for. 56 00:02:26,280 --> 00:02:28,210 And then we can solve out for what our demand 57 00:02:28,210 --> 00:02:29,510 curve is going to look like. 58 00:02:29,510 --> 00:02:32,230 So let's go ahead and walk through that process together. 59 00:02:32,230 --> 00:02:49,510 Substituting in for the elasticity of demand P and Q, 60 00:02:49,510 --> 00:02:50,909 we're gonna have this equation. 61 00:02:50,909 --> 00:02:52,700 And the one thing that I want you to notice 62 00:02:52,700 --> 00:02:57,810 is since the derivative of the demand curve with respect to P 63 00:02:57,810 --> 00:03:01,670 is negative 0.25, in this case, we know that it's linear. 64 00:03:01,670 --> 00:03:03,480 But just because it's linear at the point 65 00:03:03,480 --> 00:03:05,120 where price is 20 and quantity is 66 00:03:05,120 --> 00:03:07,620 100, that doesn't necessarily mean it's 67 00:03:07,620 --> 00:03:08,880 gonna be linear throughout. 68 00:03:08,880 --> 00:03:12,470 So it's useful to know that at any point on this line, 69 00:03:12,470 --> 00:03:15,660 it's always going to have the derivative equal 70 00:03:15,660 --> 00:03:16,682 to negative 0.25. 71 00:03:16,682 --> 00:03:17,390 So that's useful. 72 00:03:17,390 --> 00:03:21,170 We know we can integrate and have the correct answer. 73 00:03:21,170 --> 00:03:35,450 Solving for dQD dP, we're gonna have negative 1.25. 74 00:03:35,450 --> 00:03:37,580 And we're just going to integrate this with respect 75 00:03:37,580 --> 00:03:40,825 to P. And after we integrate, we're 76 00:03:40,825 --> 00:03:42,200 going to be left with a constant. 77 00:03:54,190 --> 00:03:57,834 In this case, we're going to call the constant a. 78 00:03:57,834 --> 00:03:59,750 This is how much the demand curve has actually 79 00:03:59,750 --> 00:04:03,220 shifted up to begin with, shifted up or down. 80 00:04:03,220 --> 00:04:05,390 And to solve for a, all we have to do 81 00:04:05,390 --> 00:04:09,690 is we can just plug back in for the $20 and the 100 quantity, 82 00:04:09,690 --> 00:04:12,970 and we can solve through for what a is going to be equal. 83 00:04:12,970 --> 00:04:16,250 When you solve through plugging in Q and P, 84 00:04:16,250 --> 00:04:32,610 you're going to find that a is equal to 125. 85 00:04:32,610 --> 00:04:35,260 So we're gonna have that our final demand function is gonna 86 00:04:35,260 --> 00:04:40,500 be negative 1.25P plus 125. 87 00:04:40,500 --> 00:04:43,830 Now we can go through this exact same process 88 00:04:43,830 --> 00:04:47,340 with the elasticity of supply now. 89 00:04:47,340 --> 00:04:51,000 And all we have to do now is use the number 0.5 instead, 90 00:04:51,000 --> 00:04:52,130 and we can solve through. 91 00:04:52,130 --> 00:04:53,270 We can integrate. 92 00:04:53,270 --> 00:04:55,500 And then we're going to solve for the other constant 93 00:04:55,500 --> 00:04:58,400 to, again, get our supply curve. 94 00:04:58,400 --> 00:05:10,020 Substituting in the information we have, 95 00:05:10,020 --> 00:05:11,770 we're going to be left with this equation. 96 00:05:11,770 --> 00:05:14,400 And we're gonna go ahead and isolate 97 00:05:14,400 --> 00:05:15,620 the derivative that we have. 98 00:05:27,410 --> 00:05:28,910 And when we integrate again, we have 99 00:05:28,910 --> 00:05:31,160 to remember that we are going to have a constant that we're 100 00:05:31,160 --> 00:05:32,180 gonna have to solve for. 101 00:05:40,850 --> 00:05:43,270 And I'm gonna just call this constant c. 102 00:05:43,270 --> 00:05:46,990 Again, plug in the price of 20 and the quantity equal to 100 103 00:05:46,990 --> 00:06:02,300 and you're gonna find that the supply curve is gonna be equal 104 00:06:02,300 --> 00:06:05,880 to 2.5P plus 50. 105 00:06:05,880 --> 00:06:09,141 And we can do a quick sketch of this on our axes here. 106 00:06:09,141 --> 00:06:11,640 We're just gonna go ahead and draw our upward-sloping supply 107 00:06:11,640 --> 00:06:19,050 curve, our downward-sloping demand curve. 108 00:06:19,050 --> 00:06:23,960 And we're gonna mark the equilibrium point 109 00:06:23,960 --> 00:06:26,870 and label the equilibrium quantities and the equilibrium 110 00:06:26,870 --> 00:06:27,620 prices, as well. 111 00:06:33,346 --> 00:06:35,720 And before we move on to the second part of this problem, 112 00:06:35,720 --> 00:06:36,510 we can pause here. 113 00:06:36,510 --> 00:06:38,843 And we can think about what did the elasticities that we 114 00:06:38,843 --> 00:06:40,590 started with actually mean. 115 00:06:40,590 --> 00:06:43,230 Well, if we were to look at this point of intersection 116 00:06:43,230 --> 00:06:46,540 at the equilibrium of the demand curve, 117 00:06:46,540 --> 00:06:48,680 we're looking at the percentage change 118 00:06:48,680 --> 00:06:53,960 at this point in quantity per percentage change in price. 119 00:06:53,960 --> 00:06:59,850 So we're basically just saying, for that tiny change, 120 00:06:59,850 --> 00:07:03,060 an infinitesimally small change at this point for the demand 121 00:07:03,060 --> 00:07:07,020 curve, how much does quantity change, percentage-wise, 122 00:07:07,020 --> 00:07:08,380 relative to price? 123 00:07:08,380 --> 00:07:09,800 And that's also what we're looking 124 00:07:09,800 --> 00:07:12,000 at with the supply curve. 125 00:07:12,000 --> 00:07:14,030 So when you're given a elasticity, 126 00:07:14,030 --> 00:07:15,690 if you have an elasticity of supply, 127 00:07:15,690 --> 00:07:19,150 it makes sense that it's gonna be positive, in this case 0.5, 128 00:07:19,150 --> 00:07:22,480 because when price increases, suppliers 129 00:07:22,480 --> 00:07:24,080 are willing to supply more. 130 00:07:24,080 --> 00:07:26,620 And it makes sense that the demand elasticity that we're 131 00:07:26,620 --> 00:07:29,980 given is negative, or negative 0.25, 132 00:07:29,980 --> 00:07:32,810 because when price begins to increase, 133 00:07:32,810 --> 00:07:37,410 the consumers are gonna want less of the product. 134 00:07:37,410 --> 00:07:38,960 Now, the second part of this problem 135 00:07:38,960 --> 00:07:41,120 is going to give us new elasticities of demand 136 00:07:41,120 --> 00:07:41,820 and supply. 137 00:07:41,820 --> 00:07:44,484 And I'm gonna just quickly run through the actual calculation, 138 00:07:44,484 --> 00:07:46,525 because it's gonna be the same as our calculation 139 00:07:46,525 --> 00:07:47,960 that we just did. 140 00:07:47,960 --> 00:07:52,020 And instead, we're gonna think about possible causes 141 00:07:52,020 --> 00:07:54,360 for the shifts that we see in the supply and the demand 142 00:07:54,360 --> 00:07:56,220 curve. 143 00:07:56,220 --> 00:07:58,410 Part B says, suppose that you discover 144 00:07:58,410 --> 00:08:01,820 that the current price of steel is $15 per ton 145 00:08:01,820 --> 00:08:04,590 and the current level of worldwide sales of steel 146 00:08:04,590 --> 00:08:07,130 is 150 million tons. 147 00:08:07,130 --> 00:08:09,890 The most recent elasticity estimates 148 00:08:09,890 --> 00:08:11,540 from the trade association this year 149 00:08:11,540 --> 00:08:16,820 are negative 0.125 for demand and 0.25 for supply. 150 00:08:16,820 --> 00:08:19,140 Describe the change in the supply and the demand curves 151 00:08:19,140 --> 00:08:21,280 over the past year using your diagram 152 00:08:21,280 --> 00:08:26,410 from part A. What sort of events might explain the change? 153 00:08:26,410 --> 00:08:28,670 Now, I've given us the information for this part 154 00:08:28,670 --> 00:08:31,260 of the problem on this board. 155 00:08:31,260 --> 00:08:33,850 And you'll notice that our inputs, or our variables, 156 00:08:33,850 --> 00:08:34,570 have changed now. 157 00:08:34,570 --> 00:08:37,380 The price has dropped from $20. 158 00:08:37,380 --> 00:08:39,600 Now it's gonna be down to $15. 159 00:08:39,600 --> 00:08:42,419 You're gonna notice that the quantity has actually 160 00:08:42,419 --> 00:08:45,720 increased from 100 up to 150. 161 00:08:45,720 --> 00:08:49,005 And our elasticities of demand and elasticities of supply 162 00:08:49,005 --> 00:08:51,630 have changed, because the price and the quantity are different, 163 00:08:51,630 --> 00:08:55,210 and we're at a different point on our graph. 164 00:08:55,210 --> 00:08:57,800 The process we're gonna do to solve for our demand 165 00:08:57,800 --> 00:09:01,340 curves and our supply curves are going to be exactly identical. 166 00:09:01,340 --> 00:09:03,100 And when you follow the same process-- 167 00:09:03,100 --> 00:09:05,020 I'll just do the first step up here-- 168 00:09:05,020 --> 00:09:07,726 you're gonna substitute in for the information that's 169 00:09:07,726 --> 00:09:08,600 given in the problem. 170 00:09:44,869 --> 00:09:46,410 And all you're going to do is you're, 171 00:09:46,410 --> 00:09:48,170 again, gonna solve through for the derivative. 172 00:09:48,170 --> 00:09:49,128 You're gonna integrate. 173 00:09:49,128 --> 00:09:50,960 And you're gonna find the constants. 174 00:09:50,960 --> 00:09:53,120 After you do that entire process, 175 00:09:53,120 --> 00:09:59,370 you're gonna find that the demand curve is 176 00:09:59,370 --> 00:10:00,420 given by this equation. 177 00:10:05,094 --> 00:10:07,010 And you're gonna find that the supply curve is 178 00:10:07,010 --> 00:10:08,135 given by this new equation. 179 00:10:24,100 --> 00:10:26,990 Now, if we look at this new demand 180 00:10:26,990 --> 00:10:30,310 curve and this new supply curve, we'll 181 00:10:30,310 --> 00:10:33,870 actually notice that the slope, with respect to P, 182 00:10:33,870 --> 00:10:35,910 is going to be identical in both of the cases 183 00:10:35,910 --> 00:10:39,020 that we solved for, both the beginning case and the case 184 00:10:39,020 --> 00:10:40,860 in the end of the problem. 185 00:10:40,860 --> 00:10:43,780 The only thing that's shifted between our quantities demanded 186 00:10:43,780 --> 00:10:46,760 and our quantities supplied, or the curves, 187 00:10:46,760 --> 00:10:48,240 is there's been a shift. 188 00:10:48,240 --> 00:10:51,610 And the shift for the demand curve-- 189 00:10:51,610 --> 00:10:57,270 it went from an intercept of 125 now to an intercept of 168.75. 190 00:10:57,270 --> 00:11:02,200 So our demand curve is shifting up and out. 191 00:11:02,200 --> 00:11:08,330 So we can represent this shift in demand like this. 192 00:11:08,330 --> 00:11:11,040 Notice that the slope is going to be exactly identical. 193 00:11:11,040 --> 00:11:14,597 I'm going to write a small db for part B. 194 00:11:14,597 --> 00:11:16,680 And then we can do the same sort of interpretation 195 00:11:16,680 --> 00:11:18,150 for our supply curve. 196 00:11:18,150 --> 00:11:22,420 Looking at our supply curve, the intercepts, now, is 112.5. 197 00:11:22,420 --> 00:11:26,070 But before, it was only at 50. 198 00:11:26,070 --> 00:11:28,920 And what this means, this means that the supply curve 199 00:11:28,920 --> 00:11:32,320 is going to shift in and down. 200 00:11:40,150 --> 00:11:42,554 And so my graph with the equilibrium price 201 00:11:42,554 --> 00:11:44,220 that I've drawn-- it's a little bit off, 202 00:11:44,220 --> 00:11:49,690 but what you should see-- you should 203 00:11:49,690 --> 00:11:54,960 see that the new equilibrium price has fallen. 204 00:11:54,960 --> 00:11:58,790 In this case, it's fallen to 15. 205 00:11:58,790 --> 00:12:04,090 And the equilibrium quantity has increased from 100 to 150. 206 00:12:04,090 --> 00:12:08,130 So since we had both a shift in supply and a shift in demand, 207 00:12:08,130 --> 00:12:12,060 necessarily we see that quantity is going to increase. 208 00:12:12,060 --> 00:12:18,310 But if the demand curve had shifted way up here, 209 00:12:18,310 --> 00:12:20,300 we could see that price could have increased. 210 00:12:20,300 --> 00:12:23,370 So the effect on the price in this market is ambiguous. 211 00:12:23,370 --> 00:12:26,250 We can say that, necessarily, the effect on quantity 212 00:12:26,250 --> 00:12:29,170 is going to be clearly towards an increase. 213 00:12:29,170 --> 00:12:30,990 So to wrap up this problem, we saw 214 00:12:30,990 --> 00:12:34,190 that changes in elasticities can also 215 00:12:34,190 --> 00:12:36,990 represent changes in the underlying demand and supply 216 00:12:36,990 --> 00:12:37,922 functions. 217 00:12:37,922 --> 00:12:39,630 Let's wrap up by just thinking about what 218 00:12:39,630 --> 00:12:43,040 could have caused the demand shift that we've seen. 219 00:12:43,040 --> 00:12:47,210 And what could have caused the supply shift that we saw? 220 00:12:47,210 --> 00:12:50,305 Now, there are a couple of ideas that we can have for demand. 221 00:12:53,646 --> 00:12:55,270 The first idea that we could have is we 222 00:12:55,270 --> 00:12:58,750 could just have had an increase in the income of a consumer. 223 00:12:58,750 --> 00:13:01,090 If a consumer has more income, then they 224 00:13:01,090 --> 00:13:03,940 might be willing to spend more on steel. 225 00:13:03,940 --> 00:13:06,120 A second idea that we have, we could 226 00:13:06,120 --> 00:13:08,420 have that the price of a substitute-- perhaps you're 227 00:13:08,420 --> 00:13:11,450 considering building a bridge out of iron instead of steel-- 228 00:13:11,450 --> 00:13:14,460 if the price of the substitute has increased, 229 00:13:14,460 --> 00:13:15,960 then perhaps the consumers are going 230 00:13:15,960 --> 00:13:19,650 to be willing to pay more to get the steel since the iron 231 00:13:19,650 --> 00:13:21,460 is more expensive. 232 00:13:21,460 --> 00:13:23,400 A third possible idea is that the number 233 00:13:23,400 --> 00:13:27,690 of goods that you need to make from steel is increasing. 234 00:13:27,690 --> 00:13:30,200 So if you suddenly find new uses for steel, 235 00:13:30,200 --> 00:13:34,060 then the price that you're willing to pay at any given 236 00:13:34,060 --> 00:13:37,640 point is going to be higher. 237 00:13:37,640 --> 00:13:39,704 Basically, to affect the demand curve, 238 00:13:39,704 --> 00:13:41,370 you have to think about why would people 239 00:13:41,370 --> 00:13:44,280 be more willing to pay more for a fixed quantity. 240 00:13:44,280 --> 00:13:47,340 And I just listed off a couple of ideas there. 241 00:13:47,340 --> 00:13:51,930 We can also think about reasons about why the supply 242 00:13:51,930 --> 00:13:56,850 curve could be shifting in. 243 00:13:56,850 --> 00:13:59,075 In this case, why is it-- why are 244 00:13:59,075 --> 00:14:02,550 sellers willing to offer a cheaper price at any fixed 245 00:14:02,550 --> 00:14:03,594 quantity? 246 00:14:03,594 --> 00:14:05,260 And one idea that we could have for this 247 00:14:05,260 --> 00:14:08,100 is just that there are more firms in this market. 248 00:14:08,100 --> 00:14:11,440 If this market isn't perfectly competitive to start off with, 249 00:14:11,440 --> 00:14:12,950 then increasing the number of firms 250 00:14:12,950 --> 00:14:15,470 is gonna increase competition, and the producers 251 00:14:15,470 --> 00:14:17,790 are gonna have to drop their prices. 252 00:14:17,790 --> 00:14:20,700 A second idea for why we've seen the supply curve 253 00:14:20,700 --> 00:14:26,570 shift out and down could be the fact that input price for steel 254 00:14:26,570 --> 00:14:27,340 has dropped. 255 00:14:27,340 --> 00:14:30,329 Perhaps the way of manufacturing or getting the raw material 256 00:14:30,329 --> 00:14:32,870 is cheaper because the machine they're using to get the steel 257 00:14:32,870 --> 00:14:34,030 is cheaper. 258 00:14:34,030 --> 00:14:36,280 Basically, when you're thinking about the shift that's 259 00:14:36,280 --> 00:14:39,540 making it cheaper for suppliers to produce the good, 260 00:14:39,540 --> 00:14:41,000 all you need to think about is what 261 00:14:41,000 --> 00:14:43,208 could make it so that they're more willing to produce 262 00:14:43,208 --> 00:14:45,090 at a lower price. 263 00:14:45,090 --> 00:14:46,970 So again, with this problem, we went 264 00:14:46,970 --> 00:14:49,350 through working with elasticities and demands. 265 00:14:49,350 --> 00:14:53,000 We've seen that we can go from a demand curve or supply curve 266 00:14:53,000 --> 00:14:56,650 to elasticities, or we can go from elasticities to demands. 267 00:14:56,650 --> 00:14:59,880 And then, once we've had the supply and the demand curves, 268 00:14:59,880 --> 00:15:02,667 we looked at how do we interpret the shifts and shocks? 269 00:15:02,667 --> 00:15:04,250 And we looked at possible explanations 270 00:15:04,250 --> 00:15:05,910 for those shift and shocks. 271 00:15:05,910 --> 00:15:08,320 I hope you found this problem helpful.