WEBVTT 00:00:00.040 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 Commons license. 00:00:03.870 --> 00:00:06.320 Your support will help MIT OpenCourseWare 00:00:06.320 --> 00:00:10.560 continue to offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.300 To make a donation or view additional materials 00:00:13.300 --> 00:00:17.210 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.210 --> 00:00:17.865 at ocw.mit.edu. 00:00:22.170 --> 00:00:25.300 PROFESSOR: Hi, and welcome back to the 14.01 problem solving 00:00:25.300 --> 00:00:25.990 videos. 00:00:25.990 --> 00:00:29.620 Today, we're going to work on Fall 2010, problem set one, 00:00:29.620 --> 00:00:30.727 problem number four. 00:00:30.727 --> 00:00:32.310 And in this problem, we're going to be 00:00:32.310 --> 00:00:34.062 working with elasticities. 00:00:34.062 --> 00:00:36.020 But instead of starting with a demand function, 00:00:36.020 --> 00:00:37.980 and starting with a supply function, 00:00:37.980 --> 00:00:41.210 and calculating the elasticity given those functions, 00:00:41.210 --> 00:00:43.210 we're going to be given the elasticity of demand 00:00:43.210 --> 00:00:44.672 and the elasticity of supply. 00:00:44.672 --> 00:00:46.130 And we're going to have to back out 00:00:46.130 --> 00:00:48.741 what the demand functions and the supply functions 00:00:48.741 --> 00:00:49.740 should have looked like. 00:00:49.740 --> 00:00:51.620 So we're basically just working in reverse 00:00:51.620 --> 00:00:53.570 from what we did in lecture. 00:00:53.570 --> 00:00:57.210 Let's go ahead and read the full problem up through part A. 00:00:57.210 --> 00:01:00.040 You have been asked to analyze the market for steel. 00:01:00.040 --> 00:01:01.870 From public sources, you are able to find 00:01:01.870 --> 00:01:05.120 that last year's price for steel was $20 per ton. 00:01:05.120 --> 00:01:08.750 At this price, 100 million tons were sold on the world market. 00:01:08.750 --> 00:01:10.810 From trade association data, you are 00:01:10.810 --> 00:01:14.170 able to obtain estimates for their own price elasticities 00:01:14.170 --> 00:01:18.690 of demand and supply on the world markets as negative 0.25 00:01:18.690 --> 00:01:21.610 for demand and 0.5 for supply. 00:01:21.610 --> 00:01:24.440 Assume the steel has linear demand and supply 00:01:24.440 --> 00:01:25.960 curves throughout. 00:01:25.960 --> 00:01:29.850 Part A asks us to solve for the equations of demand and supply 00:01:29.850 --> 00:01:34.010 in this market, and to sketch the demand and supply curves. 00:01:34.010 --> 00:01:37.580 So looking at the formal definition of elasticity 00:01:37.580 --> 00:01:40.300 of demand and elasticity of supply, 00:01:40.300 --> 00:01:43.500 we basically are going to have three different parts to it. 00:01:43.500 --> 00:01:46.360 We have the derivative of either demand or supply 00:01:46.360 --> 00:01:50.180 function with respect to P, in this case the own price of P, 00:01:50.180 --> 00:01:52.300 or the price of steel. 00:01:52.300 --> 00:01:54.470 And we also have the equilibrium price, 00:01:54.470 --> 00:01:57.270 or any price at the point on the curve, and a quantity. 00:01:57.270 --> 00:02:00.420 In this case, it's going to be the equilibrium quantity. 00:02:00.420 --> 00:02:02.820 So basically, what we have now is 00:02:02.820 --> 00:02:05.740 we are given-- for the elasticity of demand, 00:02:05.740 --> 00:02:07.080 we're given three variables. 00:02:07.080 --> 00:02:09.509 We're given the price, the quantity, 00:02:09.509 --> 00:02:11.390 and the elasticity of demand. 00:02:11.390 --> 00:02:14.730 And that means the only thing that we don't know 00:02:14.730 --> 00:02:19.260 is the derivative of the demand curve with respect to P. 00:02:19.260 --> 00:02:21.940 So if we can isolate this derivative, 00:02:21.940 --> 00:02:24.750 then we can integrate the number that we're 00:02:24.750 --> 00:02:26.280 able to solve through for. 00:02:26.280 --> 00:02:28.210 And then we can solve out for what our demand 00:02:28.210 --> 00:02:29.510 curve is going to look like. 00:02:29.510 --> 00:02:32.230 So let's go ahead and walk through that process together. 00:02:32.230 --> 00:02:49.510 Substituting in for the elasticity of demand P and Q, 00:02:49.510 --> 00:02:50.909 we're gonna have this equation. 00:02:50.909 --> 00:02:52.700 And the one thing that I want you to notice 00:02:52.700 --> 00:02:57.810 is since the derivative of the demand curve with respect to P 00:02:57.810 --> 00:03:01.670 is negative 0.25, in this case, we know that it's linear. 00:03:01.670 --> 00:03:03.480 But just because it's linear at the point 00:03:03.480 --> 00:03:05.120 where price is 20 and quantity is 00:03:05.120 --> 00:03:07.620 100, that doesn't necessarily mean it's 00:03:07.620 --> 00:03:08.880 gonna be linear throughout. 00:03:08.880 --> 00:03:12.470 So it's useful to know that at any point on this line, 00:03:12.470 --> 00:03:15.660 it's always going to have the derivative equal 00:03:15.660 --> 00:03:16.682 to negative 0.25. 00:03:16.682 --> 00:03:17.390 So that's useful. 00:03:17.390 --> 00:03:21.170 We know we can integrate and have the correct answer. 00:03:21.170 --> 00:03:35.450 Solving for dQD dP, we're gonna have negative 1.25. 00:03:35.450 --> 00:03:37.580 And we're just going to integrate this with respect 00:03:37.580 --> 00:03:40.825 to P. And after we integrate, we're 00:03:40.825 --> 00:03:42.200 going to be left with a constant. 00:03:54.190 --> 00:03:57.834 In this case, we're going to call the constant a. 00:03:57.834 --> 00:03:59.750 This is how much the demand curve has actually 00:03:59.750 --> 00:04:03.220 shifted up to begin with, shifted up or down. 00:04:03.220 --> 00:04:05.390 And to solve for a, all we have to do 00:04:05.390 --> 00:04:09.690 is we can just plug back in for the $20 and the 100 quantity, 00:04:09.690 --> 00:04:12.970 and we can solve through for what a is going to be equal. 00:04:12.970 --> 00:04:16.250 When you solve through plugging in Q and P, 00:04:16.250 --> 00:04:32.610 you're going to find that a is equal to 125. 00:04:32.610 --> 00:04:35.260 So we're gonna have that our final demand function is gonna 00:04:35.260 --> 00:04:40.500 be negative 1.25P plus 125. 00:04:40.500 --> 00:04:43.830 Now we can go through this exact same process 00:04:43.830 --> 00:04:47.340 with the elasticity of supply now. 00:04:47.340 --> 00:04:51.000 And all we have to do now is use the number 0.5 instead, 00:04:51.000 --> 00:04:52.130 and we can solve through. 00:04:52.130 --> 00:04:53.270 We can integrate. 00:04:53.270 --> 00:04:55.500 And then we're going to solve for the other constant 00:04:55.500 --> 00:04:58.400 to, again, get our supply curve. 00:04:58.400 --> 00:05:10.020 Substituting in the information we have, 00:05:10.020 --> 00:05:11.770 we're going to be left with this equation. 00:05:11.770 --> 00:05:14.400 And we're gonna go ahead and isolate 00:05:14.400 --> 00:05:15.620 the derivative that we have. 00:05:27.410 --> 00:05:28.910 And when we integrate again, we have 00:05:28.910 --> 00:05:31.160 to remember that we are going to have a constant that we're 00:05:31.160 --> 00:05:32.180 gonna have to solve for. 00:05:40.850 --> 00:05:43.270 And I'm gonna just call this constant c. 00:05:43.270 --> 00:05:46.990 Again, plug in the price of 20 and the quantity equal to 100 00:05:46.990 --> 00:06:02.300 and you're gonna find that the supply curve is gonna be equal 00:06:02.300 --> 00:06:05.880 to 2.5P plus 50. 00:06:05.880 --> 00:06:09.141 And we can do a quick sketch of this on our axes here. 00:06:09.141 --> 00:06:11.640 We're just gonna go ahead and draw our upward-sloping supply 00:06:11.640 --> 00:06:19.050 curve, our downward-sloping demand curve. 00:06:19.050 --> 00:06:23.960 And we're gonna mark the equilibrium point 00:06:23.960 --> 00:06:26.870 and label the equilibrium quantities and the equilibrium 00:06:26.870 --> 00:06:27.620 prices, as well. 00:06:33.346 --> 00:06:35.720 And before we move on to the second part of this problem, 00:06:35.720 --> 00:06:36.510 we can pause here. 00:06:36.510 --> 00:06:38.843 And we can think about what did the elasticities that we 00:06:38.843 --> 00:06:40.590 started with actually mean. 00:06:40.590 --> 00:06:43.230 Well, if we were to look at this point of intersection 00:06:43.230 --> 00:06:46.540 at the equilibrium of the demand curve, 00:06:46.540 --> 00:06:48.680 we're looking at the percentage change 00:06:48.680 --> 00:06:53.960 at this point in quantity per percentage change in price. 00:06:53.960 --> 00:06:59.850 So we're basically just saying, for that tiny change, 00:06:59.850 --> 00:07:03.060 an infinitesimally small change at this point for the demand 00:07:03.060 --> 00:07:07.020 curve, how much does quantity change, percentage-wise, 00:07:07.020 --> 00:07:08.380 relative to price? 00:07:08.380 --> 00:07:09.800 And that's also what we're looking 00:07:09.800 --> 00:07:12.000 at with the supply curve. 00:07:12.000 --> 00:07:14.030 So when you're given a elasticity, 00:07:14.030 --> 00:07:15.690 if you have an elasticity of supply, 00:07:15.690 --> 00:07:19.150 it makes sense that it's gonna be positive, in this case 0.5, 00:07:19.150 --> 00:07:22.480 because when price increases, suppliers 00:07:22.480 --> 00:07:24.080 are willing to supply more. 00:07:24.080 --> 00:07:26.620 And it makes sense that the demand elasticity that we're 00:07:26.620 --> 00:07:29.980 given is negative, or negative 0.25, 00:07:29.980 --> 00:07:32.810 because when price begins to increase, 00:07:32.810 --> 00:07:37.410 the consumers are gonna want less of the product. 00:07:37.410 --> 00:07:38.960 Now, the second part of this problem 00:07:38.960 --> 00:07:41.120 is going to give us new elasticities of demand 00:07:41.120 --> 00:07:41.820 and supply. 00:07:41.820 --> 00:07:44.484 And I'm gonna just quickly run through the actual calculation, 00:07:44.484 --> 00:07:46.525 because it's gonna be the same as our calculation 00:07:46.525 --> 00:07:47.960 that we just did. 00:07:47.960 --> 00:07:52.020 And instead, we're gonna think about possible causes 00:07:52.020 --> 00:07:54.360 for the shifts that we see in the supply and the demand 00:07:54.360 --> 00:07:56.220 curve. 00:07:56.220 --> 00:07:58.410 Part B says, suppose that you discover 00:07:58.410 --> 00:08:01.820 that the current price of steel is $15 per ton 00:08:01.820 --> 00:08:04.590 and the current level of worldwide sales of steel 00:08:04.590 --> 00:08:07.130 is 150 million tons. 00:08:07.130 --> 00:08:09.890 The most recent elasticity estimates 00:08:09.890 --> 00:08:11.540 from the trade association this year 00:08:11.540 --> 00:08:16.820 are negative 0.125 for demand and 0.25 for supply. 00:08:16.820 --> 00:08:19.140 Describe the change in the supply and the demand curves 00:08:19.140 --> 00:08:21.280 over the past year using your diagram 00:08:21.280 --> 00:08:26.410 from part A. What sort of events might explain the change? 00:08:26.410 --> 00:08:28.670 Now, I've given us the information for this part 00:08:28.670 --> 00:08:31.260 of the problem on this board. 00:08:31.260 --> 00:08:33.850 And you'll notice that our inputs, or our variables, 00:08:33.850 --> 00:08:34.570 have changed now. 00:08:34.570 --> 00:08:37.380 The price has dropped from $20. 00:08:37.380 --> 00:08:39.600 Now it's gonna be down to $15. 00:08:39.600 --> 00:08:42.419 You're gonna notice that the quantity has actually 00:08:42.419 --> 00:08:45.720 increased from 100 up to 150. 00:08:45.720 --> 00:08:49.005 And our elasticities of demand and elasticities of supply 00:08:49.005 --> 00:08:51.630 have changed, because the price and the quantity are different, 00:08:51.630 --> 00:08:55.210 and we're at a different point on our graph. 00:08:55.210 --> 00:08:57.800 The process we're gonna do to solve for our demand 00:08:57.800 --> 00:09:01.340 curves and our supply curves are going to be exactly identical. 00:09:01.340 --> 00:09:03.100 And when you follow the same process-- 00:09:03.100 --> 00:09:05.020 I'll just do the first step up here-- 00:09:05.020 --> 00:09:07.726 you're gonna substitute in for the information that's 00:09:07.726 --> 00:09:08.600 given in the problem. 00:09:44.869 --> 00:09:46.410 And all you're going to do is you're, 00:09:46.410 --> 00:09:48.170 again, gonna solve through for the derivative. 00:09:48.170 --> 00:09:49.128 You're gonna integrate. 00:09:49.128 --> 00:09:50.960 And you're gonna find the constants. 00:09:50.960 --> 00:09:53.120 After you do that entire process, 00:09:53.120 --> 00:09:59.370 you're gonna find that the demand curve is 00:09:59.370 --> 00:10:00.420 given by this equation. 00:10:05.094 --> 00:10:07.010 And you're gonna find that the supply curve is 00:10:07.010 --> 00:10:08.135 given by this new equation. 00:10:24.100 --> 00:10:26.990 Now, if we look at this new demand 00:10:26.990 --> 00:10:30.310 curve and this new supply curve, we'll 00:10:30.310 --> 00:10:33.870 actually notice that the slope, with respect to P, 00:10:33.870 --> 00:10:35.910 is going to be identical in both of the cases 00:10:35.910 --> 00:10:39.020 that we solved for, both the beginning case and the case 00:10:39.020 --> 00:10:40.860 in the end of the problem. 00:10:40.860 --> 00:10:43.780 The only thing that's shifted between our quantities demanded 00:10:43.780 --> 00:10:46.760 and our quantities supplied, or the curves, 00:10:46.760 --> 00:10:48.240 is there's been a shift. 00:10:48.240 --> 00:10:51.610 And the shift for the demand curve-- 00:10:51.610 --> 00:10:57.270 it went from an intercept of 125 now to an intercept of 168.75. 00:10:57.270 --> 00:11:02.200 So our demand curve is shifting up and out. 00:11:02.200 --> 00:11:08.330 So we can represent this shift in demand like this. 00:11:08.330 --> 00:11:11.040 Notice that the slope is going to be exactly identical. 00:11:11.040 --> 00:11:14.597 I'm going to write a small db for part B. 00:11:14.597 --> 00:11:16.680 And then we can do the same sort of interpretation 00:11:16.680 --> 00:11:18.150 for our supply curve. 00:11:18.150 --> 00:11:22.420 Looking at our supply curve, the intercepts, now, is 112.5. 00:11:22.420 --> 00:11:26.070 But before, it was only at 50. 00:11:26.070 --> 00:11:28.920 And what this means, this means that the supply curve 00:11:28.920 --> 00:11:32.320 is going to shift in and down. 00:11:40.150 --> 00:11:42.554 And so my graph with the equilibrium price 00:11:42.554 --> 00:11:44.220 that I've drawn-- it's a little bit off, 00:11:44.220 --> 00:11:49.690 but what you should see-- you should 00:11:49.690 --> 00:11:54.960 see that the new equilibrium price has fallen. 00:11:54.960 --> 00:11:58.790 In this case, it's fallen to 15. 00:11:58.790 --> 00:12:04.090 And the equilibrium quantity has increased from 100 to 150. 00:12:04.090 --> 00:12:08.130 So since we had both a shift in supply and a shift in demand, 00:12:08.130 --> 00:12:12.060 necessarily we see that quantity is going to increase. 00:12:12.060 --> 00:12:18.310 But if the demand curve had shifted way up here, 00:12:18.310 --> 00:12:20.300 we could see that price could have increased. 00:12:20.300 --> 00:12:23.370 So the effect on the price in this market is ambiguous. 00:12:23.370 --> 00:12:26.250 We can say that, necessarily, the effect on quantity 00:12:26.250 --> 00:12:29.170 is going to be clearly towards an increase. 00:12:29.170 --> 00:12:30.990 So to wrap up this problem, we saw 00:12:30.990 --> 00:12:34.190 that changes in elasticities can also 00:12:34.190 --> 00:12:36.990 represent changes in the underlying demand and supply 00:12:36.990 --> 00:12:37.922 functions. 00:12:37.922 --> 00:12:39.630 Let's wrap up by just thinking about what 00:12:39.630 --> 00:12:43.040 could have caused the demand shift that we've seen. 00:12:43.040 --> 00:12:47.210 And what could have caused the supply shift that we saw? 00:12:47.210 --> 00:12:50.305 Now, there are a couple of ideas that we can have for demand. 00:12:53.646 --> 00:12:55.270 The first idea that we could have is we 00:12:55.270 --> 00:12:58.750 could just have had an increase in the income of a consumer. 00:12:58.750 --> 00:13:01.090 If a consumer has more income, then they 00:13:01.090 --> 00:13:03.940 might be willing to spend more on steel. 00:13:03.940 --> 00:13:06.120 A second idea that we have, we could 00:13:06.120 --> 00:13:08.420 have that the price of a substitute-- perhaps you're 00:13:08.420 --> 00:13:11.450 considering building a bridge out of iron instead of steel-- 00:13:11.450 --> 00:13:14.460 if the price of the substitute has increased, 00:13:14.460 --> 00:13:15.960 then perhaps the consumers are going 00:13:15.960 --> 00:13:19.650 to be willing to pay more to get the steel since the iron 00:13:19.650 --> 00:13:21.460 is more expensive. 00:13:21.460 --> 00:13:23.400 A third possible idea is that the number 00:13:23.400 --> 00:13:27.690 of goods that you need to make from steel is increasing. 00:13:27.690 --> 00:13:30.200 So if you suddenly find new uses for steel, 00:13:30.200 --> 00:13:34.060 then the price that you're willing to pay at any given 00:13:34.060 --> 00:13:37.640 point is going to be higher. 00:13:37.640 --> 00:13:39.704 Basically, to affect the demand curve, 00:13:39.704 --> 00:13:41.370 you have to think about why would people 00:13:41.370 --> 00:13:44.280 be more willing to pay more for a fixed quantity. 00:13:44.280 --> 00:13:47.340 And I just listed off a couple of ideas there. 00:13:47.340 --> 00:13:51.930 We can also think about reasons about why the supply 00:13:51.930 --> 00:13:56.850 curve could be shifting in. 00:13:56.850 --> 00:13:59.075 In this case, why is it-- why are 00:13:59.075 --> 00:14:02.550 sellers willing to offer a cheaper price at any fixed 00:14:02.550 --> 00:14:03.594 quantity? 00:14:03.594 --> 00:14:05.260 And one idea that we could have for this 00:14:05.260 --> 00:14:08.100 is just that there are more firms in this market. 00:14:08.100 --> 00:14:11.440 If this market isn't perfectly competitive to start off with, 00:14:11.440 --> 00:14:12.950 then increasing the number of firms 00:14:12.950 --> 00:14:15.470 is gonna increase competition, and the producers 00:14:15.470 --> 00:14:17.790 are gonna have to drop their prices. 00:14:17.790 --> 00:14:20.700 A second idea for why we've seen the supply curve 00:14:20.700 --> 00:14:26.570 shift out and down could be the fact that input price for steel 00:14:26.570 --> 00:14:27.340 has dropped. 00:14:27.340 --> 00:14:30.329 Perhaps the way of manufacturing or getting the raw material 00:14:30.329 --> 00:14:32.870 is cheaper because the machine they're using to get the steel 00:14:32.870 --> 00:14:34.030 is cheaper. 00:14:34.030 --> 00:14:36.280 Basically, when you're thinking about the shift that's 00:14:36.280 --> 00:14:39.540 making it cheaper for suppliers to produce the good, 00:14:39.540 --> 00:14:41.000 all you need to think about is what 00:14:41.000 --> 00:14:43.208 could make it so that they're more willing to produce 00:14:43.208 --> 00:14:45.090 at a lower price. 00:14:45.090 --> 00:14:46.970 So again, with this problem, we went 00:14:46.970 --> 00:14:49.350 through working with elasticities and demands. 00:14:49.350 --> 00:14:53.000 We've seen that we can go from a demand curve or supply curve 00:14:53.000 --> 00:14:56.650 to elasticities, or we can go from elasticities to demands. 00:14:56.650 --> 00:14:59.880 And then, once we've had the supply and the demand curves, 00:14:59.880 --> 00:15:02.667 we looked at how do we interpret the shifts and shocks? 00:15:02.667 --> 00:15:04.250 And we looked at possible explanations 00:15:04.250 --> 00:15:05.910 for those shift and shocks. 00:15:05.910 --> 00:15:08.320 I hope you found this problem helpful.