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 --> 00:00:06.910 Your support will help MIT OpenCourseWare continue to 00:00:06.910 --> 00:00:10.560 offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.460 To make a donation or view additional materials from 00:00:13.460 --> 00:00:17.390 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.390 --> 00:00:18.640 ocw.mit.edu. 00:00:21.708 --> 00:00:24.990 PROFESSOR: Today, we're going to be working on fall 2010, 00:00:24.990 --> 00:00:27.360 problem set number 2, and we're going to be doing 00:00:27.360 --> 00:00:29.200 problem number 4. 00:00:29.200 --> 00:00:31.030 I'm going to start off by reading the problem. 00:00:31.030 --> 00:00:32.970 There's a lot of information here, so I've written up some 00:00:32.970 --> 00:00:36.220 of the key points on the board for you guys already. 00:00:36.220 --> 00:00:40.240 It is exactly 24 hours before Lauren's physics final. 00:00:40.240 --> 00:00:42.830 She has an economics final directly after her physics 00:00:42.830 --> 00:00:45.810 final, and has no time to study in between. 00:00:45.810 --> 00:00:48.390 Lauren wants to be a physicist, so she places more 00:00:48.390 --> 00:00:50.630 weight on her physics test score. 00:00:50.630 --> 00:00:55.560 Her utility function is given right here. 00:00:55.560 --> 00:00:58.550 Where p is the score of her physics final and e is the 00:00:58.550 --> 00:01:00.740 score of her economics final. 00:01:00.740 --> 00:01:02.665 Although, she cares more about physics, she 00:01:02.665 --> 00:01:04.280 is better at economics. 00:01:04.280 --> 00:01:07.280 For each hour spent studying economics, she will increase 00:01:07.280 --> 00:01:09.560 her score by three points. 00:01:09.560 --> 00:01:12.640 But her physics score will only increase by two points 00:01:12.640 --> 00:01:15.830 for every hour spent studying physics. 00:01:15.830 --> 00:01:18.170 Studying zero hours results in a score of 00:01:18.170 --> 00:01:20.310 zero on both subjects. 00:01:20.310 --> 00:01:22.960 Although natural log of zero is not defined, assume her 00:01:22.960 --> 00:01:27.070 utility for a score of zero is negative infinity. 00:01:27.070 --> 00:01:28.730 Now, we're going to go ahead and we're going to work 00:01:28.730 --> 00:01:33.300 through parts A, B, and C, A, B, C, and D and then we'll do 00:01:33.300 --> 00:01:36.510 part E, which is a new scenario afterwards. 00:01:36.510 --> 00:01:38.740 Part A, we're going to find the constraints that Lauren 00:01:38.740 --> 00:01:41.790 faces in her test score maximization problem. 00:01:41.790 --> 00:01:44.260 And part B, we're going to find how many hours does 00:01:44.260 --> 00:01:47.340 Lauren optimally spend studying physics, how many 00:01:47.340 --> 00:01:49.950 hours does she spent studying economics. 00:01:49.950 --> 00:01:52.270 And hours are divisible, so we don't need whole number 00:01:52.270 --> 00:01:55.120 solutions for the hours spent studying physics and the hours 00:01:55.120 --> 00:01:58.550 spent studying economics. 00:01:58.550 --> 00:02:00.740 So for part A, all we're looking for is we're looking 00:02:00.740 --> 00:02:01.810 for the constraints. 00:02:01.810 --> 00:02:04.020 And it sounds like the constraint that Lauren is 00:02:04.020 --> 00:02:06.310 really facing in this scenario is the amount of 00:02:06.310 --> 00:02:07.770 time that she has. 00:02:07.770 --> 00:02:10.509 It seems like Lauren's pretty intense about studying, so in 00:02:10.509 --> 00:02:13.630 the 24 hours before the test, she's not getting any sleep. 00:02:13.630 --> 00:02:16.370 She's going to spend all of this time studying. 00:02:16.370 --> 00:02:19.980 So we're going to have two hours variables to represent 00:02:19.980 --> 00:02:22.970 the hours she spends studying physics and the hours she 00:02:22.970 --> 00:02:25.350 spends studying economics. 00:02:25.350 --> 00:02:33.360 And we know that the hours, when we add them together, 00:02:33.360 --> 00:02:38.610 they're going to have to be less than or equal to 24. 00:02:38.610 --> 00:02:41.160 Now, if this is our constraint, we really know 00:02:41.160 --> 00:02:43.190 that if she's trying to maximize her scores and if 00:02:43.190 --> 00:02:45.650 that's all she really cares about, she's not going to 00:02:45.650 --> 00:02:48.120 spend less than 24 hours studying. 00:02:48.120 --> 00:02:55.320 So we can actually just say that the sum of those two 00:02:55.320 --> 00:02:57.200 hours-- her hours spent studying physics and her hours 00:02:57.200 --> 00:03:01.780 spent studying economics are going to be equal to 24. 00:03:01.780 --> 00:03:05.000 So that's the first constraint that Lauren is going to face. 00:03:05.000 --> 00:03:07.040 The second constraint that she's going to face, and it's 00:03:07.040 --> 00:03:11.190 not necessarily an intuitive one-- 00:03:11.190 --> 00:03:14.940 or it is actually really intuitive, but it's also a 00:03:14.940 --> 00:03:16.190 trivial one. 00:03:19.360 --> 00:03:22.810 It's just that she can't spend less than zero hours studying 00:03:22.810 --> 00:03:24.880 physics or studying economics. 00:03:24.880 --> 00:03:28.210 So we're going to add those constraints in as well. 00:03:28.210 --> 00:03:30.530 And the final constraints are actually production 00:03:30.530 --> 00:03:32.100 constraints. 00:03:32.100 --> 00:03:33.640 If the hours spent studying-- 00:03:33.640 --> 00:03:35.430 these aren't actually what she's interested in. 00:03:35.430 --> 00:03:38.160 What she's interested in are the p and the e. 00:03:38.160 --> 00:03:39.870 That's how she's going to get her utilities-- 00:03:39.870 --> 00:03:41.520 through the test scores. 00:03:41.520 --> 00:03:45.360 So what we need is we need to find how she produces her 00:03:45.360 --> 00:03:46.720 physics score. 00:03:46.720 --> 00:03:50.530 If she gets two points for every hour she spent studying, 00:03:50.530 --> 00:03:57.140 her physics score is going to be p equals 2 times Hp-- 00:03:57.140 --> 00:03:59.690 that's the production of her physics score. 00:03:59.690 --> 00:04:03.640 The production of her economics score is going to 00:04:03.640 --> 00:04:08.190 equal 3 times He, since we know that she's a little bit 00:04:08.190 --> 00:04:10.820 better at economics than physics. 00:04:10.820 --> 00:04:12.650 So these are the constraints that we're going to face in 00:04:12.650 --> 00:04:14.370 the problem. 00:04:14.370 --> 00:04:19.680 Let's go ahead and move on to part B. And for part B, what 00:04:19.680 --> 00:04:21.870 we're going to try to do is we're going to try to take 00:04:21.870 --> 00:04:25.620 these constraints and we're trying to maximize the amount 00:04:25.620 --> 00:04:30.180 of utility she's going to get from studying. 00:04:30.180 --> 00:04:32.720 So what we're doing for part B is we're going to maximize 00:04:32.720 --> 00:04:35.710 utility, and we're going to plug in some of these 00:04:35.710 --> 00:04:36.840 constraints. 00:04:36.840 --> 00:04:39.530 Before we can do this, we really need to get down to 00:04:39.530 --> 00:04:40.650 first-order conditions. 00:04:40.650 --> 00:04:44.620 We need to be able to take the derivative with regards to 00:04:44.620 --> 00:04:45.670 only one variable. 00:04:45.670 --> 00:04:48.890 So we have to get-- instead of p and e here, we're going to 00:04:48.890 --> 00:04:50.400 try to get only one variable. 00:04:50.400 --> 00:04:52.940 And the variable we're going to get in there is going to be 00:04:52.940 --> 00:04:54.560 the hours spent study economics. 00:04:54.560 --> 00:04:58.240 So we're going to get it all in terms of He. 00:04:58.240 --> 00:05:04.810 So we need to find a way to replace both e and p with He. 00:05:04.810 --> 00:05:08.740 In this case, replacing e with He is really easy. 00:05:08.740 --> 00:05:11.740 We can just say that e is going to equal 3He. 00:05:14.830 --> 00:05:19.150 Replacing p with the He or the hours spent studying economics 00:05:19.150 --> 00:05:21.060 is a little bit more difficult. 00:05:21.060 --> 00:05:23.120 We're going to go to this equation. 00:05:23.120 --> 00:05:27.260 And we can say that Hp is going to be 00:05:27.260 --> 00:05:33.470 equal to 24 minus He. 00:05:33.470 --> 00:05:36.930 And then for this Hp, we're going to plug this into our 00:05:36.930 --> 00:05:39.310 production function for the physics score. 00:05:39.310 --> 00:05:50.250 So we get that p is equal to 2 times 24 minus He. 00:05:50.250 --> 00:05:55.650 So we're going to take these two equations that we have in 00:05:55.650 --> 00:05:57.316 and we're going to substitute for e and p 00:05:57.316 --> 00:05:58.566 in our utility function. 00:06:30.740 --> 00:06:32.430 And so we're going to maximize, and the way we're 00:06:32.430 --> 00:06:35.720 going to maximize is by just taking the first-order 00:06:35.720 --> 00:06:36.970 conditions or the FOC. 00:06:39.490 --> 00:06:41.970 And the FOC in this case is just going to be the 00:06:41.970 --> 00:06:44.590 derivative with respect to He. 00:06:44.590 --> 00:06:47.040 When you take the derivative with respect to He, what 00:06:47.040 --> 00:06:56.400 you're going to get is 0.6 times negative 2 all over (48 00:06:56.400 --> 00:07:06.983 minus 2 He) plus 0.4 times 3 all over 3He. 00:07:10.220 --> 00:07:12.350 And the reason this first-order condition is going 00:07:12.350 --> 00:07:14.800 to help us is because we know to maximize. 00:07:14.800 --> 00:07:18.030 We just set the first-order condition or the derivative 00:07:18.030 --> 00:07:22.810 with respect to He equal to zero. 00:07:22.810 --> 00:07:25.514 And from here, we can solve out for He. 00:07:29.780 --> 00:07:32.530 And when we solve for He, we find that the hour she spends 00:07:32.530 --> 00:07:35.766 studying economics is going to equal to 9.6. 00:07:38.390 --> 00:07:41.960 And we know that any time she spends not studying the 00:07:41.960 --> 00:07:44.770 economics, she's studying physics, since all she cares 00:07:44.770 --> 00:07:46.020 about is her test scores. 00:07:50.450 --> 00:07:53.070 She's going to spend the remaining 14.4 00:07:53.070 --> 00:07:54.570 hours studying physics. 00:07:58.610 --> 00:08:02.960 This is our answer for part B. 00:08:02.960 --> 00:08:09.340 Now, part C just asks us, "What economics and physics 00:08:09.340 --> 00:08:12.145 test scores will she achieve?" so we're looking for the e 00:08:12.145 --> 00:08:13.840 star and the p star. 00:08:13.840 --> 00:08:16.950 We can just take the hour she spent studying physics and the 00:08:16.950 --> 00:08:19.930 hours she spends studying economics and we can plug 00:08:19.930 --> 00:08:23.550 these back into her production functions for her physics 00:08:23.550 --> 00:08:27.300 score and an economics score. 00:08:27.300 --> 00:08:33.330 So, for part C, her economics score is going to be equal to 00:08:33.330 --> 00:08:35.565 3 times 9.6. 00:08:38.320 --> 00:08:45.620 She's going to get a 28.8 on her economics test. 00:08:45.620 --> 00:08:54.040 And for her physics test, she's also 00:08:54.040 --> 00:09:04.350 going to get a 28.8. 00:09:04.350 --> 00:09:09.310 Probably the not the best test scores in the world. 00:09:09.310 --> 00:09:12.940 Let's go ahead and look at part D. Part D asks us, "What 00:09:12.940 --> 00:09:16.400 utility level will she achieve with these test scores that 00:09:16.400 --> 00:09:20.170 she has?" Now, to find the utility levels, all we have to 00:09:20.170 --> 00:09:22.790 do is we have to go back to our utility function that we 00:09:22.790 --> 00:09:26.460 wrote down in part A, and we're just going to plug in 00:09:26.460 --> 00:09:29.850 the two test scores that she received, and we can solve 00:09:29.850 --> 00:09:31.620 through for overall utility. 00:09:57.900 --> 00:09:59.940 And so when you grab a calculator, and you solve 00:09:59.940 --> 00:10:02.350 through for this equation, what you're going to find is 00:10:02.350 --> 00:10:12.850 you're going to find her overall utility level is going 00:10:12.850 --> 00:10:15.210 to be equal to 3.36. 00:10:15.210 --> 00:10:17.090 And when we're measuring utility, we don't have any 00:10:17.090 --> 00:10:18.190 units on this. 00:10:18.190 --> 00:10:21.010 The units, we usually think of as utils, but we can just keep 00:10:21.010 --> 00:10:21.730 it like this. 00:10:21.730 --> 00:10:24.660 Just know that in relativistic terms, her 00:10:24.660 --> 00:10:26.970 utility is that 3.36. 00:10:26.970 --> 00:10:32.290 And this is going to be useful for part D. Because in part E, 00:10:32.290 --> 00:10:35.160 we have the option of sending Lauren off to 00:10:35.160 --> 00:10:36.630 an economics tutor. 00:10:36.630 --> 00:10:39.090 And we're going to be comparing her utility after 00:10:39.090 --> 00:10:44.040 going to the economics tutor to this utility of 3.36. 00:10:44.040 --> 00:10:47.270 And by comparing her new utility with the old utility, 00:10:47.270 --> 00:10:49.740 we can see if her utility's higher after going the 00:10:49.740 --> 00:10:50.900 economics tutor. 00:10:50.900 --> 00:10:53.400 We can see if that's a good decision for her. 00:10:53.400 --> 00:10:56.090 So now that we finished calculating the utility for 00:10:56.090 --> 00:10:59.080 Lauren in the initial case where she had 24 hours and no 00:10:59.080 --> 00:11:03.040 help, now we're going to move on to part E. Part E says, 00:11:03.040 --> 00:11:05.660 "Suppose Lauren can get an economics tutor. 00:11:05.660 --> 00:11:08.300 If she goes to the tutor, she will increase her economics 00:11:08.300 --> 00:11:11.710 test score by five points for every hour spent studying, 00:11:11.710 --> 00:11:13.290 instead of three points. 00:11:13.290 --> 00:11:15.380 But will lose four hours of study time by 00:11:15.380 --> 00:11:16.810 going to the tutor. 00:11:16.810 --> 00:11:19.570 She cannot study while at the tutor, and going to the tutor 00:11:19.570 --> 00:11:22.050 does not directly improve her test score. 00:11:22.050 --> 00:11:25.090 Should Lauren go to the tutor?" 00:11:25.090 --> 00:11:26.630 Now, for this problem, we can go back 00:11:26.630 --> 00:11:28.730 to our initial scenario. 00:11:28.730 --> 00:11:31.560 Where we had our constraint for the amount of 00:11:31.560 --> 00:11:33.010 time Lauren can spend. 00:11:33.010 --> 00:11:35.380 If Lauren goes to the tutor, it sounds like she's going to 00:11:35.380 --> 00:11:38.540 spend about four hours in the car driving to the tutor's 00:11:38.540 --> 00:11:40.470 house and back. 00:11:40.470 --> 00:11:44.150 So instead of having 24 total hours of study time, her new 00:11:44.150 --> 00:11:48.630 constraint is that her total study time Hp plus 00:11:48.630 --> 00:11:51.740 He is equal to 20. 00:11:51.740 --> 00:11:53.090 That's the downside. 00:11:53.090 --> 00:11:56.030 The upshot of it is that her production function for 00:11:56.030 --> 00:11:59.970 economics test scores is no longer three points for every 00:11:59.970 --> 00:12:06.210 hour she spent studying, now she gets five point for every 00:12:06.210 --> 00:12:08.150 hour she spends studying. 00:12:08.150 --> 00:12:11.000 So we're going to go through the exact same process for A, 00:12:11.000 --> 00:12:14.520 B, C, and D that we did before, only now we're going 00:12:14.520 --> 00:12:17.090 to use our new constraints to solve for the 00:12:17.090 --> 00:12:18.340 maximization problem. 00:12:24.160 --> 00:12:26.580 Before we start off for our maximization problem and 00:12:26.580 --> 00:12:29.150 solving for the first-order conditions, what we have to do 00:12:29.150 --> 00:12:31.690 is we have to get it all in terms of one variable. 00:12:31.690 --> 00:12:34.000 And the variable we're going to get it in terms of is the 00:12:34.000 --> 00:12:36.870 hours of economic studying or He. 00:12:36.870 --> 00:12:45.340 So from the utility function, to solve or to plug in for e, 00:12:45.340 --> 00:12:53.290 the He, we have that e is going to equal five times He. 00:12:53.290 --> 00:13:00.510 For p, we know that Hp is equal to 20 minus He. 00:13:00.510 --> 00:13:03.490 That just says that any time she's not studying economics, 00:13:03.490 --> 00:13:05.770 she's going to be studying physics. 00:13:05.770 --> 00:13:12.580 We can plug this into our p equals 2 times Hp. 00:13:12.580 --> 00:13:21.200 So we know that p is going to equal 2 times 20 minus He. 00:13:21.200 --> 00:13:25.500 Let's go ahead and plug this and this into our utility 00:13:25.500 --> 00:13:27.830 maximization problem and solve for the first-order 00:13:27.830 --> 00:13:29.080 conditions. 00:13:53.780 --> 00:13:55.230 So we're maximizing utilities-- 00:13:55.230 --> 00:13:56.510 so we're going to take the derivative with 00:13:56.510 --> 00:13:58.020 respect to He again. 00:13:58.020 --> 00:13:59.610 When we do that, we get our FOC. 00:14:25.030 --> 00:14:27.660 And again, for the maximization problem, we can 00:14:27.660 --> 00:14:30.310 just set that equal to 0. 00:14:30.310 --> 00:14:33.000 Solving out for He, we find that she's going to spend 00:14:33.000 --> 00:14:39.560 eight hours studying economics. 00:14:39.560 --> 00:14:46.430 And we also find that any time she's not spending studying 00:14:46.430 --> 00:14:50.320 economics, she's going to be studying physics. 00:14:50.320 --> 00:14:56.080 So she's going to spend 12 hours studying physics. 00:14:56.080 --> 00:14:58.750 Now, we're not done with part E because really all we have 00:14:58.750 --> 00:15:01.580 to do for this problem is we have to solve for her new 00:15:01.580 --> 00:15:05.130 utility using these hours that she spends studying. 00:15:05.130 --> 00:15:08.380 And we have to compare her new utility to her old utility of 00:15:08.380 --> 00:15:11.980 3.36 to see if she's better off or worse with the 00:15:11.980 --> 00:15:14.720 economics tutor. 00:15:14.720 --> 00:15:18.360 In her production function for economics test score, we know 00:15:18.360 --> 00:15:22.980 that for each of those eight hours you spend studying, her 00:15:22.980 --> 00:15:25.670 test score's going to improve eight points, so her new 00:15:25.670 --> 00:15:29.850 economic score is 40. 00:15:29.850 --> 00:15:33.350 And we know that for each of those 12 hours she spends 00:15:33.350 --> 00:15:38.910 studying physics, her score's going to improve two points. 00:15:38.910 --> 00:15:44.510 So her physics score has dropped a little bit to 24. 00:15:44.510 --> 00:15:51.280 When we plug in for e and p into our utility function, we 00:15:51.280 --> 00:15:54.290 can find her utility level with the economics tutor. 00:16:06.890 --> 00:16:08.960 When we solve through for these natural logs, we're 00:16:08.960 --> 00:16:19.150 going to find that her new utility is equal to 3.38. 00:16:19.150 --> 00:16:20.990 And since the problem asks us to actually make 00:16:20.990 --> 00:16:22.220 a qualitative judgment-- 00:16:22.220 --> 00:16:25.840 is she better off or worse off, we need to compare her 00:16:25.840 --> 00:16:36.240 utility before of 3.36 to her new utility of 3.38. 00:16:36.240 --> 00:16:38.460 This is the comparison that we're interested in. 00:16:38.460 --> 00:16:40.910 And when we make this comparison, we can see that in 00:16:40.910 --> 00:16:43.900 relative terms, she's a little bit better off going to be 00:16:43.900 --> 00:16:47.415 economics tutor because her utility has risen two 00:16:47.415 --> 00:16:50.010 hundredths of a point. 00:16:50.010 --> 00:16:53.840 So in this case, for part E, we found that her utility 00:16:53.840 --> 00:16:56.640 increases even though she's lost time because her 00:16:56.640 --> 00:16:59.680 production function for her economics to score has gone 00:16:59.680 --> 00:17:03.050 up, and that she's better off going to the economics tutor 00:17:03.050 --> 00:17:05.369 because of that increase in utility. 00:17:05.369 --> 00:17:08.010 So the point of this problem was to look at the different 00:17:08.010 --> 00:17:10.880 constraints that consumers face when deciding where to 00:17:10.880 --> 00:17:14.079 spend their time or where to spend their money, and to see 00:17:14.079 --> 00:17:16.530 how those different constraints affect their 00:17:16.530 --> 00:17:17.780 utility levels.