WEBVTT 00:00:00.000 --> 00:00:01.920 align:middle line:90% [SQUEAKING] 00:00:01.920 --> 00:00:03.840 align:middle line:90% [RUSTLING] 00:00:03.840 --> 00:00:05.280 align:middle line:90% [CLICKING] 00:00:05.280 --> 00:00:11.540 align:middle line:90% 00:00:11.540 --> 00:00:13.880 align:middle line:84% ROBERT M. TOWNSEND: Let me remind you from the reading 00:00:13.880 --> 00:00:15.570 align:middle line:90% list where we are. 00:00:15.570 --> 00:00:18.660 align:middle line:84% We did the introductory lecture, of course, 00:00:18.660 --> 00:00:21.170 align:middle line:90% about economic science. 00:00:21.170 --> 00:00:25.490 align:middle line:84% And last time, we did our first nuts and bolts lecture 00:00:25.490 --> 00:00:31.160 align:middle line:84% on preferences, consumer theory, with the associated starred 00:00:31.160 --> 00:00:32.600 align:middle line:90% readings. 00:00:32.600 --> 00:00:36.020 align:middle line:84% And today, we're going to do more 00:00:36.020 --> 00:00:39.710 align:middle line:84% the way we ended the last class on consumer behavior. 00:00:39.710 --> 00:00:42.980 align:middle line:84% I'm showing this reading list because I just 00:00:42.980 --> 00:00:46.430 align:middle line:84% want to remind you that these starred articles do exist. 00:00:46.430 --> 00:00:49.010 align:middle line:84% The Nicholson and Schneider chapter 5, 00:00:49.010 --> 00:00:51.530 align:middle line:84% from which this is drawn, although not entirely. 00:00:51.530 --> 00:00:55.440 align:middle line:84% And also, this Jensen and Miller paper, 00:00:55.440 --> 00:00:58.610 align:middle line:84% which I will say a few words about today because it fits 00:00:58.610 --> 00:01:01.610 align:middle line:84% into this lecture, but also, as I probably 00:01:01.610 --> 00:01:04.129 align:middle line:84% mentioned too many times, that's also 00:01:04.129 --> 00:01:08.750 align:middle line:84% a big part of your first problem set and involves 00:01:08.750 --> 00:01:13.370 align:middle line:84% getting that paper, reading the yellow markings as highlights, 00:01:13.370 --> 00:01:16.820 align:middle line:84% accessing the data that these guys used, 00:01:16.820 --> 00:01:20.910 align:middle line:84% and running through some of their calculations. 00:01:20.910 --> 00:01:24.590 align:middle line:84% So that's the part of the starred readings 00:01:24.590 --> 00:01:26.550 align:middle line:90% for this lecture. 00:01:26.550 --> 00:01:31.690 align:middle line:84% So in terms of the study guide, Consumer Theory, 00:01:31.690 --> 00:01:35.040 align:middle line:84% as I was alerting you last time, there's 00:01:35.040 --> 00:01:39.660 align:middle line:84% a pretty good thorough review of what you should have learned 00:01:39.660 --> 00:01:43.410 align:middle line:84% or are learning from the topic of last time. 00:01:43.410 --> 00:01:45.570 align:middle line:84% And I will not go through all of this, 00:01:45.570 --> 00:01:47.910 align:middle line:84% but I will ask a few questions, get 00:01:47.910 --> 00:01:52.150 align:middle line:84% a few volunteers, maybe even call on a couple of you. 00:01:52.150 --> 00:01:57.330 align:middle line:84% So this one, give a plausible example of a commodity point 00:01:57.330 --> 00:02:01.400 align:middle line:84% that could not be in a typical consumption set. 00:02:01.400 --> 00:02:03.020 align:middle line:84% And I'll take volunteers on that. 00:02:03.020 --> 00:02:06.620 align:middle line:84% It's partly remembering the slide where I showed 00:02:06.620 --> 00:02:08.419 align:middle line:90% examples of consumption sets. 00:02:08.419 --> 00:02:10.840 align:middle line:90% Does anyone remember? 00:02:10.840 --> 00:02:13.692 align:middle line:90% The question is-- yes? 00:02:13.692 --> 00:02:15.400 align:middle line:84% AUDIENCE: One infeasible consumption site 00:02:15.400 --> 00:02:18.940 align:middle line:84% is consuming 25 hours in a day or something like that. 00:02:18.940 --> 00:02:20.800 align:middle line:90% ROBERT M. TOWNSEND: That's good. 00:02:20.800 --> 00:02:23.590 align:middle line:84% Yep, that was one of the diagrams where 00:02:23.590 --> 00:02:28.630 align:middle line:84% the amount of leisure available for work or leisure, 00:02:28.630 --> 00:02:31.330 align:middle line:90% the sum could not exceed 25. 00:02:31.330 --> 00:02:33.130 align:middle line:84% There was another one, which you might not 00:02:33.130 --> 00:02:35.530 align:middle line:84% have thought of given the way this is worded, 00:02:35.530 --> 00:02:39.310 align:middle line:84% which is consumption of bread in New York at noon 00:02:39.310 --> 00:02:42.580 align:middle line:84% and consumption in Boston at noon of bread. 00:02:42.580 --> 00:02:47.710 align:middle line:84% And you're not allowed to have bread in the interior 00:02:47.710 --> 00:02:51.040 align:middle line:84% because you can't consume simultaneously 00:02:51.040 --> 00:02:52.790 align:middle line:90% in both New York and Boston. 00:02:52.790 --> 00:02:54.220 align:middle line:90% So that was another example. 00:02:54.220 --> 00:02:57.590 align:middle line:90% But your answer was great. 00:02:57.590 --> 00:03:00.890 align:middle line:84% So can someone volunteer to give me 00:03:00.890 --> 00:03:05.870 align:middle line:84% the definition of local non-satiation? 00:03:05.870 --> 00:03:07.830 align:middle line:84% AUDIENCE: That means that there's 00:03:07.830 --> 00:03:13.440 align:middle line:84% in that-- so if you have a bundle, in any neighborhood 00:03:13.440 --> 00:03:16.890 align:middle line:84% of that bundle, there is some other bundle that's 00:03:16.890 --> 00:03:19.230 align:middle line:84% preferred to the bundle you originally had. 00:03:19.230 --> 00:03:21.600 align:middle line:90% There are no bliss points. 00:03:21.600 --> 00:03:24.180 align:middle line:84% ROBERT M. TOWNSEND: That's right. 00:03:24.180 --> 00:03:26.360 align:middle line:90% That's an excellent answer. 00:03:26.360 --> 00:03:28.920 align:middle line:84% Yeah, I can't remember whether we did X or Y. 00:03:28.920 --> 00:03:34.380 align:middle line:84% But for any X in the consumption set, 00:03:34.380 --> 00:03:39.150 align:middle line:84% there's some other Y, no matter how close 00:03:39.150 --> 00:03:44.430 align:middle line:84% we are to X, which is preferred to X and-- well, 00:03:44.430 --> 00:03:46.500 align:middle line:84% and Y has to be in the consumption set. 00:03:46.500 --> 00:03:48.430 align:middle line:90% So both things have to be true. 00:03:48.430 --> 00:03:53.250 align:middle line:84% It has to be available in this little ball or circle around X, 00:03:53.250 --> 00:03:57.180 align:middle line:90% and it has to be preferred. 00:03:57.180 --> 00:03:59.450 align:middle line:84% And you gave a good counter-example. 00:03:59.450 --> 00:04:00.980 align:middle line:84% If there were a bliss point, that 00:04:00.980 --> 00:04:07.650 align:middle line:84% means there's something that is maximal utility 00:04:07.650 --> 00:04:09.310 align:middle line:90% among all possible points. 00:04:09.310 --> 00:04:12.210 align:middle line:84% So if we do a circle around that, 00:04:12.210 --> 00:04:14.145 align:middle line:84% everything else would be inferior. 00:04:14.145 --> 00:04:16.850 align:middle line:90% 00:04:16.850 --> 00:04:21.529 align:middle line:84% If you went beyond the bliss point, northeast of it, 00:04:21.529 --> 00:04:24.860 align:middle line:84% you have too much of all the goods. 00:04:24.860 --> 00:04:29.240 align:middle line:84% But then, ironically, you also have local non-satiation 00:04:29.240 --> 00:04:36.080 align:middle line:84% because, given any point X, there's another point Y, which 00:04:36.080 --> 00:04:39.530 align:middle line:90% involves less than both goods. 00:04:39.530 --> 00:04:41.510 align:middle line:84% That would be strictly preferred because you're 00:04:41.510 --> 00:04:44.150 align:middle line:90% moving to the bliss point. 00:04:44.150 --> 00:04:46.520 align:middle line:84% So anyway, the point of this little discussion 00:04:46.520 --> 00:04:51.380 align:middle line:84% is that local non-satiation and global non-satiation 00:04:51.380 --> 00:04:54.500 align:middle line:90% are two different things. 00:04:54.500 --> 00:04:56.120 align:middle line:90% Thank you. 00:04:56.120 --> 00:05:00.610 align:middle line:84% OK, let's-- what does it mean for a preference relation to be 00:05:00.610 --> 00:05:02.665 align:middle line:84% represented by a utility function? 00:05:02.665 --> 00:05:05.380 align:middle line:90% 00:05:05.380 --> 00:05:07.540 align:middle line:84% Given a preference relation, is the utility 00:05:07.540 --> 00:05:10.420 align:middle line:84% function that represents it unique? 00:05:10.420 --> 00:05:14.810 align:middle line:90% Yes or no, and explain. 00:05:14.810 --> 00:05:20.210 align:middle line:84% AUDIENCE: Yeah, so we say that a utility function represents 00:05:20.210 --> 00:05:25.370 align:middle line:84% a preference if, for example, for any consumption bundle, X, 00:05:25.370 --> 00:05:29.610 align:middle line:84% any two consumption bundles, X and Y, if X is preferred to Y, 00:05:29.610 --> 00:05:34.040 align:middle line:84% then U of X is greater than U of Y. 00:05:34.040 --> 00:05:36.270 align:middle line:84% And the utility function is not unique. 00:05:36.270 --> 00:05:39.050 align:middle line:84% It just has to respect the given ordering of the preferences. 00:05:39.050 --> 00:05:42.800 align:middle line:84% So we could have U, or U squared, or U cubed, 00:05:42.800 --> 00:05:44.550 align:middle line:90% and all of those would work. 00:05:44.550 --> 00:05:45.800 align:middle line:90% ROBERT M. TOWNSEND: Excellent. 00:05:45.800 --> 00:05:48.860 align:middle line:90% Perfect answer. 00:05:48.860 --> 00:05:51.635 align:middle line:84% Any other comments or questions about that? 00:05:51.635 --> 00:05:55.450 align:middle line:90% 00:05:55.450 --> 00:05:57.070 align:middle line:90% AUDIENCE: I had a question. 00:05:57.070 --> 00:05:59.800 align:middle line:84% On this slide, it had said that it 00:05:59.800 --> 00:06:03.520 align:middle line:84% was unique only up to an order preserving transformation. 00:06:03.520 --> 00:06:08.030 align:middle line:84% What exactly does that, I guess, mean? 00:06:08.030 --> 00:06:11.260 align:middle line:84% ROBERT M. TOWNSEND: Well, if you had three points, 00:06:11.260 --> 00:06:13.360 align:middle line:84% ordering them from low to high in terms 00:06:13.360 --> 00:06:15.730 align:middle line:90% of preference relation-- 00:06:15.730 --> 00:06:20.560 align:middle line:84% say, X1, X2, X3, where X3 is preferred to X2 00:06:20.560 --> 00:06:23.800 align:middle line:84% is preferred to X1, all we need to do 00:06:23.800 --> 00:06:27.940 align:middle line:84% is to assign a higher number to the best one, a lower number 00:06:27.940 --> 00:06:29.140 align:middle line:90% to the intermediate one. 00:06:29.140 --> 00:06:35.140 align:middle line:84% So something like 15, 5, 1 preserves the rank ordering. 00:06:35.140 --> 00:06:37.960 align:middle line:84% Those are utility number assignments. 00:06:37.960 --> 00:06:46.950 align:middle line:84% But we could multiply by 2, and then we have 30, 10, 2. 00:06:46.950 --> 00:06:51.810 align:middle line:84% But we're still assigning numbers from high to low 00:06:51.810 --> 00:06:53.550 align:middle line:84% to preserve the same rank ordering. 00:06:53.550 --> 00:06:54.610 align:middle line:90% Does that help? 00:06:54.610 --> 00:06:56.610 align:middle line:90% AUDIENCE: Yeah, thank you. 00:06:56.610 --> 00:06:59.340 align:middle line:90% ROBERT M. TOWNSEND: OK. 00:06:59.340 --> 00:07:00.540 align:middle line:90% Yes, go on. 00:07:00.540 --> 00:07:03.240 align:middle line:84% AUDIENCE: Do you eventually need to, I guess, 00:07:03.240 --> 00:07:06.450 align:middle line:84% reintroduce some concept of magnitude-- 00:07:06.450 --> 00:07:08.160 align:middle line:84% or not magnitude, but have something 00:07:08.160 --> 00:07:11.040 align:middle line:84% beyond just ordering if you want to start talking about gambling 00:07:11.040 --> 00:07:12.420 align:middle line:90% or expected value? 00:07:12.420 --> 00:07:14.010 align:middle line:84% ROBERT M. TOWNSEND: Yes, absolutely. 00:07:14.010 --> 00:07:16.810 align:middle line:84% And we're going to do that in two lectures. 00:07:16.810 --> 00:07:21.490 align:middle line:84% So there, the utility function will-- 00:07:21.490 --> 00:07:25.820 align:middle line:84% the ordinal representation matters. 00:07:25.820 --> 00:07:29.840 align:middle line:84% And the question will be, how much more utility do you get 00:07:29.840 --> 00:07:35.330 align:middle line:84% or how much less, depending on an adverse event? 00:07:35.330 --> 00:07:42.150 align:middle line:84% So that the curvature literally represents your willingness 00:07:42.150 --> 00:07:44.590 align:middle line:90% to take gambles. 00:07:44.590 --> 00:07:46.780 align:middle line:84% Whereas with an ordinal function, 00:07:46.780 --> 00:07:48.610 align:middle line:90% a linear function is fine. 00:07:48.610 --> 00:07:51.010 align:middle line:84% Linear function when you're facing 00:07:51.010 --> 00:07:53.470 align:middle line:84% gambles means you're risk-neutral, 00:07:53.470 --> 00:07:57.310 align:middle line:84% and you just rank based on expected values. 00:07:57.310 --> 00:07:58.583 align:middle line:90% That's all a little bit early. 00:07:58.583 --> 00:07:59.500 align:middle line:90% It's a great question. 00:07:59.500 --> 00:08:03.280 align:middle line:84% And we're going to get there in two lectures. 00:08:03.280 --> 00:08:05.335 align:middle line:90% We got time for one more here. 00:08:05.335 --> 00:08:09.000 align:middle line:90% 00:08:09.000 --> 00:08:12.060 align:middle line:84% What does it mean for preferences to be homothetic? 00:08:12.060 --> 00:08:14.960 align:middle line:90% 00:08:14.960 --> 00:08:19.790 align:middle line:84% AUDIENCE: The MRS is just depending on the ratio 00:08:19.790 --> 00:08:22.310 align:middle line:90% of two goods, right? 00:08:22.310 --> 00:08:28.220 align:middle line:84% Not depending on the absolute quantity of each good. 00:08:28.220 --> 00:08:31.150 align:middle line:84% ROBERT M. TOWNSEND: Yep, perfect. 00:08:31.150 --> 00:08:34.150 align:middle line:84% Wow, I'm very happy, very impressed. 00:08:34.150 --> 00:08:35.200 align:middle line:90% Thank you, all. 00:08:35.200 --> 00:08:36.049 align:middle line:90% That's tremendous. 00:08:36.049 --> 00:08:40.120 align:middle line:84% OK, so again, these are just somewhat randomly selected. 00:08:40.120 --> 00:08:43.960 align:middle line:84% I do go over these before class to help 00:08:43.960 --> 00:08:46.930 align:middle line:84% me decide, since I can't ask them all, 00:08:46.930 --> 00:08:49.630 align:middle line:90% which one I would ask in class. 00:08:49.630 --> 00:08:51.850 align:middle line:84% There are some of these that say draw a graph 00:08:51.850 --> 00:08:54.610 align:middle line:90% or so on and so forth. 00:08:54.610 --> 00:08:56.450 align:middle line:90% So I didn't ask about those. 00:08:56.450 --> 00:09:00.520 align:middle line:84% But none of them are supposed to be terribly hard. 00:09:00.520 --> 00:09:05.290 align:middle line:84% But it is always a good review, at a minimum. 00:09:05.290 --> 00:09:11.320 align:middle line:84% OK, so today is a lecture on income and substitution 00:09:11.320 --> 00:09:12.370 align:middle line:90% effects. 00:09:12.370 --> 00:09:17.690 align:middle line:84% Or really, it's about consumer-maximizing behavior. 00:09:17.690 --> 00:09:24.890 align:middle line:84% And so when we solve the consumers utility maximization 00:09:24.890 --> 00:09:28.740 align:middle line:84% problem, we get individual demand functions. 00:09:28.740 --> 00:09:32.750 align:middle line:84% It should say function with an S because if there 00:09:32.750 --> 00:09:36.830 align:middle line:84% are n goods, little n, then there's a demand for good 1, 00:09:36.830 --> 00:09:40.320 align:middle line:90% good 2, good n, and so on. 00:09:40.320 --> 00:09:43.580 align:middle line:84% And what are the arguments in these demand functions? 00:09:43.580 --> 00:09:46.850 align:middle line:84% The entire vector of Prices, denoted P1 00:09:46.850 --> 00:09:51.350 align:middle line:84% through Pn, and Income, denoted as I. 00:09:51.350 --> 00:09:54.530 align:middle line:84% And you can think about prices and income denominated 00:09:54.530 --> 00:09:58.290 align:middle line:84% in dollars or some unit of account. 00:09:58.290 --> 00:10:00.720 align:middle line:84% We're going to be spending time looking 00:10:00.720 --> 00:10:03.900 align:middle line:84% at how the price of good k changes when 00:10:03.900 --> 00:10:06.510 align:middle line:90% we change the price of good k. 00:10:06.510 --> 00:10:11.610 align:middle line:84% But of necessity, other goods will be changing as well. 00:10:11.610 --> 00:10:16.500 align:middle line:84% It's at best, the implicit happenings in the diagrams. 00:10:16.500 --> 00:10:19.560 align:middle line:84% So here is a little bit of a repeat 00:10:19.560 --> 00:10:21.750 align:middle line:90% but expanded from last time. 00:10:21.750 --> 00:10:24.390 align:middle line:84% We've got two goods denoted, X and Y. 00:10:24.390 --> 00:10:30.270 align:middle line:84% And this household has a certain amount of income, I, say, I2. 00:10:30.270 --> 00:10:32.100 align:middle line:84% If income were I2, to this would be 00:10:32.100 --> 00:10:37.800 align:middle line:84% the budget line, which has a slope of PX over PY minus. 00:10:37.800 --> 00:10:41.610 align:middle line:84% And this is the tangency which is the maximum utility point 00:10:41.610 --> 00:10:46.680 align:middle line:84% denoted X2,Y2, associated with income I2. 00:10:46.680 --> 00:10:48.670 align:middle line:84% What this diagram is featuring, though, 00:10:48.670 --> 00:10:51.090 align:middle line:90% is not a single maximization. 00:10:51.090 --> 00:10:55.120 align:middle line:84% It's how the maximization point changes if you increase income. 00:10:55.120 --> 00:11:00.720 align:middle line:84% So you would move from X2,Y2 to X3,Y3, for example. 00:11:00.720 --> 00:11:04.320 align:middle line:84% And likewise, if your income were less, 00:11:04.320 --> 00:11:09.540 align:middle line:84% if this household's income were less, they would move to Y1,X1. 00:11:09.540 --> 00:11:12.990 align:middle line:90% So you could connect these dots. 00:11:12.990 --> 00:11:16.800 align:middle line:84% And as only income is varying and the price ratio 00:11:16.800 --> 00:11:19.860 align:middle line:84% is held fixed, we could call the connected dots 00:11:19.860 --> 00:11:21.990 align:middle line:90% an income expansion path. 00:11:21.990 --> 00:11:26.160 align:middle line:84% And I'll show you some examples of that momentarily. 00:11:26.160 --> 00:11:29.400 align:middle line:90% Here is a different experiment. 00:11:29.400 --> 00:11:30.720 align:middle line:90% Did we miss something? 00:11:30.720 --> 00:11:33.970 align:middle line:84% Well, in this one, you'll get lulled 00:11:33.970 --> 00:11:36.370 align:middle line:84% into thinking that, as income changes, 00:11:36.370 --> 00:11:40.000 align:middle line:84% the amount spent on both goods has to change, increase. 00:11:40.000 --> 00:11:42.860 align:middle line:84% But that's not necessarily the case. 00:11:42.860 --> 00:11:48.100 align:middle line:84% So here, again, we're going from I1 to I2 to I3 00:11:48.100 --> 00:11:56.230 align:middle line:84% and tracing out the quantities from ZY1, ZY2 to ZY3. 00:11:56.230 --> 00:12:02.350 align:middle line:84% And here, the quantity demanded of good Y is increasing, 00:12:02.350 --> 00:12:06.500 align:middle line:84% but the quantity demanded of good Z is actually decreasing. 00:12:06.500 --> 00:12:10.300 align:middle line:84% So there's nothing terribly peculiar about the shape 00:12:10.300 --> 00:12:11.760 align:middle line:90% of these indifference curves. 00:12:11.760 --> 00:12:15.040 align:middle line:84% We didn't have to struggle too hard to get this phenomenon 00:12:15.040 --> 00:12:16.270 align:middle line:90% to happen. 00:12:16.270 --> 00:12:19.330 align:middle line:84% But anyway, this phenomenon has a name. 00:12:19.330 --> 00:12:24.850 align:middle line:84% Namely, the good Y over here would be called a normal good, 00:12:24.850 --> 00:12:30.820 align:middle line:84% and good Z would be called an inferior good or a necessity 00:12:30.820 --> 00:12:32.980 align:middle line:84% because the amount that you consume of it 00:12:32.980 --> 00:12:37.190 align:middle line:90% diminishes as income goes up. 00:12:37.190 --> 00:12:39.940 align:middle line:84% So here are those definitions, a good 00:12:39.940 --> 00:12:43.390 align:middle line:84% for which the demand of good i goes down 00:12:43.390 --> 00:12:48.130 align:middle line:84% as income-- capital I-- goes up is said to be an inferior good. 00:12:48.130 --> 00:12:51.970 align:middle line:84% It doesn't have to happen over the entire range of income 00:12:51.970 --> 00:12:52.870 align:middle line:90% and prices. 00:12:52.870 --> 00:12:56.300 align:middle line:84% But if it happens locally, it's called an inferior good 00:12:56.300 --> 00:12:56.800 align:middle line:90% locally. 00:12:56.800 --> 00:12:59.920 align:middle line:84% And likewise, if the derivative with respect to income 00:12:59.920 --> 00:13:02.770 align:middle line:84% is positive, then we call, in some range, 00:13:02.770 --> 00:13:07.370 align:middle line:84% the good to be a normal good or noninferior good. 00:13:07.370 --> 00:13:10.010 align:middle line:84% So there's something called Engel's Law. 00:13:10.010 --> 00:13:13.970 align:middle line:84% And with the caveat that this is not one household, 00:13:13.970 --> 00:13:17.420 align:middle line:84% it's actually comparing across households that 00:13:17.420 --> 00:13:20.250 align:middle line:90% differ in their annual income. 00:13:20.250 --> 00:13:26.540 align:middle line:84% So here, we're going from an annual income of $225 00:13:26.540 --> 00:13:28.940 align:middle line:84% to-- this is written in 1920, by the way. 00:13:28.940 --> 00:13:30.110 align:middle line:90% Don't be too alarmed. 00:13:30.110 --> 00:13:33.120 align:middle line:90% There's been a lot of inflation. 00:13:33.120 --> 00:13:34.970 align:middle line:84% So this, in real purchasing power, 00:13:34.970 --> 00:13:38.150 align:middle line:84% was more than you thought, $300-- 00:13:38.150 --> 00:13:44.210 align:middle line:84% or this range, 450 to 600 or 750 to 1,000. 00:13:44.210 --> 00:13:46.257 align:middle line:84% And what you can see is the percent 00:13:46.257 --> 00:13:52.220 align:middle line:84% spent of the budget on food declines 00:13:52.220 --> 00:13:55.130 align:middle line:84% as we move to these income groups 00:13:55.130 --> 00:13:57.000 align:middle line:90% with higher and higher income. 00:13:57.000 --> 00:14:00.330 align:middle line:84% So in some sense, if all consumers were alike, 00:14:00.330 --> 00:14:02.630 align:middle line:84% and some had more income than others, 00:14:02.630 --> 00:14:05.810 align:middle line:84% we'd be looking at what an individual household would 00:14:05.810 --> 00:14:08.670 align:middle line:90% do as its income changed. 00:14:08.670 --> 00:14:11.750 align:middle line:84% And in this case, food would be an inferior good. 00:14:11.750 --> 00:14:13.130 align:middle line:90% You're saying, what's going up? 00:14:13.130 --> 00:14:15.350 align:middle line:90% Clothing, not by much. 00:14:15.350 --> 00:14:17.720 align:middle line:90% Lodging, not at all. 00:14:17.720 --> 00:14:20.480 align:middle line:84% What really expands as income goes up 00:14:20.480 --> 00:14:24.190 align:middle line:84% is services, education, health, legal, and other services. 00:14:24.190 --> 00:14:27.110 align:middle line:84% In fact, in the US economy, looking 00:14:27.110 --> 00:14:32.150 align:middle line:84% over decades, the fraction of services produced in GDP 00:14:32.150 --> 00:14:33.720 align:middle line:84% just keeps going up, and up, and up, 00:14:33.720 --> 00:14:36.710 align:middle line:84% which is reflective of the fact that, as households 00:14:36.710 --> 00:14:40.085 align:middle line:84% have more income, they demand more services than less food. 00:14:40.085 --> 00:14:42.760 align:middle line:90% 00:14:42.760 --> 00:14:44.740 align:middle line:90% So that's pretty dated, 1920. 00:14:44.740 --> 00:14:48.520 align:middle line:84% So I went on the web and looked this thing up, Engel's Law 00:14:48.520 --> 00:14:49.750 align:middle line:90% today. 00:14:49.750 --> 00:14:52.900 align:middle line:84% Here, we're dividing households into quintiles, 00:14:52.900 --> 00:14:57.440 align:middle line:84% five groups, an equal number of households in each group. 00:14:57.440 --> 00:15:01.600 align:middle line:84% And we plot the fraction of so-called luxuries 00:15:01.600 --> 00:15:03.800 align:middle line:90% and necessities for each group. 00:15:03.800 --> 00:15:06.190 align:middle line:84% Now, this is a bit problematic in the sense 00:15:06.190 --> 00:15:09.400 align:middle line:84% that necessities are defined as goods or services 00:15:09.400 --> 00:15:12.970 align:middle line:84% where consumption is proportionately lower 00:15:12.970 --> 00:15:15.050 align:middle line:90% as a person's income increases. 00:15:15.050 --> 00:15:18.580 align:middle line:84% So again, necessity or non-normal goods 00:15:18.580 --> 00:15:21.790 align:middle line:84% or as we defined it in the last three slides. 00:15:21.790 --> 00:15:25.480 align:middle line:84% Not too surprisingly, the highest income groups, 00:15:25.480 --> 00:15:30.400 align:middle line:84% the top 20% of the population, the proportion of expenditure 00:15:30.400 --> 00:15:33.880 align:middle line:84% on luxuries is a lot higher than on necessities. 00:15:33.880 --> 00:15:37.030 align:middle line:84% And that gap lowers until you get 00:15:37.030 --> 00:15:40.010 align:middle line:84% to the lowest group, in which case, 00:15:40.010 --> 00:15:42.520 align:middle line:84% the fraction spent on necessities 00:15:42.520 --> 00:15:45.460 align:middle line:90% is a lot higher than luxuries. 00:15:45.460 --> 00:15:47.800 align:middle line:84% Now, when I grabbed this, there was an whole article 00:15:47.800 --> 00:15:50.470 align:middle line:84% about how poor people are extravagant 00:15:50.470 --> 00:15:53.560 align:middle line:84% because they're still spending money on luxury goods. 00:15:53.560 --> 00:15:57.340 align:middle line:84% Well, I mean, narrowly speaking, yes, that's true. 00:15:57.340 --> 00:15:58.933 align:middle line:84% But again, it's a little bit harsh 00:15:58.933 --> 00:16:00.850 align:middle line:84% because it doesn't mean they're driving around 00:16:00.850 --> 00:16:01.990 align:middle line:90% in a Mercedes Benz. 00:16:01.990 --> 00:16:05.050 align:middle line:84% It just means they're eating some fraction of what 00:16:05.050 --> 00:16:08.650 align:middle line:90% they spend on are normal goods. 00:16:08.650 --> 00:16:10.840 align:middle line:90% AUDIENCE: Wait. 00:16:10.840 --> 00:16:17.960 align:middle line:84% Can you-- is there a little bit of issue with how those two-- 00:16:17.960 --> 00:16:19.840 align:middle line:84% I guess, how the two lines are defined? 00:16:19.840 --> 00:16:22.420 align:middle line:84% Because the way you defined them, 00:16:22.420 --> 00:16:25.390 align:middle line:84% it seems like you said luxuries are just what rich people buy, 00:16:25.390 --> 00:16:27.490 align:middle line:84% and necessities are what poor people buy. 00:16:27.490 --> 00:16:30.280 align:middle line:84% So doesn't this kind of get confirmed just 00:16:30.280 --> 00:16:31.300 align:middle line:90% by how you define it? 00:16:31.300 --> 00:16:33.550 align:middle line:84% ROBERT M. TOWNSEND: Yes, I think that's exactly right. 00:16:33.550 --> 00:16:35.710 align:middle line:90% It's almost tautological. 00:16:35.710 --> 00:16:37.480 align:middle line:84% I mean, the way they defined it, it 00:16:37.480 --> 00:16:41.110 align:middle line:84% has to be true that this blue line is going up and up as you 00:16:41.110 --> 00:16:42.940 align:middle line:90% go across the quantiles. 00:16:42.940 --> 00:16:44.440 align:middle line:90% So I agree with that comment. 00:16:44.440 --> 00:16:46.990 align:middle line:90% But I still think the graphs-- 00:16:46.990 --> 00:16:49.430 align:middle line:84% how much is it going up, what's the difference, 00:16:49.430 --> 00:16:52.030 align:middle line:84% and so on, could go up just a teeny little bit, 00:16:52.030 --> 00:16:55.630 align:middle line:84% in which case, that blue line would not move much. 00:16:55.630 --> 00:16:59.040 align:middle line:90% Other questions? 00:16:59.040 --> 00:17:03.330 align:middle line:84% Well, one takeaway from this is different households 00:17:03.330 --> 00:17:06.990 align:middle line:84% that differ in income do not spend the same fraction 00:17:06.990 --> 00:17:10.020 align:middle line:90% of their money on goods. 00:17:10.020 --> 00:17:13.829 align:middle line:84% So let me take a chance at the risk of-- 00:17:13.829 --> 00:17:16.319 align:middle line:84% preview of coming attractions, this 00:17:16.319 --> 00:17:19.859 align:middle line:84% is way back in lecture 18, where we're 00:17:19.859 --> 00:17:24.270 align:middle line:84% going to talk about aggregation, which has to do with whether we 00:17:24.270 --> 00:17:27.569 align:middle line:84% can represent demand functions as if they came 00:17:27.569 --> 00:17:30.900 align:middle line:84% from a single maximizing representative 00:17:30.900 --> 00:17:34.830 align:middle line:84% household or weaker, that there is a group of households, 00:17:34.830 --> 00:17:37.770 align:middle line:84% like the lower quintile, and they're all alike. 00:17:37.770 --> 00:17:40.920 align:middle line:84% And we could take a representative low-quintile 00:17:40.920 --> 00:17:43.260 align:middle line:84% household and talk about the demand 00:17:43.260 --> 00:17:46.470 align:middle line:84% as if it were coming from that one person. 00:17:46.470 --> 00:17:49.290 align:middle line:84% What is a necessary and sufficient assumption 00:17:49.290 --> 00:17:52.770 align:middle line:84% for us to be able to do that kind of aggregation? 00:17:52.770 --> 00:17:54.810 align:middle line:90% It is that these income-- 00:17:54.810 --> 00:17:57.540 align:middle line:84% in this case, denoted wealth expansion paths-- 00:17:57.540 --> 00:18:02.650 align:middle line:84% are linear and have exactly the same slope. 00:18:02.650 --> 00:18:05.100 align:middle line:90% So these guys will spend-- 00:18:05.100 --> 00:18:07.890 align:middle line:84% and I'm not trying to go through the entire lecture. 00:18:07.890 --> 00:18:10.080 align:middle line:84% I'm just really trying to motivate you 00:18:10.080 --> 00:18:12.070 align:middle line:84% that we looked at some micro data, 00:18:12.070 --> 00:18:13.980 align:middle line:84% and later, we'll make some assumptions. 00:18:13.980 --> 00:18:17.160 align:middle line:84% It's good to put the microdata about the assumptions 00:18:17.160 --> 00:18:21.090 align:middle line:84% on the table at the same time and compare them. 00:18:21.090 --> 00:18:24.840 align:middle line:84% What's going on here is that, if these income expansion 00:18:24.840 --> 00:18:28.890 align:middle line:84% paths are parallel, then when we increase income 00:18:28.890 --> 00:18:31.980 align:middle line:84% for one household, it's expenditure would go up. 00:18:31.980 --> 00:18:34.330 align:middle line:84% But because this line is parallel, 00:18:34.330 --> 00:18:37.320 align:middle line:84% when we decrease expenditure for another one, 00:18:37.320 --> 00:18:40.860 align:middle line:84% it's expenditures will go down in exactly the same amount 00:18:40.860 --> 00:18:42.310 align:middle line:90% that this guy's went up. 00:18:42.310 --> 00:18:45.060 align:middle line:84% So those income effects are completely offsetting. 00:18:45.060 --> 00:18:48.990 align:middle line:84% And that turns out to be necessary and sufficient to add 00:18:48.990 --> 00:18:50.970 align:middle line:84% up all of income and drive demand 00:18:50.970 --> 00:18:54.450 align:middle line:84% as if we've given all of income to one household. 00:18:54.450 --> 00:18:54.950 align:middle line:90% All right. 00:18:54.950 --> 00:18:56.870 align:middle line:84% Well, it was with some trepidation 00:18:56.870 --> 00:18:59.030 align:middle line:84% that I showed you this picture, but I 00:18:59.030 --> 00:19:01.550 align:middle line:84% thought you might like to know what's coming. 00:19:01.550 --> 00:19:04.390 align:middle line:84% And we'll review all of that when we get to 18. 00:19:04.390 --> 00:19:08.540 align:middle line:84% OK, so let's talk about income and substitution effects now. 00:19:08.540 --> 00:19:13.250 align:middle line:84% Instead of changing income, we want to change a price. 00:19:13.250 --> 00:19:16.460 align:middle line:84% We're going to focus on good X. We're going to look 00:19:16.460 --> 00:19:22.270 align:middle line:84% at an initial point, which is Y*,X*. 00:19:22.270 --> 00:19:26.320 align:middle line:84% And then we're going to lower the price of good X 00:19:26.320 --> 00:19:28.660 align:middle line:90% and not touch good Y. 00:19:28.660 --> 00:19:31.990 align:middle line:84% One way to quickly convince yourself 00:19:31.990 --> 00:19:35.680 align:middle line:84% that the diagram is correct is that this point over here 00:19:35.680 --> 00:19:38.410 align:middle line:84% on the Y-axis is taking all your income 00:19:38.410 --> 00:19:41.770 align:middle line:84% and devoting it to the purchase of good Y. The price of Y 00:19:41.770 --> 00:19:44.350 align:middle line:84% hasn't changed, so this point hasn't changed. 00:19:44.350 --> 00:19:47.530 align:middle line:84% But the price of X is going down. 00:19:47.530 --> 00:19:52.480 align:middle line:84% And logically, the set of things you can buy is expanding. 00:19:52.480 --> 00:19:55.180 align:middle line:84% So this budget line shifts to the right, 00:19:55.180 --> 00:19:57.530 align:middle line:90% but not in a parallel fashion. 00:19:57.530 --> 00:20:00.070 align:middle line:90% So what is the new purchase? 00:20:00.070 --> 00:20:04.510 align:middle line:84% The new purchase is a tangency, Y** and X**. 00:20:04.510 --> 00:20:08.350 align:middle line:84% And we're going to break that movement from the initial point 00:20:08.350 --> 00:20:11.890 align:middle line:84% to the subsequent point into two pieces-- 00:20:11.890 --> 00:20:16.760 align:middle line:84% one, a price effect, and two, an income effect. 00:20:16.760 --> 00:20:19.840 align:middle line:84% So since the price of good X goes down, 00:20:19.840 --> 00:20:24.880 align:middle line:84% PX over PY in absolute value is going down. 00:20:24.880 --> 00:20:28.450 align:middle line:84% So there's a new budget line with a lower 00:20:28.450 --> 00:20:35.960 align:middle line:84% slope that is pivoting on the initial indifference curve, U1. 00:20:35.960 --> 00:20:41.240 align:middle line:84% So setting aside income effects, which we'll get to momentarily, 00:20:41.240 --> 00:20:45.320 align:middle line:84% the natural impact of lowering the price of good X 00:20:45.320 --> 00:20:48.080 align:middle line:84% is to increase the expenditure of good X 00:20:48.080 --> 00:20:50.900 align:middle line:84% and decrease the expenditure of good Y. 00:20:50.900 --> 00:20:53.210 align:middle line:84% So that's called the substitution effect. 00:20:53.210 --> 00:20:59.010 align:middle line:84% And it's a movement from X*,Y* to B. But in fact, of course, 00:20:59.010 --> 00:20:59.780 align:middle line:90% income-- 00:20:59.780 --> 00:21:02.390 align:middle line:84% effectively, purchasing power has gone up. 00:21:02.390 --> 00:21:05.990 align:middle line:84% Even though nominal income has not moved, one good, at least, 00:21:05.990 --> 00:21:07.040 align:middle line:90% is cheaper. 00:21:07.040 --> 00:21:09.840 align:middle line:84% So their effect of income has gone up, 00:21:09.840 --> 00:21:13.790 align:middle line:84% and that's represented by this parallel shift from B 00:21:13.790 --> 00:21:16.880 align:middle line:90% to X**,Y**. 00:21:16.880 --> 00:21:19.400 align:middle line:84% And that last bit from B to there 00:21:19.400 --> 00:21:21.420 align:middle line:90% is called the income effect. 00:21:21.420 --> 00:21:27.650 align:middle line:84% So the global movement is from X* to X** with a corresponding 00:21:27.650 --> 00:21:29.930 align:middle line:90% movement from Y* to Y**. 00:21:29.930 --> 00:21:34.340 align:middle line:84% And it's broken into two pieces, this substitution effect, which 00:21:34.340 --> 00:21:37.670 align:middle line:84% corresponds with movement along the original indifference 00:21:37.670 --> 00:21:40.520 align:middle line:84% curve, and the income effect of having higher 00:21:40.520 --> 00:21:43.530 align:middle line:90% effective purchasing power. 00:21:43.530 --> 00:21:45.050 align:middle line:90% OK? 00:21:45.050 --> 00:21:47.180 align:middle line:84% A lot of today's lecture is just going 00:21:47.180 --> 00:21:49.940 align:middle line:90% over this in various ways. 00:21:49.940 --> 00:21:54.020 align:middle line:84% You can get mesmerized with the price decrease. 00:21:54.020 --> 00:21:57.170 align:middle line:90% Here is a price increase. 00:21:57.170 --> 00:22:01.020 align:middle line:84% They're starting out with a tangency. 00:22:01.020 --> 00:22:06.940 align:middle line:84% The price of good X still goes, in this case, up. 00:22:06.940 --> 00:22:09.520 align:middle line:84% Their purchasing power goes down. 00:22:09.520 --> 00:22:13.270 align:middle line:84% The budget line rotates from this initial point, inward. 00:22:13.270 --> 00:22:17.020 align:middle line:90% 00:22:17.020 --> 00:22:22.150 align:middle line:84% So the total demand of the price increase is going down, 00:22:22.150 --> 00:22:23.740 align:middle line:90% a reduction in X. 00:22:23.740 --> 00:22:27.290 align:middle line:84% But again, we can break that into two pieces. 00:22:27.290 --> 00:22:31.120 align:middle line:84% One is a substitution effect, the movement away from good X-- 00:22:31.120 --> 00:22:34.180 align:middle line:84% because it has a higher price relative to good Y-- 00:22:34.180 --> 00:22:38.200 align:middle line:84% to point B. And now we take into account 00:22:38.200 --> 00:22:41.260 align:middle line:84% the loss of effective purchasing power, which 00:22:41.260 --> 00:22:45.820 align:middle line:84% is a shift in this budget line, the dotted tangency line, which 00:22:45.820 --> 00:22:51.910 align:middle line:84% is tangent at B, to the actual new, lower budget line. 00:22:51.910 --> 00:22:55.800 align:middle line:84% So here, we denote the combination of the substitution 00:22:55.800 --> 00:22:57.310 align:middle line:90% and income effect. 00:22:57.310 --> 00:23:00.720 align:middle line:84% This is really the previous diagram in reverse. 00:23:00.720 --> 00:23:04.500 align:middle line:84% But I think it's useful to show both. 00:23:04.500 --> 00:23:08.460 align:middle line:84% Of course, we can also start changing the price of Y. Let's 00:23:08.460 --> 00:23:09.730 align:middle line:90% not do that right away. 00:23:09.730 --> 00:23:11.050 align:middle line:90% I'll come back to that later. 00:23:11.050 --> 00:23:15.450 align:middle line:84% But the principles remain the same. 00:23:15.450 --> 00:23:19.980 align:middle line:84% So let's construct some demand curves, 00:23:19.980 --> 00:23:23.040 align:middle line:84% as in the equation system we had initially. 00:23:23.040 --> 00:23:27.930 align:middle line:84% If we keep lowering the price, moving in this direction, 00:23:27.930 --> 00:23:31.730 align:middle line:90% we trace out this tangency. 00:23:31.730 --> 00:23:35.520 align:middle line:84% And those will be the demand at the associated prices. 00:23:35.520 --> 00:23:41.290 align:middle line:84% So here, we have, say, X prime at price, 00:23:41.290 --> 00:23:48.110 align:middle line:84% P prime, X double prime at price P double prime-- 00:23:48.110 --> 00:23:51.230 align:middle line:84% and this would have been even better 00:23:51.230 --> 00:23:55.400 align:middle line:84% if these things were lined up as they were intended to be-- 00:23:55.400 --> 00:24:02.200 align:middle line:84% X triple prime at price P triple prime. 00:24:02.200 --> 00:24:09.020 align:middle line:84% Now, again, price is going down, and demand is going up. 00:24:09.020 --> 00:24:12.980 align:middle line:90% 00:24:12.980 --> 00:24:15.890 align:middle line:84% Now, just to alert you, I don't know 00:24:15.890 --> 00:24:19.250 align:middle line:84% where this got started, with Marshall or whatever. 00:24:19.250 --> 00:24:23.510 align:middle line:84% But usually, we plot Y is a function of X, where 00:24:23.510 --> 00:24:26.720 align:middle line:84% X is on the Y-axis, and Y, the dependent variable, 00:24:26.720 --> 00:24:28.730 align:middle line:90% is on the Y-axis. 00:24:28.730 --> 00:24:30.740 align:middle line:84% This is actually plotted in reverse 00:24:30.740 --> 00:24:34.400 align:middle line:84% because we're varying price as an exogenous variable 00:24:34.400 --> 00:24:37.010 align:middle line:90% and plotting the demand. 00:24:37.010 --> 00:24:38.310 align:middle line:90% No big deal. 00:24:38.310 --> 00:24:41.000 align:middle line:84% I don't think I can plot it the other way. 00:24:41.000 --> 00:24:41.960 align:middle line:90% Actually, I could. 00:24:41.960 --> 00:24:46.200 align:middle line:84% But you'll see why this can get confusing. 00:24:46.200 --> 00:24:49.750 align:middle line:84% I just thought I'd point it out along the way. 00:24:49.750 --> 00:24:50.950 align:middle line:90% OK. 00:24:50.950 --> 00:24:53.800 align:middle line:90% Now we come to Giffen's Paradox. 00:24:53.800 --> 00:24:57.560 align:middle line:84% If the income effect of a price change is strong enough, 00:24:57.560 --> 00:25:00.220 align:middle line:84% the change in the price and the resulting change 00:25:00.220 --> 00:25:02.500 align:middle line:84% in the quantity demanded could actually 00:25:02.500 --> 00:25:05.200 align:middle line:90% move in the same direction. 00:25:05.200 --> 00:25:06.750 align:middle line:84% So let me give you an example that's 00:25:06.750 --> 00:25:10.080 align:middle line:84% attributed to Robert Giffen, who claimed 00:25:10.080 --> 00:25:13.870 align:middle line:84% to observe this in Ireland, where they had a famine. 00:25:13.870 --> 00:25:17.220 align:middle line:84% So the potato crop was failing, and the price of potatoes 00:25:17.220 --> 00:25:18.540 align:middle line:90% was rising. 00:25:18.540 --> 00:25:22.480 align:middle line:84% And despite the rise in price, people consumed more. 00:25:22.480 --> 00:25:24.540 align:middle line:84% So the demand curve was upward-sloping, not 00:25:24.540 --> 00:25:26.130 align:middle line:90% downward-sloping. 00:25:26.130 --> 00:25:29.820 align:middle line:84% His explanation was, potatoes are a big part of the budget. 00:25:29.820 --> 00:25:33.930 align:middle line:84% When their price goes up, the effect of income goes down. 00:25:33.930 --> 00:25:36.300 align:middle line:84% And that income effect dominates. 00:25:36.300 --> 00:25:40.470 align:middle line:84% The lower income effect is what lowers the demand for the good. 00:25:40.470 --> 00:25:44.400 align:middle line:84% But in effect, prices go up, and demand goes-- 00:25:44.400 --> 00:25:46.930 align:middle line:90% well, income's going down. 00:25:46.930 --> 00:25:49.390 align:middle line:90% But this good is a necessity. 00:25:49.390 --> 00:25:52.600 align:middle line:84% So you get a negative sign on the income effect, 00:25:52.600 --> 00:25:54.330 align:middle line:90% so the demand is going up. 00:25:54.330 --> 00:25:58.180 align:middle line:84% So price and income are going up at the same time. 00:25:58.180 --> 00:25:59.580 align:middle line:90% It's a bit of a mouthful. 00:25:59.580 --> 00:26:00.780 align:middle line:90% No pun intended. 00:26:00.780 --> 00:26:03.100 align:middle line:84% Actually, this could not have happened. 00:26:03.100 --> 00:26:05.460 align:middle line:84% It would require that potatoes be 00:26:05.460 --> 00:26:08.760 align:middle line:84% an important part of the budget, that potatoes be a necessity. 00:26:08.760 --> 00:26:11.790 align:middle line:84% But what doesn't fit the facts is, how are they consuming more 00:26:11.790 --> 00:26:14.970 align:middle line:84% potatoes when, in fact, there was a blight, 00:26:14.970 --> 00:26:18.580 align:middle line:84% and there were no more potatoes to be had? 00:26:18.580 --> 00:26:21.630 align:middle line:84% So something doesn't add up, literally. 00:26:21.630 --> 00:26:25.200 align:middle line:84% Anyway, all intermediate micro classes 00:26:25.200 --> 00:26:29.940 align:middle line:84% typically feature Giffen goods and then move on. 00:26:29.940 --> 00:26:32.070 align:middle line:84% And you may be left wondering if it ever 00:26:32.070 --> 00:26:33.750 align:middle line:90% really happens in practice. 00:26:33.750 --> 00:26:39.270 align:middle line:84% So searching around, I discovered Jensen and Miller, 00:26:39.270 --> 00:26:43.380 align:middle line:90% who actually ran an experiment. 00:26:43.380 --> 00:26:47.550 align:middle line:84% So this is literally a randomized controlled trial. 00:26:47.550 --> 00:26:50.700 align:middle line:84% They went to China, two different provinces, 00:26:50.700 --> 00:26:53.430 align:middle line:84% looked at relatively poor households, 00:26:53.430 --> 00:26:57.870 align:middle line:84% and changed the price of the staple food item that 00:26:57.870 --> 00:27:00.480 align:middle line:90% was a big part of their budget. 00:27:00.480 --> 00:27:03.030 align:middle line:84% Now, in order to make the comparison, 00:27:03.030 --> 00:27:04.690 align:middle line:90% you've got to change the price. 00:27:04.690 --> 00:27:07.120 align:middle line:84% So it was a randomized controlled trial, 00:27:07.120 --> 00:27:10.710 align:middle line:84% meaning, randomly, some households were exposed 00:27:10.710 --> 00:27:13.350 align:middle line:84% to a lower price because they were given vouchers 00:27:13.350 --> 00:27:15.450 align:middle line:84% so that when they went to the store, 00:27:15.450 --> 00:27:18.660 align:middle line:84% the sum of U unpaid plus vouchers 00:27:18.660 --> 00:27:20.715 align:middle line:90% equaled the stated price. 00:27:20.715 --> 00:27:22.590 align:middle line:84% But they didn't have to pay for the vouchers. 00:27:22.590 --> 00:27:24.870 align:middle line:84% So effectively, from their point of view, 00:27:24.870 --> 00:27:27.990 align:middle line:84% the price was lowered, given the vouchers. 00:27:27.990 --> 00:27:31.260 align:middle line:84% And the control group did not receive vouchers. 00:27:31.260 --> 00:27:34.800 align:middle line:84% So the control group faced the normal market price. 00:27:34.800 --> 00:27:37.080 align:middle line:84% And what Jensen and Miller are doing 00:27:37.080 --> 00:27:40.680 align:middle line:84% is comparing the control group with the treatment group. 00:27:40.680 --> 00:27:44.430 align:middle line:84% And what they discovered was, in Hunan, in the South, 00:27:44.430 --> 00:27:46.590 align:middle line:90% rice was the staple good. 00:27:46.590 --> 00:27:49.080 align:middle line:90% In Guangzhou, it's wheat. 00:27:49.080 --> 00:27:52.660 align:middle line:84% They did surveys, like Engel, before, 00:27:52.660 --> 00:27:56.010 align:middle line:84% during, and after the subsidy on both treatment and control 00:27:56.010 --> 00:27:56.740 align:middle line:90% groups. 00:27:56.740 --> 00:28:00.660 align:middle line:84% And they found strong evidence that poor households in Hunan 00:28:00.660 --> 00:28:03.990 align:middle line:84% experienced this Giffen behavior with respect to rice. 00:28:03.990 --> 00:28:07.980 align:middle line:84% That is to say, when the price of rice went down, 00:28:07.980 --> 00:28:11.460 align:middle line:84% households reduced their demand for rice. 00:28:11.460 --> 00:28:15.270 align:middle line:84% The evidence was mixed in the other province. 00:28:15.270 --> 00:28:20.190 align:middle line:84% And I think I will choose not to say very much about this. 00:28:20.190 --> 00:28:23.700 align:middle line:84% There were groups, the very poorest and the least poor 00:28:23.700 --> 00:28:27.300 align:middle line:84% that responded in what you might think to be the ordinary way, 00:28:27.300 --> 00:28:28.200 align:middle line:90% with price-- 00:28:28.200 --> 00:28:29.760 align:middle line:90% with demand down-sloping. 00:28:29.760 --> 00:28:33.600 align:middle line:84% But there was a group in the middle between the very poorest 00:28:33.600 --> 00:28:37.470 align:middle line:84% and the least poor which exhibited Giffen behavior. 00:28:37.470 --> 00:28:40.950 align:middle line:84% As it's written on the slide, this is a bit of a mystery. 00:28:40.950 --> 00:28:42.870 align:middle line:84% Why is it that we're getting this difference 00:28:42.870 --> 00:28:46.830 align:middle line:84% across the three groups, or two groups plus one? 00:28:46.830 --> 00:28:50.640 align:middle line:84% And I leave that for you because, again, the problem set 00:28:50.640 --> 00:28:52.500 align:middle line:90% features the entire article. 00:28:52.500 --> 00:28:56.520 align:middle line:84% We try to make it easy by highlighting the text. 00:28:56.520 --> 00:28:59.280 align:middle line:84% But somewhere in that paper is a discussion 00:28:59.280 --> 00:29:03.360 align:middle line:84% of why this result, which is a bit hard to interpret here, 00:29:03.360 --> 00:29:04.770 align:middle line:90% does happen. 00:29:04.770 --> 00:29:07.450 align:middle line:84% The authors did give it an interpretation. 00:29:07.450 --> 00:29:11.280 align:middle line:84% So that's advertising to go and grab that paper, which 00:29:11.280 --> 00:29:12.510 align:middle line:90% I think I already have. 00:29:12.510 --> 00:29:14.010 align:middle line:90% Yes, question? 00:29:14.010 --> 00:29:18.390 align:middle line:84% AUDIENCE: Yeah, so is the demand the amount of potatoes 00:29:18.390 --> 00:29:20.790 align:middle line:84% or rice you're consuming, or the proportion 00:29:20.790 --> 00:29:24.307 align:middle line:84% that is of your income that's increasing? 00:29:24.307 --> 00:29:25.890 align:middle line:84% ROBERT M. TOWNSEND: It's the quantity. 00:29:25.890 --> 00:29:28.050 align:middle line:84% But obviously, they have the prices. 00:29:28.050 --> 00:29:31.680 align:middle line:90% So you can look at both. 00:29:31.680 --> 00:29:37.995 align:middle line:84% But the definition we've been using of a Giffen good, it's-- 00:29:37.995 --> 00:29:41.780 align:middle line:90% 00:29:41.780 --> 00:29:44.500 align:middle line:90% well, actually, it never showed. 00:29:44.500 --> 00:29:48.310 align:middle line:84% But it is the quantity-- in this case, the quantity of potatoes, 00:29:48.310 --> 00:29:49.945 align:middle line:90% not expenditure on potatoes. 00:29:49.945 --> 00:29:56.660 align:middle line:90% 00:29:56.660 --> 00:29:59.240 align:middle line:90% All right. 00:29:59.240 --> 00:30:02.750 align:middle line:84% OK, so now I want to get to something, effectively, 00:30:02.750 --> 00:30:06.650 align:middle line:84% we've already been doing, which is compensated 00:30:06.650 --> 00:30:10.390 align:middle line:90% versus uncompensated demands. 00:30:10.390 --> 00:30:13.650 align:middle line:84% So the uncompensated demand is the total demand, 00:30:13.650 --> 00:30:18.160 align:middle line:84% sometimes called the Marshallian demand from Marshall, 00:30:18.160 --> 00:30:20.610 align:middle line:84% who was an early English economist. 00:30:20.610 --> 00:30:23.130 align:middle line:84% It shows the relationship between price of the good 00:30:23.130 --> 00:30:26.340 align:middle line:84% and the quantity purchased, assuming that all other prices 00:30:26.340 --> 00:30:28.230 align:middle line:90% and income are held constant. 00:30:28.230 --> 00:30:31.200 align:middle line:84% The compensated demand, sometimes called the Hicksian 00:30:31.200 --> 00:30:34.860 align:middle line:84% demand, shows the relationship between price and quantity. 00:30:34.860 --> 00:30:38.070 align:middle line:84% It continues to assume that all the other prices are 00:30:38.070 --> 00:30:38.730 align:middle line:90% held fixed. 00:30:38.730 --> 00:30:42.580 align:middle line:84% But instead of holding income fixed, utility is held fixed. 00:30:42.580 --> 00:30:44.820 align:middle line:84% So to go back to the earlier slides, 00:30:44.820 --> 00:30:47.790 align:middle line:84% we were featuring the substitution effect 00:30:47.790 --> 00:30:50.970 align:middle line:84% as the movement along an indifference curve-- 00:30:50.970 --> 00:30:53.310 align:middle line:90% i.e., holding utility fixed. 00:30:53.310 --> 00:30:56.340 align:middle line:84% So when we looked at how much demand would change 00:30:56.340 --> 00:30:59.520 align:middle line:84% one way or the other as a function of the price change, 00:30:59.520 --> 00:31:03.930 align:middle line:84% and we went along the initial baseline indifference curve, 00:31:03.930 --> 00:31:07.880 align:middle line:90% that was the compensated demand. 00:31:07.880 --> 00:31:12.830 align:middle line:84% So more dramatically, if we fix the initial point 00:31:12.830 --> 00:31:17.750 align:middle line:84% and change the price, we would be, for a lower price, 00:31:17.750 --> 00:31:21.590 align:middle line:84% pivoting along this indifference curve, U2. 00:31:21.590 --> 00:31:26.570 align:middle line:84% By construction, we keep utility constant at U2. 00:31:26.570 --> 00:31:30.080 align:middle line:84% And then plot the demands, which are going up, 00:31:30.080 --> 00:31:32.990 align:middle line:84% and we put that on the price quantity diagram, 00:31:32.990 --> 00:31:35.310 align:middle line:84% and the demand curve is sloping down. 00:31:35.310 --> 00:31:38.630 align:middle line:84% But this is the Hicksian demand at little h 00:31:38.630 --> 00:31:42.140 align:middle line:90% to remind us that it's Hicks. 00:31:42.140 --> 00:31:47.710 align:middle line:84% So putting both types of demand on the same diagram, 00:31:47.710 --> 00:31:52.300 align:middle line:84% we'd have the normal demand, or the Marshallian demand, 00:31:52.300 --> 00:31:54.860 align:middle line:90% and h for the Hicksian demand. 00:31:54.860 --> 00:31:58.720 align:middle line:84% So as price goes down, demand goes up, 00:31:58.720 --> 00:32:01.730 align:middle line:84% but the demand goes up from two different effects. 00:32:01.730 --> 00:32:05.530 align:middle line:84% One is the substitution effect holding utility constant, 00:32:05.530 --> 00:32:09.470 align:middle line:84% and that's a point on the Hicksian demand. 00:32:09.470 --> 00:32:13.480 align:middle line:84% And then the rest of it is this movement to the Marshallian 00:32:13.480 --> 00:32:16.810 align:middle line:84% demand, or ordinary demand, which is the income effect that 00:32:16.810 --> 00:32:19.830 align:middle line:90% comes from the lower price. 00:32:19.830 --> 00:32:22.650 align:middle line:84% And obviously, we can go in the other direction. 00:32:22.650 --> 00:32:28.500 align:middle line:84% Starting here as a baseline, if we increase the price, 00:32:28.500 --> 00:32:30.930 align:middle line:84% we would pivot along the indifference curve 00:32:30.930 --> 00:32:34.210 align:middle line:84% and get a lower demand due to the substitution effect. 00:32:34.210 --> 00:32:36.330 align:middle line:84% But effective income has gone down, 00:32:36.330 --> 00:32:39.480 align:middle line:84% so this negative income effect kicks in, 00:32:39.480 --> 00:32:41.710 align:middle line:90% and demand is even less. 00:32:41.710 --> 00:32:45.930 align:middle line:84% So kind of intuitively, the Hicksian demand 00:32:45.930 --> 00:32:49.950 align:middle line:84% is steeper than the ordinary demand curve 00:32:49.950 --> 00:32:53.850 align:middle line:84% because it has included those income effects. 00:32:53.850 --> 00:32:57.570 align:middle line:84% Or putting in another way, the ordinary demand curve 00:32:57.570 --> 00:33:00.990 align:middle line:84% is, quote, flatter with a lower slope 00:33:00.990 --> 00:33:05.970 align:middle line:84% because there's a more dramatic larger change in the quantity 00:33:05.970 --> 00:33:07.740 align:middle line:90% as we change the price. 00:33:07.740 --> 00:33:09.960 align:middle line:84% Either the income effect is positive, 00:33:09.960 --> 00:33:12.330 align:middle line:84% and demand goes up all the more, or the income 00:33:12.330 --> 00:33:17.050 align:middle line:84% effect is negative, and the demand decreases all the more. 00:33:17.050 --> 00:33:19.450 align:middle line:84% So we could say-- be careful about this-- 00:33:19.450 --> 00:33:21.850 align:middle line:84% that, in some sense, the ordinary demand 00:33:21.850 --> 00:33:23.470 align:middle line:90% curve is more elastic. 00:33:23.470 --> 00:33:28.140 align:middle line:84% At least here, it has a higher derivative. 00:33:28.140 --> 00:33:32.120 align:middle line:84% So there's a mathematical expression 00:33:32.120 --> 00:33:35.060 align:middle line:84% for the difference between the Hicksian demand 00:33:35.060 --> 00:33:37.250 align:middle line:90% and the ordinary demand. 00:33:37.250 --> 00:33:41.000 align:middle line:84% Namely, the derivative of demand of good X 00:33:41.000 --> 00:33:44.510 align:middle line:84% as we change the price of X, the slope of the ordinary demand 00:33:44.510 --> 00:33:49.100 align:middle line:84% curve, is first term, the substitution effect 00:33:49.100 --> 00:33:53.120 align:middle line:84% of the price change, PX, but then this income effect 00:33:53.120 --> 00:33:54.290 align:middle line:90% kicks in. 00:33:54.290 --> 00:33:57.470 align:middle line:84% And there's two pieces to the income effect. 00:33:57.470 --> 00:34:01.970 align:middle line:84% One has to do with how high is the marginal change in X 00:34:01.970 --> 00:34:03.750 align:middle line:90% as you change income. 00:34:03.750 --> 00:34:07.610 align:middle line:84% The other is, how much X you were eating to begin with. 00:34:07.610 --> 00:34:10.969 align:middle line:84% Think of potatoes, and so on, or those budget 00:34:10.969 --> 00:34:13.699 align:middle line:84% shares, either Engel or-- but this is X. 00:34:13.699 --> 00:34:15.030 align:middle line:90% This is the quantity. 00:34:15.030 --> 00:34:18.770 align:middle line:84% So it looks, from this, since we're subtracting-- say, 00:34:18.770 --> 00:34:22.080 align:middle line:84% the income effect is positive, and X is positive. 00:34:22.080 --> 00:34:24.420 align:middle line:84% Then we're subtracting a positive number. 00:34:24.420 --> 00:34:28.440 align:middle line:84% So it looks like this minus this must be lower than this. 00:34:28.440 --> 00:34:30.290 align:middle line:84% So it looks like the ordinary demand 00:34:30.290 --> 00:34:34.050 align:middle line:84% should have a lower slope if this equation is accurate. 00:34:34.050 --> 00:34:36.920 align:middle line:84% But in fact, as I just said, it's the other way around. 00:34:36.920 --> 00:34:38.360 align:middle line:90% Puzzle, puzzle, puzzle. 00:34:38.360 --> 00:34:39.560 align:middle line:90% What's going on? 00:34:39.560 --> 00:34:42.920 align:middle line:84% Well, it's this stupid way that we traditionally 00:34:42.920 --> 00:34:46.350 align:middle line:84% plot the dependent variable on the X-axis. 00:34:46.350 --> 00:34:50.989 align:middle line:84% So this is dX/dP, the response of the dependent variable 00:34:50.989 --> 00:34:52.880 align:middle line:84% to the independent variable, which 00:34:52.880 --> 00:34:55.610 align:middle line:90% is 1 over the slope of this. 00:34:55.610 --> 00:34:58.790 align:middle line:84% So this diagram is consistent with this equation. 00:34:58.790 --> 00:35:02.510 align:middle line:84% It just takes a little bit of juggling to realize why. 00:35:02.510 --> 00:35:05.540 align:middle line:84% Anyway, this is called Slutsky's equation. 00:35:05.540 --> 00:35:08.690 align:middle line:84% And again, at the risk of trying to do too much too 00:35:08.690 --> 00:35:11.000 align:middle line:84% fast, but on the other hand, providing 00:35:11.000 --> 00:35:13.280 align:middle line:84% some motivation for where we're going, 00:35:13.280 --> 00:35:16.670 align:middle line:84% eventually, we'll get to lecture 19 and talk about 00:35:16.670 --> 00:35:21.110 align:middle line:84% whether there are retractable restrictions on data, as 00:35:21.110 --> 00:35:22.460 align:middle line:90% in the science lecture. 00:35:22.460 --> 00:35:23.570 align:middle line:90% We have a model. 00:35:23.570 --> 00:35:25.430 align:middle line:84% Does the model fit the data or not? 00:35:25.430 --> 00:35:27.920 align:middle line:90% Can you even reject the model? 00:35:27.920 --> 00:35:29.720 align:middle line:90% Is it providing some guidance? 00:35:29.720 --> 00:35:33.390 align:middle line:84% Or you can always tweak things, so the model is always true. 00:35:33.390 --> 00:35:36.680 align:middle line:84% So this lecture 19 is about restrictions 00:35:36.680 --> 00:35:38.270 align:middle line:90% that can be tested. 00:35:38.270 --> 00:35:42.260 align:middle line:84% And there's something called Slutsky, same guy, 00:35:42.260 --> 00:35:45.470 align:middle line:84% Slutsky's Decomposition this equation is 00:35:45.470 --> 00:35:48.170 align:middle line:84% a generalization of the one I was just showing you, 00:35:48.170 --> 00:35:51.200 align:middle line:84% with the exception that we put this substitution 00:35:51.200 --> 00:35:54.320 align:middle line:84% effect on the left and the regular derivative 00:35:54.320 --> 00:35:55.170 align:middle line:90% on the right. 00:35:55.170 --> 00:35:57.170 align:middle line:84% So it looks like the signs are changing. 00:35:57.170 --> 00:35:58.950 align:middle line:90% This becomes a plus. 00:35:58.950 --> 00:36:02.390 align:middle line:84% This is a generalization in a number of respects. 00:36:02.390 --> 00:36:05.450 align:middle line:84% We're looking at the demand for good L 00:36:05.450 --> 00:36:08.990 align:middle line:84% as we change the price of another good, good K, not 00:36:08.990 --> 00:36:12.440 align:middle line:84% the own effect of one good with respect to its own price, 00:36:12.440 --> 00:36:14.690 align:middle line:90% but a cross-partial derivative. 00:36:14.690 --> 00:36:17.180 align:middle line:84% Although we still have this income effect over here, 00:36:17.180 --> 00:36:20.330 align:middle line:84% and we're still weighting it by the initial quantity purchased. 00:36:20.330 --> 00:36:23.600 align:middle line:84% So it turns out that what we want to do 00:36:23.600 --> 00:36:27.500 align:middle line:90% is ask whether observe demands-- 00:36:27.500 --> 00:36:30.650 align:middle line:90% this guy, these guys, really-- 00:36:30.650 --> 00:36:33.720 align:middle line:84% how demand is changing with prices and income, 00:36:33.720 --> 00:36:37.010 align:middle line:84% whether there are patterns which could be rejected as being 00:36:37.010 --> 00:36:38.760 align:middle line:90% inconsistent with a model. 00:36:38.760 --> 00:36:40.910 align:middle line:84% And the problem is that the restrictions 00:36:40.910 --> 00:36:44.060 align:middle line:84% are on these Hicksian demands, and we don't see those. 00:36:44.060 --> 00:36:45.440 align:middle line:90% We see the regular demands. 00:36:45.440 --> 00:36:47.720 align:middle line:84% We don't see the Hicksian demands. 00:36:47.720 --> 00:36:50.240 align:middle line:84% And I'll say more about this in a minute. 00:36:50.240 --> 00:36:52.460 align:middle line:84% It turns out that, with some math, 00:36:52.460 --> 00:36:54.320 align:middle line:84% you can actually go back and forth 00:36:54.320 --> 00:36:57.050 align:middle line:84% between the regular demand and the Hicksian in demand, 00:36:57.050 --> 00:36:58.670 align:middle line:90% and there are restrictions. 00:36:58.670 --> 00:37:01.190 align:middle line:84% So anyway, I'm running an experiment 00:37:01.190 --> 00:37:03.710 align:middle line:84% now because I'm showing you things 00:37:03.710 --> 00:37:06.620 align:middle line:84% that are more toward the end of the class 00:37:06.620 --> 00:37:08.690 align:middle line:84% set of lectures than at the beginning. 00:37:08.690 --> 00:37:12.410 align:middle line:84% But since we're introducing Slutsky's equation today, 00:37:12.410 --> 00:37:14.840 align:middle line:84% I thought I'd show you that we will come back to it 00:37:14.840 --> 00:37:17.600 align:middle line:84% and use it in a subsequent lecture. 00:37:17.600 --> 00:37:20.120 align:middle line:90% This is a lot simpler notation. 00:37:20.120 --> 00:37:23.720 align:middle line:84% And don't worry about either the income expansion paths 00:37:23.720 --> 00:37:26.360 align:middle line:84% or this generalization of Slutsky. 00:37:26.360 --> 00:37:29.330 align:middle line:84% That was meant to be a preview of coming attractions 00:37:29.330 --> 00:37:31.820 align:middle line:84% and motivation, not something you should be worried 00:37:31.820 --> 00:37:34.470 align:middle line:90% about learning right away. 00:37:34.470 --> 00:37:38.180 align:middle line:84% So now I want to introduce something called duality. 00:37:38.180 --> 00:37:41.120 align:middle line:84% This looks like an income expansion path. 00:37:41.120 --> 00:37:44.960 align:middle line:84% As we increase income, we get more and more points. 00:37:44.960 --> 00:37:47.660 align:middle line:84% I think this is identical with the figure I showed you 00:37:47.660 --> 00:37:49.300 align:middle line:90% earlier. 00:37:49.300 --> 00:37:51.760 align:middle line:84% But I want to use it for a different purpose. 00:37:51.760 --> 00:37:55.110 align:middle line:84% I want to say, suppose we're at this point 00:37:55.110 --> 00:37:58.470 align:middle line:84% with this intermediate income, I2. 00:37:58.470 --> 00:38:02.190 align:middle line:84% And we could be maximizing utility subject to that income 00:38:02.190 --> 00:38:04.620 align:middle line:90% level holding prices fixed. 00:38:04.620 --> 00:38:09.720 align:middle line:84% And X2,Y2 would be the maximum point of utility. 00:38:09.720 --> 00:38:12.310 align:middle line:84% But suppose we did a different experiment. 00:38:12.310 --> 00:38:15.630 align:middle line:84% Suppose we wanted to achieve this utility 00:38:15.630 --> 00:38:19.600 align:middle line:84% level, U2, by spending as little as possible. 00:38:19.600 --> 00:38:21.270 align:middle line:84% The amount spent would be captured 00:38:21.270 --> 00:38:23.850 align:middle line:84% by these various possible budget lines. 00:38:23.850 --> 00:38:27.480 align:middle line:84% So something called a minimizing expenditure problem, 00:38:27.480 --> 00:38:29.880 align:middle line:84% minimal expenditure problem, would 00:38:29.880 --> 00:38:32.010 align:middle line:84% be to find the budget line, which 00:38:32.010 --> 00:38:36.750 align:middle line:84% is lowest, closest to 0,0, and still allows the household 00:38:36.750 --> 00:38:39.030 align:middle line:90% to get the utility level U2. 00:38:39.030 --> 00:38:41.940 align:middle line:90% So at this point, it's clear. 00:38:41.940 --> 00:38:44.880 align:middle line:84% You could think of this as maximizing utility subject 00:38:44.880 --> 00:38:48.300 align:middle line:84% to a fixed budget or minimizing the expenses to get 00:38:48.300 --> 00:38:50.340 align:middle line:90% to a fixed level of utility. 00:38:50.340 --> 00:38:53.650 align:middle line:84% Both things are true and, you can go back and forth. 00:38:53.650 --> 00:38:58.320 align:middle line:84% And that is the essential idea of duality. 00:38:58.320 --> 00:39:02.730 align:middle line:84% OK, let's do it a bit more with notation. 00:39:02.730 --> 00:39:04.940 align:middle line:84% Let's take a consumer with a consumption 00:39:04.940 --> 00:39:08.540 align:middle line:84% set in L-dimensional commodity space 00:39:08.540 --> 00:39:13.340 align:middle line:84% with the preferences represented by a continuous utility 00:39:13.340 --> 00:39:14.970 align:middle line:90% function. 00:39:14.970 --> 00:39:19.020 align:middle line:84% And let's look at the Utility Maximization Problem, 00:39:19.020 --> 00:39:22.980 align:middle line:84% UMP for short, which is to maximize 00:39:22.980 --> 00:39:26.640 align:middle line:84% utility subject to being within the budget. 00:39:26.640 --> 00:39:29.685 align:middle line:84% So this says, choose a point in the consumption set, 00:39:29.685 --> 00:39:35.790 align:middle line:84% all of the positive [INAUDIBLE],, which maximizes that utility 00:39:35.790 --> 00:39:39.870 align:middle line:84% function subject to, the points under consideration have to be 00:39:39.870 --> 00:39:43.080 align:middle line:84% such that, when you take the dot product of P and X, 00:39:43.080 --> 00:39:48.570 align:middle line:84% P1,X1 plus P2,X2, dot, dot, dot, it cannot exceed wealth 00:39:48.570 --> 00:39:49.690 align:middle line:90% or income. 00:39:49.690 --> 00:39:54.390 align:middle line:84% So that's the utility maximization problem. 00:39:54.390 --> 00:39:58.860 align:middle line:84% The result of solving the maximization problem 00:39:58.860 --> 00:40:01.320 align:middle line:84% can be substituted into utility-- 00:40:01.320 --> 00:40:05.440 align:middle line:84% in other words, utility at the maximal solution. 00:40:05.440 --> 00:40:08.160 align:middle line:84% And we're going to call that the value function 00:40:08.160 --> 00:40:10.960 align:middle line:90% or indirect utility function. 00:40:10.960 --> 00:40:15.210 align:middle line:84% Note that the maximized utility is a function of the parameters 00:40:15.210 --> 00:40:18.090 align:middle line:84% or the exogenous variables of the problem-- 00:40:18.090 --> 00:40:23.620 align:middle line:84% namely, the Price vector, p, and Wealth, w. 00:40:23.620 --> 00:40:27.550 align:middle line:84% Now, in contrast, the Expenditure Minimization 00:40:27.550 --> 00:40:32.050 align:middle line:90% Problem, EMP, is to-- 00:40:32.050 --> 00:40:34.990 align:middle line:84% given a certain level of utility-- 00:40:34.990 --> 00:40:41.180 align:middle line:84% minimize expenditures, the dot product P.X, 00:40:41.180 --> 00:40:45.640 align:middle line:84% that allows you to achieve at least that utility. 00:40:45.640 --> 00:40:46.940 align:middle line:90% Getting more is fine. 00:40:46.940 --> 00:40:50.050 align:middle line:84% But in most of the figures we show, 00:40:50.050 --> 00:40:51.580 align:middle line:90% this will be at an equality. 00:40:51.580 --> 00:40:55.870 align:middle line:84% But just like you could spend less than your income 00:40:55.870 --> 00:40:58.300 align:middle line:84% in principle, here, you could achieve 00:40:58.300 --> 00:41:00.070 align:middle line:90% a higher level of utility. 00:41:00.070 --> 00:41:04.180 align:middle line:84% But whatever you're doing here in the expenditure minimization 00:41:04.180 --> 00:41:06.910 align:middle line:84% problem, you're trying to minimize total expenditures. 00:41:06.910 --> 00:41:13.070 align:middle line:84% And we call the solution to that the E for Expenditure-- 00:41:13.070 --> 00:41:16.580 align:middle line:84% minimized expenditure at the optimum-- 00:41:16.580 --> 00:41:20.330 align:middle line:84% is a function of the parameters of the problem, the same price 00:41:20.330 --> 00:41:21.470 align:middle line:90% vector p. 00:41:21.470 --> 00:41:27.170 align:middle line:84% But now what's taken as a given is u, not w, but u. 00:41:27.170 --> 00:41:31.160 align:middle line:84% So we'll have this notation for value function and expenditure 00:41:31.160 --> 00:41:32.240 align:middle line:90% functions. 00:41:32.240 --> 00:41:35.480 align:middle line:84% Expenditure functions and demands, for that matter, 00:41:35.480 --> 00:41:37.460 align:middle line:90% have certain properties. 00:41:37.460 --> 00:41:40.880 align:middle line:84% It turns out, the expenditure function is concave, 00:41:40.880 --> 00:41:45.600 align:middle line:84% and the derivative of the expenditure function is demand. 00:41:45.600 --> 00:41:48.680 align:middle line:84% So when you take the second derivative of demand, 00:41:48.680 --> 00:41:53.310 align:middle line:84% it picks up the concavity of the level expenditure function. 00:41:53.310 --> 00:41:56.660 align:middle line:84% And that's why that Slutsky equation 00:41:56.660 --> 00:41:59.810 align:middle line:84% holds because we're looking at the matrix 00:41:59.810 --> 00:42:02.680 align:middle line:84% of second-order conditions, effectively. 00:42:02.680 --> 00:42:05.950 align:middle line:84% But again, just ignore me if that's confusing. 00:42:05.950 --> 00:42:08.440 align:middle line:84% We'll get there in that lecture 19. 00:42:08.440 --> 00:42:12.190 align:middle line:84% For now, I just want to focus on this relationship, 00:42:12.190 --> 00:42:13.300 align:middle line:90% this duality. 00:42:13.300 --> 00:42:15.460 align:middle line:84% And there's a number of ways to put it. 00:42:15.460 --> 00:42:17.830 align:middle line:84% The notation is surprisingly subtle. 00:42:17.830 --> 00:42:19.870 align:middle line:84% But if you work through it, it's fine. 00:42:19.870 --> 00:42:22.780 align:middle line:84% So suppose we're given a continuous utility function 00:42:22.780 --> 00:42:26.860 align:middle line:84% representing locally nonsatiated preferences with a consumption 00:42:26.860 --> 00:42:29.530 align:middle line:84% set, strictly positive price vector. 00:42:29.530 --> 00:42:34.180 align:middle line:84% If x* star is optimal under the utility max problem for a given 00:42:34.180 --> 00:42:38.410 align:middle line:84% wealth at those prices, then x* is optimal in the expenditure 00:42:38.410 --> 00:42:42.880 align:middle line:84% minimization problem when the required utility level is 00:42:42.880 --> 00:42:47.020 align:middle line:84% the utility you achieve under the utility maximization 00:42:47.020 --> 00:42:48.070 align:middle line:90% problem. 00:42:48.070 --> 00:42:51.500 align:middle line:84% I showed you a picture of that just a minute ago. 00:42:51.500 --> 00:42:55.900 align:middle line:84% And the minimized expenditure when you solve the EMP 00:42:55.900 --> 00:43:01.270 align:middle line:84% is exactly the wealth that you had in the utility maximization 00:43:01.270 --> 00:43:02.540 align:middle line:90% problem. 00:43:02.540 --> 00:43:03.800 align:middle line:90% Let me go back. 00:43:03.800 --> 00:43:05.510 align:middle line:90% So I started here. 00:43:05.510 --> 00:43:08.710 align:middle line:84% We had income I. Prices were given. 00:43:08.710 --> 00:43:15.190 align:middle line:84% You maximized utility, ended up at U2, given I and P. 00:43:15.190 --> 00:43:22.000 align:middle line:84% We had a certain optimizing solution here called X2,Y2 that 00:43:22.000 --> 00:43:25.060 align:middle line:84% solves the utility maximization problem. 00:43:25.060 --> 00:43:30.550 align:middle line:84% And then I said, what if we fix the utility level instead 00:43:30.550 --> 00:43:33.550 align:middle line:84% and solve the expenditure minimization problem, which 00:43:33.550 --> 00:43:36.880 align:middle line:84% is finding the expenditure level earmarked by these budget 00:43:36.880 --> 00:43:41.620 align:middle line:84% lines, which is minimal and, yet, allows us to achieve 00:43:41.620 --> 00:43:43.780 align:middle line:90% the utility level U2. 00:43:43.780 --> 00:43:47.320 align:middle line:84% So this budget line, which was a budget line for the utility 00:43:47.320 --> 00:43:51.910 align:middle line:84% maximization problem, is kind of the minimized expenditure 00:43:51.910 --> 00:43:56.268 align:middle line:84% line for the expenditure minimization problem. 00:43:56.268 --> 00:43:56.810 align:middle line:90% I don't know. 00:43:56.810 --> 00:43:59.180 align:middle line:84% Maybe the picture is just easier. 00:43:59.180 --> 00:44:01.520 align:middle line:90% But the notation says that. 00:44:01.520 --> 00:44:04.230 align:middle line:84% And it actually goes the other way. 00:44:04.230 --> 00:44:07.970 align:middle line:84% If x* were solving the expenditure minimization 00:44:07.970 --> 00:44:13.430 align:middle line:84% problem, when the required utility level is parameter u 00:44:13.430 --> 00:44:16.580 align:middle line:90% then x* is also optimal-- 00:44:16.580 --> 00:44:19.940 align:middle line:84% the same x-- in the utility maximization problem, 00:44:19.940 --> 00:44:22.220 align:middle line:84% where we give the guys wealth equal 00:44:22.220 --> 00:44:25.370 align:middle line:90% to the minimized expenditure. 00:44:25.370 --> 00:44:28.940 align:middle line:84% And when they solve the utility maximization problem 00:44:28.940 --> 00:44:32.150 align:middle line:84% at that level of wealth, they end up exactly 00:44:32.150 --> 00:44:35.720 align:middle line:84% with the u we started with in the expenditure minimization 00:44:35.720 --> 00:44:38.410 align:middle line:90% problem. 00:44:38.410 --> 00:44:42.400 align:middle line:84% Relatedly-- and this is the most compact notation 00:44:42.400 --> 00:44:45.680 align:middle line:90% of all that's kind of cool. 00:44:45.680 --> 00:44:47.710 align:middle line:84% Another way to state the proposition 00:44:47.710 --> 00:44:53.590 align:middle line:84% is to say that, given equation 3, when we start, given a price 00:44:53.590 --> 00:44:58.000 align:middle line:84% vector p, we are fixing prices p, 00:44:58.000 --> 00:45:01.240 align:middle line:84% we give the guy income or wealth, w. 00:45:01.240 --> 00:45:04.090 align:middle line:84% We solve the utility maximization problem. 00:45:04.090 --> 00:45:08.080 align:middle line:84% We substitute the maximizing demands into utility. 00:45:08.080 --> 00:45:10.960 align:middle line:84% We get the indirect utility function. 00:45:10.960 --> 00:45:13.540 align:middle line:84% Now, starting with that level of utility, 00:45:13.540 --> 00:45:16.570 align:middle line:84% we do the expenditure minimization problem 00:45:16.570 --> 00:45:19.240 align:middle line:84% and the minimized expenditures at that 00:45:19.240 --> 00:45:22.910 align:middle line:84% the in particular utility level is W all over again. 00:45:22.910 --> 00:45:24.655 align:middle line:84% So it's like feeding back to itself. 00:45:24.655 --> 00:45:28.110 align:middle line:90% 00:45:28.110 --> 00:45:32.550 align:middle line:84% Likewise here, we start with the parameterized level of u 00:45:32.550 --> 00:45:36.240 align:middle line:84% in the expenditure minimization problem, taking, again, p. 00:45:36.240 --> 00:45:37.980 align:middle line:90% Vector p is given. 00:45:37.980 --> 00:45:42.240 align:middle line:84% At that level u, we solve for the minimized expenditure, 00:45:42.240 --> 00:45:44.970 align:middle line:90% which we called e of p and u. 00:45:44.970 --> 00:45:48.810 align:middle line:84% And if we take that minimized expenditure and substituted it 00:45:48.810 --> 00:45:52.410 align:middle line:84% as a wealth term and solve the utility maximization 00:45:52.410 --> 00:45:54.810 align:middle line:84% problem at that wealth, we would get 00:45:54.810 --> 00:45:57.450 align:middle line:84% that the indirect utility is exactly 00:45:57.450 --> 00:45:59.650 align:middle line:90% the u that we started with. 00:45:59.650 --> 00:46:02.580 align:middle line:90% So it is a bit of a mouthful. 00:46:02.580 --> 00:46:05.670 align:middle line:84% The notation is quite compact, and you may just 00:46:05.670 --> 00:46:08.590 align:middle line:84% have a white-out moment when you see it. 00:46:08.590 --> 00:46:14.110 align:middle line:84% But it is repeating in different ways in related notation. 00:46:14.110 --> 00:46:19.110 align:middle line:84% Oh, this also means that e and v are inverses of one another 00:46:19.110 --> 00:46:23.040 align:middle line:84% in the sense that take e and multiply by some imagined e 00:46:23.040 --> 00:46:26.490 align:middle line:84% inverse, and you'd get rid of e on the left-hand side 00:46:26.490 --> 00:46:29.020 align:middle line:84% and have an e inverse on the right-hand side. 00:46:29.020 --> 00:46:31.410 align:middle line:84% So you'd have the value of function 00:46:31.410 --> 00:46:37.950 align:middle line:84% is an inverse of v. Sorry, I said that wrong, actually. 00:46:37.950 --> 00:46:42.660 align:middle line:84% If you take e and get rid of it by taking e inverse, 00:46:42.660 --> 00:46:45.390 align:middle line:84% when you take the inverse of the expenditure function, 00:46:45.390 --> 00:46:47.760 align:middle line:84% you end up with a value function. 00:46:47.760 --> 00:46:51.300 align:middle line:84% And likewise, when you take the inverse of the value function, 00:46:51.300 --> 00:46:54.300 align:middle line:84% you'll end up with the expenditure function. 00:46:54.300 --> 00:46:57.183 align:middle line:84% OK, I said it right at that time. 00:46:57.183 --> 00:46:57.975 align:middle line:90% There's a question. 00:46:57.975 --> 00:47:02.865 align:middle line:90% 00:47:02.865 --> 00:47:05.498 align:middle line:84% AUDIENCE: You keep saying indirect utility function-- 00:47:05.498 --> 00:47:07.665 align:middle line:84% or, I guess, what do you mean when you say indirect? 00:47:07.665 --> 00:47:10.062 align:middle line:90% 00:47:10.062 --> 00:47:11.520 align:middle line:84% ROBERT M. TOWNSEND: It has a label. 00:47:11.520 --> 00:47:13.830 align:middle line:90% I mean, v looks like value. 00:47:13.830 --> 00:47:15.090 align:middle line:90% You say, value function. 00:47:15.090 --> 00:47:17.640 align:middle line:84% But it's called the indirect utility function 00:47:17.640 --> 00:47:18.808 align:middle line:90% in the literature. 00:47:18.808 --> 00:47:19.350 align:middle line:90% AUDIENCE: OK. 00:47:19.350 --> 00:47:23.220 align:middle line:84% ROBERT M. TOWNSEND: That's the name that we give it. 00:47:23.220 --> 00:47:28.200 align:middle line:84% So Hicksian Demand is solving the expenditure minimization 00:47:28.200 --> 00:47:29.380 align:middle line:90% problem. 00:47:29.380 --> 00:47:32.970 align:middle line:84% And we have the Hicksian demand or demand correspondence 00:47:32.970 --> 00:47:34.980 align:middle line:84% if there's more than one solution. 00:47:34.980 --> 00:47:38.320 align:middle line:84% And then the diagram, so there's been a unique solution. 00:47:38.320 --> 00:47:40.590 align:middle line:84% This looks like it's a new slide. 00:47:40.590 --> 00:47:43.530 align:middle line:84% But actually, right from the very beginning, 00:47:43.530 --> 00:47:46.200 align:middle line:84% where we were talking about price effects and income 00:47:46.200 --> 00:47:49.170 align:middle line:84% effects, and we defined the price effect as a substitution 00:47:49.170 --> 00:47:52.030 align:middle line:84% effect, we were moving along a given indifference curve. 00:47:52.030 --> 00:47:55.170 align:middle line:84% So we were really talking about Hicksian demands all along. 00:47:55.170 --> 00:47:57.480 align:middle line:84% And again, for the third time, we 00:47:57.480 --> 00:47:59.700 align:middle line:84% have a different way of stating the problem, 00:47:59.700 --> 00:48:05.220 align:middle line:84% that the ordinary demand is equal to the Hicksian demand 00:48:05.220 --> 00:48:07.410 align:middle line:84% evaluated at a utility level, which 00:48:07.410 --> 00:48:11.310 align:middle line:84% is the indirect utility of the utility maximization problem. 00:48:11.310 --> 00:48:14.130 align:middle line:84% And likewise, for the expenditure minimization 00:48:14.130 --> 00:48:15.030 align:middle line:90% problem. 00:48:15.030 --> 00:48:20.320 align:middle line:84% This is, as I've been saying, true starting at this point. 00:48:20.320 --> 00:48:22.500 align:middle line:84% You start looking at it one way or the other. 00:48:22.500 --> 00:48:26.240 align:middle line:84% You end up with exactly the same point. 00:48:26.240 --> 00:48:30.170 align:middle line:84% But if we start actually taking discrete changes, 00:48:30.170 --> 00:48:31.470 align:middle line:90% things are a bit different. 00:48:31.470 --> 00:48:34.500 align:middle line:84% And in fact, we were actually doing that at the beginning, 00:48:34.500 --> 00:48:35.000 align:middle line:90% right? 00:48:35.000 --> 00:48:38.330 align:middle line:84% We were talking about a non-trivial decrease 00:48:38.330 --> 00:48:41.780 align:middle line:90% in the price of X. 00:48:41.780 --> 00:48:47.860 align:middle line:84% So let's talk about discrete changes and compensated demand. 00:48:47.860 --> 00:48:49.360 align:middle line:90% We start with p and w. 00:48:49.360 --> 00:48:55.540 align:middle line:84% We have the maximized demand, x, for that given w. 00:48:55.540 --> 00:48:59.350 align:middle line:84% But we change p from p to p prime. 00:48:59.350 --> 00:49:05.260 align:middle line:84% Then w prime would, say, be the minimized expenditure 00:49:05.260 --> 00:49:09.250 align:middle line:84% at the level of utility, u, the initial level 00:49:09.250 --> 00:49:14.980 align:middle line:84% of utility at the baseline but with the new different price 00:49:14.980 --> 00:49:16.630 align:middle line:90% vector, p prime. 00:49:16.630 --> 00:49:20.200 align:middle line:84% And we call that new wealth as opposed to old wealth. 00:49:20.200 --> 00:49:21.650 align:middle line:90% Call it w prime. 00:49:21.650 --> 00:49:25.240 align:middle line:84% So in words, that level of income or wealth 00:49:25.240 --> 00:49:29.800 align:middle line:84% w prime is the one that, under the new price vector p prime, 00:49:29.800 --> 00:49:34.480 align:middle line:84% would minimize the expenses at p prime of achieving 00:49:34.480 --> 00:49:35.860 align:middle line:90% the old utility level. 00:49:35.860 --> 00:49:37.390 align:middle line:84% Remember, when we change the price, 00:49:37.390 --> 00:49:40.030 align:middle line:84% we just rotate it along the indifference curve 00:49:40.030 --> 00:49:42.310 align:middle line:90% going one way or the other. 00:49:42.310 --> 00:49:46.600 align:middle line:84% That's all this English and all this notation means. 00:49:46.600 --> 00:49:50.050 align:middle line:84% We're just compensating the agent for the change 00:49:50.050 --> 00:49:52.980 align:middle line:90% in purchasing power. 00:49:52.980 --> 00:49:56.540 align:middle line:84% But of course, demand doesn't stay constant because the new 00:49:56.540 --> 00:50:01.670 align:middle line:84% demand, x*, will be different as the prices are different. 00:50:01.670 --> 00:50:06.200 align:middle line:84% And the Hicksian demand includes the price substitution effect, 00:50:06.200 --> 00:50:08.430 align:middle line:90% is defined by it. 00:50:08.430 --> 00:50:13.200 align:middle line:84% So the Hicksian demand, at the old utility level and prices p 00:50:13.200 --> 00:50:19.670 align:middle line:84% prime, is different from the original baseline purchase. 00:50:19.670 --> 00:50:22.910 align:middle line:84% Anyway, if you understood the income and substitution effects 00:50:22.910 --> 00:50:24.980 align:middle line:84% at the beginning, that's all this slide 00:50:24.980 --> 00:50:28.640 align:middle line:84% is saying, where we have nontrivial discrete changes 00:50:28.640 --> 00:50:29.690 align:middle line:90% in the price. 00:50:29.690 --> 00:50:32.420 align:middle line:90% Here's the picture again. 00:50:32.420 --> 00:50:38.000 align:middle line:84% But be forewarned that, somehow, to make it more interesting, 00:50:38.000 --> 00:50:41.990 align:middle line:84% we're going to change the price of good 2 instead of good 1. 00:50:41.990 --> 00:50:46.580 align:middle line:84% And we are going to increase the price. 00:50:46.580 --> 00:50:54.740 align:middle line:84% So the original baseline budget is from w to w over p2. 00:50:54.740 --> 00:50:56.720 align:middle line:90% Why just w? 00:50:56.720 --> 00:51:00.960 align:middle line:84% Well, there's an assumption down here that we only 00:51:00.960 --> 00:51:03.280 align:middle line:90% care about the price ratio. 00:51:03.280 --> 00:51:08.520 align:middle line:84% So let's denominate everything in units of good on the X-axis. 00:51:08.520 --> 00:51:12.390 align:middle line:90% p1, the first good, equals 1. 00:51:12.390 --> 00:51:14.040 align:middle line:84% So we don't need to worry anymore 00:51:14.040 --> 00:51:17.328 align:middle line:84% about the price of the first good because it's 1. 00:51:17.328 --> 00:51:18.870 align:middle line:84% And the question is, what's the price 00:51:18.870 --> 00:51:21.990 align:middle line:84% of the second good relative to 1? 00:51:21.990 --> 00:51:26.920 align:middle line:84% In this case, the price of the second good, p2, is going up. 00:51:26.920 --> 00:51:32.380 align:middle line:84% So you would go from w over p2 attainable to w 00:51:32.380 --> 00:51:36.550 align:middle line:84% over p2 prime attainable when you go from a price p2 00:51:36.550 --> 00:51:38.500 align:middle line:90% to p2 prime. 00:51:38.500 --> 00:51:40.990 align:middle line:84% This is like the earlier diagram, 00:51:40.990 --> 00:51:43.720 align:middle line:84% where we fixed a point on the Y-axis 00:51:43.720 --> 00:51:46.130 align:middle line:90% and pivoted on the X-axis. 00:51:46.130 --> 00:51:48.280 align:middle line:84% But this is the other way around because we're 00:51:48.280 --> 00:51:51.790 align:middle line:84% changing the price of good 2 and not the price of good 1. 00:51:51.790 --> 00:51:55.510 align:middle line:90% 00:51:55.510 --> 00:51:59.680 align:middle line:84% OK, so we start here, and we have an income, a lower budget 00:51:59.680 --> 00:52:00.640 align:middle line:90% line. 00:52:00.640 --> 00:52:03.790 align:middle line:84% Now we want to break that into two pieces, namely 00:52:03.790 --> 00:52:07.240 align:middle line:84% the substitution effect and the income effect. 00:52:07.240 --> 00:52:09.550 align:middle line:90% What is the substitution effect? 00:52:09.550 --> 00:52:13.690 align:middle line:84% At this higher price of p2, it's p1, over p2, 00:52:13.690 --> 00:52:15.530 align:middle line:90% If you remember the slope. 00:52:15.530 --> 00:52:19.070 align:middle line:84% So as p2 goes up, that slope goes down. 00:52:19.070 --> 00:52:22.720 align:middle line:84% So we're going to rotate along this indifference curve 00:52:22.720 --> 00:52:26.480 align:middle line:84% and effectively maximize utility, 00:52:26.480 --> 00:52:28.450 align:middle line:90% which is giving you this point. 00:52:28.450 --> 00:52:31.570 align:middle line:84% What's the maximum utility you can get 00:52:31.570 --> 00:52:33.520 align:middle line:90% or the minimized expenditure? 00:52:33.520 --> 00:52:36.550 align:middle line:84% Rotating along this indifference curve. 00:52:36.550 --> 00:52:40.300 align:middle line:84% And then the income effect is moving from here, 00:52:40.300 --> 00:52:43.520 align:middle line:90% a parallel shift back to here. 00:52:43.520 --> 00:52:49.300 align:middle line:84% So our initial level of wealth, w, was on this good 1 axis, 00:52:49.300 --> 00:52:54.820 align:middle line:84% and our new level of wealth is this minimized expenditure 00:52:54.820 --> 00:52:59.020 align:middle line:84% at the new prices, p prime, given utility level u. 00:52:59.020 --> 00:53:02.350 align:middle line:90% So we've gone from w to w prime. 00:53:02.350 --> 00:53:06.370 align:middle line:84% That's the change in wealth in terms of good 1. 00:53:06.370 --> 00:53:09.400 align:middle line:84% Initial wealth, new wealth in terms of good 1, 00:53:09.400 --> 00:53:12.490 align:middle line:84% this must therefore be the change in wealth. 00:53:12.490 --> 00:53:16.370 align:middle line:84% And that's the change in wealth, which is positive, 00:53:16.370 --> 00:53:18.110 align:middle line:84% which you may have been wondering about, 00:53:18.110 --> 00:53:21.940 align:middle line:84% which is necessary to compensate the agent for the fact 00:53:21.940 --> 00:53:24.710 align:middle line:84% that the price of good 2 has gone up. 00:53:24.710 --> 00:53:29.080 align:middle line:84% We have to compensate the agent for the adverse price change 00:53:29.080 --> 00:53:32.560 align:middle line:84% by giving him or her more income. 00:53:32.560 --> 00:53:34.210 align:middle line:84% And this is that level of income. 00:53:34.210 --> 00:53:37.306 align:middle line:90% 00:53:37.306 --> 00:53:42.420 align:middle line:84% OK, so as I keep saying, but it's probably been more or less 00:53:42.420 --> 00:53:44.700 align:middle line:84% an abstract statement, I'm really 00:53:44.700 --> 00:53:48.150 align:middle line:84% interested in real economies, where 00:53:48.150 --> 00:53:50.760 align:middle line:84% we have something going on over time, 00:53:50.760 --> 00:53:54.510 align:middle line:84% or something with the geography, or something having 00:53:54.510 --> 00:53:56.580 align:middle line:84% to do with uncertainty, and gambles, 00:53:56.580 --> 00:53:58.600 align:middle line:90% and states of the world. 00:53:58.600 --> 00:54:03.150 align:middle line:84% So having memorized or shown you all those figures 00:54:03.150 --> 00:54:05.730 align:middle line:84% with some very interesting concepts and a little bit 00:54:05.730 --> 00:54:09.120 align:middle line:84% of data along the way, one may, nevertheless, 00:54:09.120 --> 00:54:11.280 align:middle line:84% walk away with the impression that we're 00:54:11.280 --> 00:54:13.260 align:middle line:84% talking about a static world, where 00:54:13.260 --> 00:54:17.160 align:middle line:84% we have rice and other goods or potatoes and other goods. 00:54:17.160 --> 00:54:21.780 align:middle line:84% And we're just comparing wheat to apples as we change prices. 00:54:21.780 --> 00:54:26.010 align:middle line:84% But no, the framework applies to any goods, 00:54:26.010 --> 00:54:28.720 align:middle line:84% including goods that are earmarked by time. 00:54:28.720 --> 00:54:33.720 align:middle line:84% So suppose we have two time periods only. 00:54:33.720 --> 00:54:36.570 align:middle line:90% And we'll call consumption C1-- 00:54:36.570 --> 00:54:40.410 align:middle line:84% consumption and date 1, and C2, consumption and date 2. 00:54:40.410 --> 00:54:44.130 align:middle line:84% And we're going to give income as an endowment, 00:54:44.130 --> 00:54:49.050 align:middle line:84% meaning how much income they have in the first date, 1, 00:54:49.050 --> 00:54:52.950 align:middle line:84% and how much income they have in the second date, 2. 00:54:52.950 --> 00:54:55.170 align:middle line:84% Using income is a bit tricky here 00:54:55.170 --> 00:54:58.740 align:middle line:84% because we used that before in the static optimization 00:54:58.740 --> 00:54:59.650 align:middle line:90% problem. 00:54:59.650 --> 00:55:01.830 align:middle line:90% So think of it as money, then-- 00:55:01.830 --> 00:55:05.700 align:middle line:84% money in date 1, money at date 2. 00:55:05.700 --> 00:55:07.600 align:middle line:90% Now, what is the budget line? 00:55:07.600 --> 00:55:10.860 align:middle line:84% Well, if they just eat their money, 00:55:10.860 --> 00:55:14.070 align:middle line:84% they would have consumption equal to m1 and m2. 00:55:14.070 --> 00:55:17.200 align:middle line:84% But the way this is drawn, they don't want to do that. 00:55:17.200 --> 00:55:20.700 align:middle line:84% They want to move in one direction or another. 00:55:20.700 --> 00:55:22.640 align:middle line:90% So what is this budget line? 00:55:22.640 --> 00:55:26.060 align:middle line:84% This budget line reflects the possibility 00:55:26.060 --> 00:55:29.390 align:middle line:84% of giving up money tomorrow and getting more consumption 00:55:29.390 --> 00:55:30.440 align:middle line:90% today-- 00:55:30.440 --> 00:55:33.980 align:middle line:84% that's borrowing-- or giving up money today 00:55:33.980 --> 00:55:37.340 align:middle line:84% and getting more consumption tomorrow-- that's lending. 00:55:37.340 --> 00:55:40.280 align:middle line:84% So the slope of this heavy blue line 00:55:40.280 --> 00:55:43.850 align:middle line:84% represents the tradeoff of the price of goods 00:55:43.850 --> 00:55:47.300 align:middle line:84% today for the price of good tomorrow, which you knew 00:55:47.300 --> 00:55:49.460 align:middle line:90% all along but probably forgot. 00:55:49.460 --> 00:55:50.030 align:middle line:90% Why? 00:55:50.030 --> 00:55:53.500 align:middle line:84% Because the slope of the budget line is p1 over p2-- 00:55:53.500 --> 00:55:56.300 align:middle line:84% p1 being consumption at date one, p2 00:55:56.300 --> 00:55:59.480 align:middle line:84% to being-- sorry-- the price of consumption at date two. 00:55:59.480 --> 00:56:04.130 align:middle line:84% But the economics of the problem doesn't write p1 over p2. 00:56:04.130 --> 00:56:06.690 align:middle line:90% It writes r as an interest rate. 00:56:06.690 --> 00:56:08.150 align:middle line:90% Why is it the interest rate? 00:56:08.150 --> 00:56:11.780 align:middle line:84% Because if you gave up some of your money today 00:56:11.780 --> 00:56:15.110 align:middle line:84% and put it in the bank, you would get the principal 00:56:15.110 --> 00:56:16.910 align:middle line:90% and interest back tomorrow. 00:56:16.910 --> 00:56:21.500 align:middle line:84% So 1 plus r, which is also the slope of this line, 00:56:21.500 --> 00:56:24.980 align:middle line:84% reflects the tradeoff between consumption today 00:56:24.980 --> 00:56:29.340 align:middle line:84% and consumption tomorrow as made possible by this interest rate. 00:56:29.340 --> 00:56:32.000 align:middle line:84% So then, as per the entire lecture, 00:56:32.000 --> 00:56:34.070 align:middle line:90% we try to change a price. 00:56:34.070 --> 00:56:36.540 align:middle line:84% Over here, let's increase the interest rate. 00:56:36.540 --> 00:56:40.130 align:middle line:84% So it's still possible to eat your money in both periods. 00:56:40.130 --> 00:56:42.140 align:middle line:84% But now that the interest rate has gone up, 00:56:42.140 --> 00:56:44.090 align:middle line:90% the budget line is steeper. 00:56:44.090 --> 00:56:47.600 align:middle line:84% For a given surrendering of one unit of consumption 00:56:47.600 --> 00:56:51.560 align:middle line:84% today, you get a higher 1 plus r tomorrow. 00:56:51.560 --> 00:56:55.280 align:middle line:84% So this paler blue line crosses through the endowment point, 00:56:55.280 --> 00:56:56.300 align:middle line:90% but it's steeper. 00:56:56.300 --> 00:56:59.240 align:middle line:84% And as drawn with these indifference curves, 00:56:59.240 --> 00:57:02.400 align:middle line:84% the household would move from this point to this point. 00:57:02.400 --> 00:57:06.590 align:middle line:84% So let's talk about income and substitution effects. 00:57:06.590 --> 00:57:11.360 align:middle line:84% Consumption, tomorrow, at date 2, goes up and up. 00:57:11.360 --> 00:57:12.300 align:middle line:90% Why? 00:57:12.300 --> 00:57:15.650 align:middle line:84% Two effects, and they're both moving in the same direction. 00:57:15.650 --> 00:57:17.270 align:middle line:84% The increase in the interest rate 00:57:17.270 --> 00:57:20.070 align:middle line:84% effectively means they have more income. 00:57:20.070 --> 00:57:22.610 align:middle line:84% They can take their endowment of money tomorrow, 00:57:22.610 --> 00:57:23.990 align:middle line:90% and it will be worth more. 00:57:23.990 --> 00:57:26.030 align:middle line:84% So there's an income effect associated 00:57:26.030 --> 00:57:28.130 align:middle line:90% with the price change. 00:57:28.130 --> 00:57:30.720 align:middle line:84% Not only that, there's a substitution effect. 00:57:30.720 --> 00:57:31.370 align:middle line:90% Why? 00:57:31.370 --> 00:57:33.350 align:middle line:84% Because as the interest rate goes up, 00:57:33.350 --> 00:57:36.440 align:middle line:84% the opportunity cost of eating today goes up. 00:57:36.440 --> 00:57:39.380 align:middle line:84% It's more painful to eat today because you could 00:57:39.380 --> 00:57:42.290 align:middle line:84% get even more tomorrow by putting the money in the bank 00:57:42.290 --> 00:57:43.890 align:middle line:90% at a higher interest rate. 00:57:43.890 --> 00:57:46.760 align:middle line:84% So the substitution effect moves them 00:57:46.760 --> 00:57:48.260 align:middle line:84% with these nice, smooth indifference 00:57:48.260 --> 00:57:50.870 align:middle line:84% curves away from consumption today 00:57:50.870 --> 00:57:53.390 align:middle line:84% toward higher consumption tomorrow. 00:57:53.390 --> 00:57:56.420 align:middle line:84% So the income and substitution effects in this diagram 00:57:56.420 --> 00:57:59.060 align:middle line:84% work in the same direction for good 2, 00:57:59.060 --> 00:58:04.220 align:middle line:84% but it's not true for good 1 at date 1. 00:58:04.220 --> 00:58:06.860 align:middle line:84% Or at least, it could be ambiguous. 00:58:06.860 --> 00:58:08.090 align:middle line:90% Why? 00:58:08.090 --> 00:58:11.210 align:middle line:84% There's an income effect, which should increase consumption 00:58:11.210 --> 00:58:12.620 align:middle line:90% at date 1. 00:58:12.620 --> 00:58:14.420 align:middle line:84% But there's a price effect, which should 00:58:14.420 --> 00:58:16.940 align:middle line:90% decrease consumption at date 1. 00:58:16.940 --> 00:58:21.060 align:middle line:84% And those can, in principle, move in opposite directions. 00:58:21.060 --> 00:58:24.530 align:middle line:84% In this case, date 1 consumption is going down, 00:58:24.530 --> 00:58:29.390 align:middle line:84% so the substitution effect is dominating the income effect. 00:58:29.390 --> 00:58:31.280 align:middle line:84% But it didn't have to go that way. 00:58:31.280 --> 00:58:33.380 align:middle line:84% You could have drawn the indifference curves 00:58:33.380 --> 00:58:36.260 align:middle line:84% and gotten an increase in the consumption of date 00:58:36.260 --> 00:58:37.580 align:middle line:90% of that good. 00:58:37.580 --> 00:58:41.780 align:middle line:84% OK, let's start out with a different but related diagram. 00:58:41.780 --> 00:58:45.230 align:middle line:84% Suppose they have money today and tomorrow as endowments. 00:58:45.230 --> 00:58:47.690 align:middle line:84% And this guy is actually doing what? 00:58:47.690 --> 00:58:51.050 align:middle line:84% Originally, borrowing, getting more consumption today 00:58:51.050 --> 00:58:54.320 align:middle line:84% than the money he or she has at the expense of paying it 00:58:54.320 --> 00:58:56.270 align:middle line:84% back with interest tomorrow, therefore 00:58:56.270 --> 00:58:59.770 align:middle line:84% having less consumption than the money that he or she originally 00:58:59.770 --> 00:59:00.270 align:middle line:90% had. 00:59:00.270 --> 00:59:03.290 align:middle line:84% So we'd end up with this tangency. 00:59:03.290 --> 00:59:06.260 align:middle line:84% And now we still increase the interest rate. 00:59:06.260 --> 00:59:10.850 align:middle line:84% That makes this pale blue line steeper. 00:59:10.850 --> 00:59:14.210 align:middle line:84% And you can see consumption is moving inwards. 00:59:14.210 --> 00:59:17.540 align:middle line:84% So actually, consumption of both goods in the diagram 00:59:17.540 --> 00:59:18.720 align:middle line:90% is going down. 00:59:18.720 --> 00:59:21.620 align:middle line:84% But let's think about income and substitution effects. 00:59:21.620 --> 00:59:24.200 align:middle line:84% Because they were borrowing originally, 00:59:24.200 --> 00:59:26.750 align:middle line:84% as opposed to lending, intuitively, 00:59:26.750 --> 00:59:30.680 align:middle line:84% an increase in the interest rate is a bad situation for them. 00:59:30.680 --> 00:59:33.360 align:middle line:84% They were borrowing, and now the interest rate is higher. 00:59:33.360 --> 00:59:35.300 align:middle line:84% So the income effect is negative. 00:59:35.300 --> 00:59:38.250 align:middle line:84% And that could show up with both goods. 00:59:38.250 --> 00:59:40.220 align:middle line:84% On the other hand, the substitution effect 00:59:40.220 --> 00:59:43.160 align:middle line:84% should still push them in the direction of higher consumption 00:59:43.160 --> 00:59:43.820 align:middle line:90% tomorrow. 00:59:43.820 --> 00:59:46.140 align:middle line:90% That's the substitution effect. 00:59:46.140 --> 00:59:49.130 align:middle line:84% So for one of these goods, consumption today 00:59:49.130 --> 00:59:50.240 align:middle line:90% is unambiguous. 00:59:50.240 --> 00:59:52.700 align:middle line:84% But consumption tomorrow is ambiguous, 00:59:52.700 --> 00:59:55.160 align:middle line:84% despite the way it's drawn, because the income 00:59:55.160 --> 00:59:58.590 align:middle line:84% and substitution effects work in the opposite direction. 00:59:58.590 --> 00:59:59.990 align:middle line:90% So take a look at those. 00:59:59.990 --> 01:00:01.230 align:middle line:90% There's a lot going on. 01:00:01.230 --> 01:00:02.810 align:middle line:84% In fact, that's kind of the point, 01:00:02.810 --> 01:00:05.570 align:middle line:84% that when we move from apples, wheats, 01:00:05.570 --> 01:00:09.230 align:middle line:84% and so on with prices and income to an applied problem, 01:00:09.230 --> 01:00:12.260 align:middle line:84% like what's happening in the US with these lower 01:00:12.260 --> 01:00:14.810 align:middle line:84% and lower interest rates, well, if we 01:00:14.810 --> 01:00:17.900 align:middle line:84% look at consumer behavior, there's income effects. 01:00:17.900 --> 01:00:20.030 align:middle line:90% There's substitution effects. 01:00:20.030 --> 01:00:22.820 align:middle line:84% You can go in different directions and be in conflict. 01:00:22.820 --> 01:00:26.180 align:middle line:84% And that's an important part of understanding the impact 01:00:26.180 --> 01:00:28.790 align:middle line:84% on the households and the impact on the economy. 01:00:28.790 --> 01:00:31.370 align:middle line:84% And it's actually a lot more work 01:00:31.370 --> 01:00:33.290 align:middle line:84% to go through this slide and the previous one 01:00:33.290 --> 01:00:34.940 align:middle line:84% than you might have imagined, even 01:00:34.940 --> 01:00:37.340 align:middle line:84% though, say, you were totally sure you 01:00:37.340 --> 01:00:40.220 align:middle line:84% could shift these curves around in whatever way you wanted. 01:00:40.220 --> 01:00:43.040 align:middle line:84% When we get to the economic application, 01:00:43.040 --> 01:00:46.760 align:middle line:84% we kind of do it all over again with the new realistic labels. 01:00:46.760 --> 01:00:50.610 align:middle line:90% And it brings that tool to life. 01:00:50.610 --> 01:00:53.430 align:middle line:84% OK, well, that's the last slide for today. 01:00:53.430 --> 01:00:56.430 align:middle line:90% That's the end of this lecture. 01:00:56.430 --> 01:00:59.270 align:middle line:90% All right, thank you very much. 01:00:59.270 --> 01:01:08.000 align:middle line:90%