WEBVTT 00:00:00.000 --> 00:00:01.988 align:middle line:90% [SQUEAKING] 00:00:01.988 --> 00:00:03.976 align:middle line:90% [RUSTLING] 00:00:03.976 --> 00:00:06.958 align:middle line:90% [CLICKING] 00:00:06.958 --> 00:00:11.440 align:middle line:90% 00:00:11.440 --> 00:00:17.260 align:middle line:84% ROBERT TOWNSEND: So take a look at the reading list. 00:00:17.260 --> 00:00:19.570 align:middle line:84% And you will see there's various things that 00:00:19.570 --> 00:00:23.440 align:middle line:84% are starred on there that are on today's lecture two. 00:00:23.440 --> 00:00:28.060 align:middle line:84% There's readings by Kreps and by Nicholson and Snyder. 00:00:28.060 --> 00:00:30.580 align:middle line:84% The lecture does not overlap entirely, 00:00:30.580 --> 00:00:33.070 align:middle line:84% but it is close in many respects. 00:00:33.070 --> 00:00:37.360 align:middle line:90% So I urge you to take a look at. 00:00:37.360 --> 00:00:41.940 align:middle line:84% There's one Kreps and two Nicholson and Snyder. 00:00:41.940 --> 00:00:45.870 align:middle line:84% I also mentioned last time, but it might have been abstract, 00:00:45.870 --> 00:00:48.520 align:middle line:90% the review sheets. 00:00:48.520 --> 00:00:51.880 align:middle line:84% This is lecture one from last time. 00:00:51.880 --> 00:00:53.200 align:middle line:90% It's a review sheet-- 00:00:53.200 --> 00:00:55.940 align:middle line:84% pretty comprehensive, but also quite specific. 00:00:55.940 --> 00:00:59.890 align:middle line:84% So you should be looking at these things 00:00:59.890 --> 00:01:02.890 align:middle line:84% before, during, or especially after the lecture, 00:01:02.890 --> 00:01:05.410 align:middle line:84% even if you think you understand the material. 00:01:05.410 --> 00:01:06.880 align:middle line:90% Take a hand at this. 00:01:06.880 --> 00:01:10.810 align:middle line:84% I certainly have included questions almost verbatim 00:01:10.810 --> 00:01:14.260 align:middle line:84% like this on the midterms, so you might as well do it 00:01:14.260 --> 00:01:15.680 align:middle line:90% in advance. 00:01:15.680 --> 00:01:18.280 align:middle line:84% And I'm going to read out just one or two of these. 00:01:18.280 --> 00:01:21.800 align:middle line:84% Obviously, we don't have time to do too many of them. 00:01:21.800 --> 00:01:24.610 align:middle line:84% I would say I'm almost picking at random, but not quite. 00:01:24.610 --> 00:01:26.590 align:middle line:84% So the fact that I pick some and not others 00:01:26.590 --> 00:01:29.020 align:middle line:84% does not mean that some are more important than others, 00:01:29.020 --> 00:01:31.730 align:middle line:84% but only I want to practice with you. 00:01:31.730 --> 00:01:34.000 align:middle line:90% So for example, this one-- 00:01:34.000 --> 00:01:36.790 align:middle line:84% Angrist distinguishes randomized trials 00:01:36.790 --> 00:01:40.180 align:middle line:84% from quasi-experiments or natural experiments. 00:01:40.180 --> 00:01:43.180 align:middle line:84% Please explain what he means by this distinction 00:01:43.180 --> 00:01:46.550 align:middle line:84% and give one example from each of the two groups, 00:01:46.550 --> 00:01:51.220 align:middle line:84% meaning from quasi-randomized controlled trials and one 00:01:51.220 --> 00:01:54.040 align:middle line:90% from quasi-experiments. 00:01:54.040 --> 00:01:57.592 align:middle line:84% I know you're new to this, so I hesitate to call on you. 00:01:57.592 --> 00:01:59.050 align:middle line:84% So it would be wonderful if someone 00:01:59.050 --> 00:02:01.992 align:middle line:84% were to volunteer to answer this question. 00:02:01.992 --> 00:02:03.700 align:middle line:84% STUDENT: I think I could make a go at it. 00:02:03.700 --> 00:02:05.740 align:middle line:84% ROBERT TOWNSEND: All right, please go ahead. 00:02:05.740 --> 00:02:10.120 align:middle line:84% STUDENT: Oh, yeah, so an actual experiment is randomized 00:02:10.120 --> 00:02:12.700 align:middle line:84% controlled trials where you actually randomly take 00:02:12.700 --> 00:02:14.140 align:middle line:84% a group of people, randomly select 00:02:14.140 --> 00:02:16.723 align:middle line:84% some to be a control group, some to be the experimental group. 00:02:16.723 --> 00:02:21.550 align:middle line:84% And then you apply some change to one of them, 00:02:21.550 --> 00:02:25.960 align:middle line:84% to the experimental group, and track the outcomes. 00:02:25.960 --> 00:02:31.270 align:middle line:84% A natural quasi-experiment would be more if something happens-- 00:02:31.270 --> 00:02:34.210 align:middle line:84% for example, one state raises the minimum wage and another 00:02:34.210 --> 00:02:35.110 align:middle line:90% state doesn't. 00:02:35.110 --> 00:02:37.190 align:middle line:90% And you can track a difference. 00:02:37.190 --> 00:02:42.310 align:middle line:84% So it wasn't an actual experiment, but you can still-- 00:02:42.310 --> 00:02:45.020 align:middle line:84% but it sort of behaves a bit like an experiment. 00:02:45.020 --> 00:02:47.740 align:middle line:84% So you can try to tease apart the causality there. 00:02:47.740 --> 00:02:50.260 align:middle line:84% And then I can't remember the rest of the question. 00:02:50.260 --> 00:02:51.010 align:middle line:90% Sorry. 00:02:51.010 --> 00:02:52.302 align:middle line:90% ROBERT TOWNSEND: No, it's fine. 00:02:52.302 --> 00:02:56.062 align:middle line:84% Do you remember an example from any of those? 00:02:56.062 --> 00:02:57.520 align:middle line:84% STUDENT: From a natural experiment, 00:02:57.520 --> 00:02:59.395 align:middle line:84% that would be like the Moving to Opportunity. 00:02:59.395 --> 00:03:04.870 align:middle line:84% I remember that being mentioned, where people were effectively 00:03:04.870 --> 00:03:08.950 align:middle line:84% randomly chosen to be given housing opportunities to move. 00:03:08.950 --> 00:03:10.225 align:middle line:90% And then that was tracked. 00:03:10.225 --> 00:03:11.350 align:middle line:90% ROBERT TOWNSEND: Excellent. 00:03:11.350 --> 00:03:12.500 align:middle line:90% OK. 00:03:12.500 --> 00:03:14.680 align:middle line:84% Yeah, and the other RCT we did was 00:03:14.680 --> 00:03:20.480 align:middle line:84% the paying bicycle messengers temporarily higher wages. 00:03:20.480 --> 00:03:23.170 align:middle line:84% RCTs are where we are with these vaccines, 00:03:23.170 --> 00:03:28.990 align:middle line:84% just to have an obvious analog to the widely used RCTs 00:03:28.990 --> 00:03:30.520 align:middle line:90% in medical trials. 00:03:30.520 --> 00:03:34.240 align:middle line:84% But we like to frame questions that way. 00:03:34.240 --> 00:03:35.470 align:middle line:90% Thank you, Charles. 00:03:35.470 --> 00:03:38.320 align:middle line:84% And I also went and read some of the things 00:03:38.320 --> 00:03:39.790 align:middle line:90% you recommended last time. 00:03:39.790 --> 00:03:42.580 align:middle line:90% Thank you. 00:03:42.580 --> 00:03:45.020 align:middle line:84% Let's see, let me pick another one. 00:03:45.020 --> 00:03:49.180 align:middle line:84% So Varian speaks of big data as consisting of machine learning 00:03:49.180 --> 00:03:50.830 align:middle line:90% and data science. 00:03:50.830 --> 00:03:54.250 align:middle line:84% What do those techniques accomplish 00:03:54.250 --> 00:03:56.350 align:middle line:84% and what do they fail to accomplish? 00:03:56.350 --> 00:03:59.763 align:middle line:84% And again, let me ask for volunteers. 00:03:59.763 --> 00:04:01.180 align:middle line:84% STUDENT: So I remember you talking 00:04:01.180 --> 00:04:03.280 align:middle line:84% about what they fail to accomplish, 00:04:03.280 --> 00:04:07.600 align:middle line:84% where it's difficult to actually construct 00:04:07.600 --> 00:04:11.020 align:middle line:84% the explanation behind the phenomena that you observe. 00:04:11.020 --> 00:04:13.480 align:middle line:84% Instead, it just gives you good predictive power 00:04:13.480 --> 00:04:15.490 align:middle line:84% where you can construct accurate models, 00:04:15.490 --> 00:04:18.160 align:middle line:84% but not necessarily explain them. 00:04:18.160 --> 00:04:19.690 align:middle line:90% ROBERT TOWNSEND: Excellent. 00:04:19.690 --> 00:04:22.760 align:middle line:90% That's exactly right. 00:04:22.760 --> 00:04:26.330 align:middle line:84% The example was just because we see more police 00:04:26.330 --> 00:04:28.880 align:middle line:84% in the community, that doesn't mean it causes crime. 00:04:28.880 --> 00:04:32.060 align:middle line:84% And I said something about the example's a bit more 00:04:32.060 --> 00:04:36.680 align:middle line:84% problematic these days than it was when Varian used that 00:04:36.680 --> 00:04:37.520 align:middle line:90% as an example. 00:04:37.520 --> 00:04:40.850 align:middle line:84% But it's still the case that we need to out the causality 00:04:40.850 --> 00:04:45.620 align:middle line:84% whether it runs from crime to police or police to crime. 00:04:45.620 --> 00:04:49.100 align:middle line:84% All right, so there are other important questions in here, 00:04:49.100 --> 00:04:51.380 align:middle line:84% but I'm a little leery about taking too, too 00:04:51.380 --> 00:04:53.550 align:middle line:90% much of the lecture time. 00:04:53.550 --> 00:04:56.970 align:middle line:84% So these are called review sheets. 00:04:56.970 --> 00:04:59.600 align:middle line:84% And I think you'll find it is not hard. 00:04:59.600 --> 00:05:02.780 align:middle line:84% It really is just a matter of keeping up with the lecture 00:05:02.780 --> 00:05:05.870 align:middle line:84% and thinking about this afterwards, along 00:05:05.870 --> 00:05:07.340 align:middle line:90% with those starred readings. 00:05:07.340 --> 00:05:11.960 align:middle line:84% So today, this is about consumer choice. 00:05:11.960 --> 00:05:14.150 align:middle line:84% Remember, we're studying economies, 00:05:14.150 --> 00:05:18.290 align:middle line:84% but we need to build and analyze the building blocks, one 00:05:18.290 --> 00:05:21.620 align:middle line:90% of which is consumer behavior. 00:05:21.620 --> 00:05:24.170 align:middle line:84% So that's what this lecture is about. 00:05:24.170 --> 00:05:28.220 align:middle line:84% The whole economy consists of consumers with preferences, 00:05:28.220 --> 00:05:30.620 align:middle line:84% and endowments, and consumption sets, 00:05:30.620 --> 00:05:33.570 align:middle line:90% and firms with technology. 00:05:33.570 --> 00:05:37.790 align:middle line:84% We're going to focus on consumer choice over possible bundles 00:05:37.790 --> 00:05:40.670 align:middle line:90% they might want to buy or eat. 00:05:40.670 --> 00:05:44.300 align:middle line:84% We're going to describe some assumptions that characterize 00:05:44.300 --> 00:05:47.810 align:middle line:84% what's known as rational, as opposed to irrational consumer 00:05:47.810 --> 00:05:49.110 align:middle line:90% choice. 00:05:49.110 --> 00:05:52.580 align:middle line:84% And then we're going to go to this application, which 00:05:52.580 --> 00:05:57.290 align:middle line:84% is maximizing utility for a given budget. 00:05:57.290 --> 00:05:59.600 align:middle line:84% Let me just say now, because I may not 00:05:59.600 --> 00:06:01.400 align:middle line:84% remember to do it later, when we get 00:06:01.400 --> 00:06:04.370 align:middle line:84% to this application of maximizing utility 00:06:04.370 --> 00:06:08.690 align:middle line:84% subject to a budget, it's what we call "partial equilibrium." 00:06:08.690 --> 00:06:12.560 align:middle line:84% We take, as Matzkin would say, some things 00:06:12.560 --> 00:06:15.530 align:middle line:84% as exogenous to the consumer choice-- 00:06:15.530 --> 00:06:19.520 align:middle line:84% namely, their income and the prices that they face. 00:06:19.520 --> 00:06:21.350 align:middle line:90% Those are like data or givens. 00:06:21.350 --> 00:06:25.370 align:middle line:84% And then we're looking at what they choose to buy or to eat. 00:06:25.370 --> 00:06:27.172 align:middle line:84% It's partial equilibrium in the sense 00:06:27.172 --> 00:06:29.630 align:middle line:84% that we're not trying to explain where the prices come from 00:06:29.630 --> 00:06:32.260 align:middle line:90% or where the income comes from. 00:06:32.260 --> 00:06:33.720 align:middle line:84% But when we put this building block 00:06:33.720 --> 00:06:37.860 align:middle line:84% together with the other ones, with other households and/or 00:06:37.860 --> 00:06:40.290 align:middle line:84% with firms, then we'll have what's 00:06:40.290 --> 00:06:42.600 align:middle line:84% called "general equilibrium," which 00:06:42.600 --> 00:06:46.690 align:middle line:84% is the endogenous determination of prices, and so on. 00:06:46.690 --> 00:06:51.660 align:middle line:84% OK, so let's call capital X here a consumer's consumption set. 00:06:51.660 --> 00:06:57.120 align:middle line:84% Two elements of the set called x and y are potential consumption 00:06:57.120 --> 00:06:58.110 align:middle line:90% bundles. 00:06:58.110 --> 00:07:01.230 align:middle line:84% You should be aware and a bit leery 00:07:01.230 --> 00:07:04.170 align:middle line:84% that the notation changes from time to time. 00:07:04.170 --> 00:07:06.790 align:middle line:84% Sometimes in the slides that follow, 00:07:06.790 --> 00:07:10.740 align:middle line:84% we're about to say x is one bundle and y is another one. 00:07:10.740 --> 00:07:13.350 align:middle line:84% Many times we'll go more abstract, 00:07:13.350 --> 00:07:17.070 align:middle line:84% and talk about a vector x of dimension n 00:07:17.070 --> 00:07:21.030 align:middle line:84% where they're all x's, x1 through x n. 00:07:21.030 --> 00:07:24.360 align:middle line:84% And indeed, when we start to talk about more than one 00:07:24.360 --> 00:07:27.090 align:middle line:90% consumer, like consumer i-- 00:07:27.090 --> 00:07:28.560 align:middle line:90% that's his or her name-- 00:07:28.560 --> 00:07:32.010 align:middle line:84% we can talk about the consumption set x sub i. 00:07:32.010 --> 00:07:34.650 align:middle line:84% The bundle x i is still a vector, 00:07:34.650 --> 00:07:37.020 align:middle line:84% but it has a little i subscript in front 00:07:37.020 --> 00:07:39.930 align:middle line:90% of the 1 to n commodity labels. 00:07:39.930 --> 00:07:43.920 align:middle line:84% But there's no point in being overly detailed when 00:07:43.920 --> 00:07:47.410 align:middle line:84% we don't need it, so we'll drop the i mostly today. 00:07:47.410 --> 00:07:48.990 align:middle line:84% I don't think it appears anywhere. 00:07:48.990 --> 00:07:51.480 align:middle line:84% And as I said for the graphs, we're 00:07:51.480 --> 00:07:54.410 align:middle line:84% going to be using x and y quite a bit. 00:07:54.410 --> 00:07:58.020 align:middle line:84% Now, one warning-- should have something flashing 00:07:58.020 --> 00:07:58.980 align:middle line:90% on the screen-- 00:07:58.980 --> 00:07:59.760 align:middle line:90% stop. 00:07:59.760 --> 00:08:03.360 align:middle line:84% Pause-- which is there's something mind 00:08:03.360 --> 00:08:05.250 align:middle line:90% numbing about the notation. 00:08:05.250 --> 00:08:06.360 align:middle line:90% It's useful. 00:08:06.360 --> 00:08:08.950 align:middle line:84% We will build up all the tools with it. 00:08:08.950 --> 00:08:11.560 align:middle line:84% But we'll lose sight of the applications. 00:08:11.560 --> 00:08:15.090 align:middle line:84% So I will bend over backwards to pop in a slide here or there 00:08:15.090 --> 00:08:18.060 align:middle line:84% and say some words about how this can be used. 00:08:18.060 --> 00:08:21.240 align:middle line:84% But the point, although abstract now, 00:08:21.240 --> 00:08:24.180 align:middle line:84% is that the artillery where you're going to create 00:08:24.180 --> 00:08:28.650 align:middle line:84% applies to wheat and rice, but it also applies to locations. 00:08:28.650 --> 00:08:31.260 align:middle line:90% It applies to dates and time. 00:08:31.260 --> 00:08:34.510 align:middle line:84% It applies to shocks and states of the world. 00:08:34.510 --> 00:08:37.679 align:middle line:84% So once we do it one way with this powerful but boring 00:08:37.679 --> 00:08:42.179 align:middle line:84% notation, we can use it in many actual applications 00:08:42.179 --> 00:08:43.080 align:middle line:90% and villages. 00:08:43.080 --> 00:08:46.020 align:middle line:84% With the exception that here, we're actually spelling out 00:08:46.020 --> 00:08:47.160 align:middle line:90% the consumption set. 00:08:47.160 --> 00:08:49.880 align:middle line:90% So you can see that. 00:08:49.880 --> 00:08:53.530 align:middle line:84% Can you see it when I'm moving the cursor around? 00:08:53.530 --> 00:08:56.050 align:middle line:84% All right, I'll have to work on my tool kit. 00:08:56.050 --> 00:09:00.070 align:middle line:84% So there's two goods here-- bread and leisure. 00:09:00.070 --> 00:09:04.300 align:middle line:84% And the consumption set is all of this. 00:09:04.300 --> 00:09:05.770 align:middle line:90% It has an upper bound. 00:09:05.770 --> 00:09:06.820 align:middle line:90% Why? 00:09:06.820 --> 00:09:09.080 align:middle line:84% Because leisure is measured in hours, 00:09:09.080 --> 00:09:12.410 align:middle line:84% and there's no more than 24 hours, a day. 00:09:12.410 --> 00:09:15.790 align:middle line:84% So that's one example of a consumption set. 00:09:15.790 --> 00:09:19.900 align:middle line:84% Here's another-- there's two goods, x1 and x2. 00:09:19.900 --> 00:09:22.330 align:middle line:90% x1 can take on any value-- 00:09:22.330 --> 00:09:25.720 align:middle line:84% integers, rational, or irrational numbers from 0 00:09:25.720 --> 00:09:26.920 align:middle line:90% to infinity. 00:09:26.920 --> 00:09:30.280 align:middle line:84% But x2 takes on discrete values-- 00:09:30.280 --> 00:09:31.930 align:middle line:90% integers-- 1, 2, 3. 00:09:31.930 --> 00:09:35.020 align:middle line:84% So all these lines, including the 0 line, 00:09:35.020 --> 00:09:37.730 align:middle line:84% represent elements in the consumption set. 00:09:37.730 --> 00:09:42.260 align:middle line:84% Here's an example with some limited time and geography. 00:09:42.260 --> 00:09:46.420 align:middle line:84% You can consume a piece of bread, or bread in general, 00:09:46.420 --> 00:09:49.670 align:middle line:84% in Washington at noon or in New York at noon. 00:09:49.670 --> 00:09:52.430 align:middle line:84% But you can't be in two places at the same time. 00:09:52.430 --> 00:09:55.360 align:middle line:84% So in this case, you have the x and y-axis 00:09:55.360 --> 00:09:57.400 align:middle line:90% being the consumption set. 00:09:57.400 --> 00:10:00.220 align:middle line:90% And here's a subsistence. 00:10:00.220 --> 00:10:01.930 align:middle line:90% If you don't eat enough-- 00:10:01.930 --> 00:10:05.530 align:middle line:84% and we'll talk about starvation in medieval villages soon-- 00:10:05.530 --> 00:10:06.290 align:middle line:90% you die. 00:10:06.290 --> 00:10:10.660 align:middle line:84% So you have to have either four slices of white bread, 00:10:10.660 --> 00:10:13.660 align:middle line:84% or four slices of brown bread, or something in between. 00:10:13.660 --> 00:10:15.110 align:middle line:90% But you can't go less than that. 00:10:15.110 --> 00:10:18.370 align:middle line:84% Those things are not in the consumption set. 00:10:18.370 --> 00:10:20.710 align:middle line:84% So that's the set in which commodity bundles 00:10:20.710 --> 00:10:21.550 align:middle line:90% lie for the house. 00:10:21.550 --> 00:10:23.140 align:middle line:90% So what about their preferences? 00:10:23.140 --> 00:10:25.450 align:middle line:84% We're going to begin with this primitive, which 00:10:25.450 --> 00:10:27.790 align:middle line:84% looks like a weak inequality, but it's 00:10:27.790 --> 00:10:31.330 align:middle line:84% actually script or squiggly, to denote that it's a preference 00:10:31.330 --> 00:10:33.620 align:middle line:90% relation, which is an ordering. 00:10:33.620 --> 00:10:39.160 align:middle line:84% So if you give me two elements of the consumption set x and y, 00:10:39.160 --> 00:10:41.890 align:middle line:84% this ordering will tell me whether x 00:10:41.890 --> 00:10:47.530 align:middle line:84% is weakly preferred or at least as good as y in the consumer's 00:10:47.530 --> 00:10:48.550 align:middle line:90% ordering. 00:10:48.550 --> 00:10:52.450 align:middle line:84% From that primitive, we can create strict preference 00:10:52.450 --> 00:10:56.200 align:middle line:90% for any two goods, x and y. 00:10:56.200 --> 00:10:58.690 align:middle line:84% If x is strictly preferred to y, that 00:10:58.690 --> 00:11:01.660 align:middle line:84% means that x is at least as good as y, 00:11:01.660 --> 00:11:05.260 align:middle line:84% but y is not at least as good as x. 00:11:05.260 --> 00:11:07.720 align:middle line:84% Indifference can also be constructed 00:11:07.720 --> 00:11:09.100 align:middle line:90% from this primitive. 00:11:09.100 --> 00:11:12.500 align:middle line:84% Namely, if x is indifferent to y, 00:11:12.500 --> 00:11:17.920 align:middle line:84% then x is at least as good as y, and y is at least as good as x. 00:11:17.920 --> 00:11:20.890 align:middle line:84% This preference relation then represents the preferences 00:11:20.890 --> 00:11:23.570 align:middle line:84% of each individual-- in this case, 00:11:23.570 --> 00:11:25.700 align:middle line:84% as I said, we've deleted individual i. 00:11:25.700 --> 00:11:28.120 align:middle line:84% This would be true for all of them. 00:11:28.120 --> 00:11:29.890 align:middle line:84% Preference relations obviously can 00:11:29.890 --> 00:11:33.120 align:middle line:84% be different across different households. 00:11:33.120 --> 00:11:35.190 align:middle line:90% What do we mean by "rational?" 00:11:35.190 --> 00:11:40.590 align:middle line:84% Rational choice means that those preferences are complete, 00:11:40.590 --> 00:11:42.720 align:middle line:90% transitive, and continuous. 00:11:42.720 --> 00:11:45.810 align:middle line:84% And let me just go over them briefly. 00:11:45.810 --> 00:11:49.110 align:middle line:84% "Completeness" means that any two bundles x and y 00:11:49.110 --> 00:11:52.380 align:middle line:84% are ranked relative to each other. 00:11:52.380 --> 00:11:55.950 align:middle line:84% So given x and y, either x is strictly preferred to y, 00:11:55.950 --> 00:11:58.620 align:middle line:84% y is strictly preferred to x, or the household 00:11:58.620 --> 00:12:00.810 align:middle line:90% is indifferent between the two. 00:12:00.810 --> 00:12:04.650 align:middle line:84% And the point here is that any two bundles x and y 00:12:04.650 --> 00:12:06.990 align:middle line:84% can be ranked and compared to each other. 00:12:06.990 --> 00:12:09.060 align:middle line:90% It's a complete ordering. 00:12:09.060 --> 00:12:11.880 align:middle line:84% Transitive-- if x is at least as good 00:12:11.880 --> 00:12:14.340 align:middle line:84% as y, y is at least as good as z, 00:12:14.340 --> 00:12:18.270 align:middle line:84% then it must also be true that x is at least as good as z. 00:12:18.270 --> 00:12:20.580 align:middle line:90% So we will use this later. 00:12:20.580 --> 00:12:23.580 align:middle line:84% It's like an internal consistency criterion. 00:12:23.580 --> 00:12:28.230 align:middle line:84% You may only see choices between x and y and y and z, 00:12:28.230 --> 00:12:30.270 align:middle line:84% but if the consumer is rational, you 00:12:30.270 --> 00:12:32.430 align:middle line:84% can infer something about the way 00:12:32.430 --> 00:12:34.290 align:middle line:84% they would behave when they're faced 00:12:34.290 --> 00:12:36.780 align:middle line:84% with a choice between x and z that we actually 00:12:36.780 --> 00:12:38.850 align:middle line:90% didn't see them make. 00:12:38.850 --> 00:12:41.910 align:middle line:84% And finally, we have continuity, which intuitively 00:12:41.910 --> 00:12:44.730 align:middle line:84% means if we have x strictly preferred to y, 00:12:44.730 --> 00:12:48.120 align:middle line:84% and we have some other bundle x prime which 00:12:48.120 --> 00:12:51.810 align:middle line:84% is close to x, since x and x prime are close to each other 00:12:51.810 --> 00:12:55.080 align:middle line:84% and x is preferred to y, then x prime is also 00:12:55.080 --> 00:12:56.875 align:middle line:90% going to be preferred to y. 00:12:56.875 --> 00:12:58.500 align:middle line:84% In other words, the preference ordering 00:12:58.500 --> 00:13:02.130 align:middle line:84% isn't jumping around hugely as we move around the commodity 00:13:02.130 --> 00:13:03.060 align:middle line:90% space. 00:13:03.060 --> 00:13:06.420 align:middle line:84% So we start with this underlying preference ordering, 00:13:06.420 --> 00:13:11.040 align:middle line:84% but we can do a representation as a utility function 00:13:11.040 --> 00:13:13.350 align:middle line:84% just by preserving the rank ordering. 00:13:13.350 --> 00:13:16.990 align:middle line:84% So for example, if x is at least as good as y, 00:13:16.990 --> 00:13:19.500 align:middle line:84% then we come up with the utility function u, 00:13:19.500 --> 00:13:21.180 align:middle line:84% which maps the entire consumption 00:13:21.180 --> 00:13:23.340 align:middle line:84% set into real numbers in such a way 00:13:23.340 --> 00:13:25.770 align:middle line:84% that we give at least as high of a number 00:13:25.770 --> 00:13:29.410 align:middle line:90% to x as we assign under u to y. 00:13:29.410 --> 00:13:34.470 align:middle line:84% One thing to note is that many utility functions would work. 00:13:34.470 --> 00:13:36.930 align:middle line:84% If we start with this primitive, we 00:13:36.930 --> 00:13:41.610 align:middle line:84% could have 2 times u or some multiplicative function of u. 00:13:41.610 --> 00:13:44.100 align:middle line:84% It's going to preserve the ranked utility 00:13:44.100 --> 00:13:46.050 align:middle line:90% numbers across the bundles. 00:13:46.050 --> 00:13:48.750 align:middle line:84% So this is referred to as an ordinal rather than 00:13:48.750 --> 00:13:50.490 align:middle line:90% a cardinal utility function. 00:13:50.490 --> 00:13:53.160 align:middle line:84% In the history of economics, utility functions 00:13:53.160 --> 00:13:57.510 align:middle line:84% were invented by Jeremy Bentham in England. 00:13:57.510 --> 00:14:01.210 align:middle line:84% Jeremy left in his will that after he died, 00:14:01.210 --> 00:14:04.020 align:middle line:84% he wanted to be embalmed and preserved, 00:14:04.020 --> 00:14:07.170 align:middle line:84% and that once a year he should have tea with his friends. 00:14:07.170 --> 00:14:09.945 align:middle line:84% So Jeremy still-- or at least when I was last there-- 00:14:09.945 --> 00:14:14.970 align:middle line:84% is sitting in kind of a glass container in the lobby of one 00:14:14.970 --> 00:14:17.910 align:middle line:84% of the buildings at University of College of London. 00:14:17.910 --> 00:14:19.140 align:middle line:90% There was some vandalism. 00:14:19.140 --> 00:14:21.390 align:middle line:84% Someone removed his head at one point. 00:14:21.390 --> 00:14:23.190 align:middle line:84% But anyway, I always think of Jeremy 00:14:23.190 --> 00:14:25.800 align:middle line:84% when I think of utility functions. 00:14:25.800 --> 00:14:27.840 align:middle line:84% And forgive my morbid sense of humor. 00:14:27.840 --> 00:14:30.480 align:middle line:84% Here's a theorem-- any preference relation that 00:14:30.480 --> 00:14:33.420 align:middle line:84% can be represented by a continuous utility function 00:14:33.420 --> 00:14:34.590 align:middle line:90% is rational. 00:14:34.590 --> 00:14:36.930 align:middle line:90% OK, that's a lot to take in. 00:14:36.930 --> 00:14:39.390 align:middle line:84% Any preference relation that can be 00:14:39.390 --> 00:14:43.950 align:middle line:84% represented as a continuous utility function is rational. 00:14:43.950 --> 00:14:49.590 align:middle line:84% Well, "completeness" means we're assigning utility numbers 00:14:49.590 --> 00:14:53.580 align:middle line:84% to any bundle and any pair of bundles. 00:14:53.580 --> 00:14:56.560 align:middle line:84% So pick two, look at their utility numbers, 00:14:56.560 --> 00:14:59.070 align:middle line:84% and we have the ranking implied by the utility. 00:14:59.070 --> 00:15:01.690 align:middle line:84% Some utility numbers are higher than others. 00:15:01.690 --> 00:15:02.850 align:middle line:90% So we can rank everything. 00:15:02.850 --> 00:15:04.710 align:middle line:84% That's why when we have a utility function, 00:15:04.710 --> 00:15:06.900 align:middle line:90% the ranking is complete. 00:15:06.900 --> 00:15:11.610 align:middle line:84% The ranking is transitive because if the utility number 00:15:11.610 --> 00:15:15.240 align:middle line:84% of a is greater than or equal to the utility number of b, 00:15:15.240 --> 00:15:17.880 align:middle line:84% and b's utility number is higher than c, 00:15:17.880 --> 00:15:23.980 align:middle line:84% then we have transitivity under the ordinary weak inequality. 00:15:23.980 --> 00:15:27.150 align:middle line:84% So the one implies the other, essentially. 00:15:27.150 --> 00:15:29.805 align:middle line:84% In this case, we're going from the existence of-- oh, 00:15:29.805 --> 00:15:32.610 align:middle line:84% and we're given continuous, which 00:15:32.610 --> 00:15:35.790 align:middle line:84% was the third property of a rational preference ordering. 00:15:35.790 --> 00:15:37.590 align:middle line:84% So that's embedded in the utility. 00:15:37.590 --> 00:15:40.950 align:middle line:84% So obviously, two bundles which are close to each other 00:15:40.950 --> 00:15:44.750 align:middle line:84% are going to have very close utility values. 00:15:44.750 --> 00:15:47.990 align:middle line:84% There's something very deceptive about this lecture 00:15:47.990 --> 00:15:52.850 align:middle line:84% because no slide is particularly hard, I think. 00:15:52.850 --> 00:15:55.580 align:middle line:84% Of course you should alert me if otherwise. 00:15:55.580 --> 00:15:57.920 align:middle line:84% But yet somehow, when we go through all of them, 00:15:57.920 --> 00:16:01.530 align:middle line:84% you'll be empowered with the tools of consumer theory. 00:16:01.530 --> 00:16:05.150 align:middle line:84% So it is definitely worth doing this well. 00:16:05.150 --> 00:16:08.090 align:middle line:84% Local non-satiation is another property. 00:16:08.090 --> 00:16:11.690 align:middle line:84% It says that if you're given an x in the consumption set 00:16:11.690 --> 00:16:14.340 align:middle line:84% and an arbitrary small number epsilon, 00:16:14.340 --> 00:16:18.530 align:middle line:84% there exists some other element y in the consumption set 00:16:18.530 --> 00:16:22.670 align:middle line:84% so that when we're within an epsilon neighborhood of x, 00:16:22.670 --> 00:16:25.010 align:middle line:90% we find this y. 00:16:25.010 --> 00:16:27.170 align:middle line:90% And y is preferred to x. 00:16:27.170 --> 00:16:29.900 align:middle line:84% In other words, it's a little weaker than global, 00:16:29.900 --> 00:16:31.190 align:middle line:90% which is on the next slide. 00:16:31.190 --> 00:16:34.370 align:middle line:84% Local just means you start with x, draw a circle 00:16:34.370 --> 00:16:36.950 align:middle line:90% around it in Euclidean space-- 00:16:36.950 --> 00:16:41.330 align:middle line:84% two-dimensional, or a sphere, or in higher dimensional objects. 00:16:41.330 --> 00:16:42.740 align:middle line:84% And you can always find something 00:16:42.740 --> 00:16:47.810 align:middle line:84% in that area, however small, that is strictly preferred 00:16:47.810 --> 00:16:50.030 align:middle line:90% to the starting point x. 00:16:50.030 --> 00:16:53.270 align:middle line:84% Eventually, we'll get to competitive equilibria 00:16:53.270 --> 00:16:55.640 align:middle line:90% or entire economies. 00:16:55.640 --> 00:16:57.830 align:middle line:84% Are competitive equilibria optimal? 00:16:57.830 --> 00:17:02.630 align:middle line:84% The answer may be no if we lose this local non-satiation. 00:17:02.630 --> 00:17:04.910 align:middle line:90% So it will sneak back up on us. 00:17:04.910 --> 00:17:07.040 align:middle line:84% In other words, there's no bliss point. 00:17:07.040 --> 00:17:09.410 align:middle line:84% This is the dismal view of economics, 00:17:09.410 --> 00:17:12.520 align:middle line:84% I suppose, that more is always preferred to less, 00:17:12.520 --> 00:17:15.890 align:middle line:84% and people just can't get enough. 00:17:15.890 --> 00:17:21.050 align:middle line:84% Here's a more global view of it, which is with two goods. 00:17:21.050 --> 00:17:23.960 align:middle line:90% We start with this bundle here. 00:17:23.960 --> 00:17:28.790 align:middle line:84% This whole hatched-in area has more of one 00:17:28.790 --> 00:17:30.440 align:middle line:90% or the other or both goods. 00:17:30.440 --> 00:17:35.210 align:middle line:84% And it's the weakly or strictly preferred set relative to that. 00:17:35.210 --> 00:17:37.640 align:middle line:84% In fact, if they value both goods, 00:17:37.640 --> 00:17:39.453 align:middle line:84% then even if you keep one constant, 00:17:39.453 --> 00:17:40.620 align:middle line:90% you're increasing the other. 00:17:40.620 --> 00:17:42.380 align:middle line:84% So it's a strictly preferred set. 00:17:42.380 --> 00:17:44.780 align:middle line:84% This, which has less of one or the other 00:17:44.780 --> 00:17:48.860 align:middle line:84% of both relative to the starting point, is the worse-than set. 00:17:48.860 --> 00:17:53.160 align:middle line:84% So back to rank ordering, we start here and say, 00:17:53.160 --> 00:17:54.510 align:middle line:90% what about a bundle here? 00:17:54.510 --> 00:17:56.210 align:middle line:90% Well, that's strictly preferred. 00:17:56.210 --> 00:17:59.060 align:middle line:84% Start here, what about going southwest? 00:17:59.060 --> 00:18:01.370 align:middle line:84% Then the original bundle is strictly preferred. 00:18:01.370 --> 00:18:05.060 align:middle line:84% But we leave some question marks in these off-diagonal elements, 00:18:05.060 --> 00:18:05.780 align:middle line:90% essentially. 00:18:05.780 --> 00:18:08.460 align:middle line:84% Because it has more of one good and less than another. 00:18:08.460 --> 00:18:10.820 align:middle line:84% So a priori we're not able to say 00:18:10.820 --> 00:18:15.210 align:middle line:84% what the consumer would choose without providing more details. 00:18:15.210 --> 00:18:19.640 align:middle line:84% But we can provide those details if we go back to, say, 00:18:19.640 --> 00:18:23.070 align:middle line:84% the representation of the utility function. 00:18:23.070 --> 00:18:27.440 align:middle line:84% So we pick a bundle, and it's associated 00:18:27.440 --> 00:18:30.200 align:middle line:90% with a given level of utility. 00:18:30.200 --> 00:18:32.540 align:middle line:90% Remember Jeremy Bentham. 00:18:32.540 --> 00:18:38.780 align:middle line:84% And then we ask what other values in this consumption set 00:18:38.780 --> 00:18:43.390 align:middle line:84% would generate the same utility level as the starting point? 00:18:43.390 --> 00:18:48.960 align:middle line:84% And that's the entire set of bundles on this curve. 00:18:48.960 --> 00:18:51.260 align:middle line:84% So this is called an "indifference curve," 00:18:51.260 --> 00:18:54.420 align:middle line:84% in the sense that these two bundles-- 00:18:54.420 --> 00:18:57.330 align:middle line:90% x1, y1 and x2, y2-- 00:18:57.330 --> 00:19:01.500 align:middle line:84% under the consumer's preferences are indifferent to each other. 00:19:01.500 --> 00:19:03.450 align:middle line:84% It might have been better to call it 00:19:03.450 --> 00:19:07.560 align:middle line:84% an "isoutility curve," because that language is even closer 00:19:07.560 --> 00:19:09.000 align:middle line:90% to what it is. 00:19:09.000 --> 00:19:12.750 align:middle line:84% And later in the class, when we get to production functions, 00:19:12.750 --> 00:19:14.400 align:middle line:90% we can talk about isoquants. 00:19:14.400 --> 00:19:18.240 align:middle line:84% And later we can talk about isocosts. 00:19:18.240 --> 00:19:20.760 align:middle line:84% And this is isoutility-- same utility. 00:19:20.760 --> 00:19:25.680 align:middle line:84% Or traditionally, it's called the "indifference curve." 00:19:25.680 --> 00:19:28.670 align:middle line:84% Now, this is drawn with a certain diminishing slope, 00:19:28.670 --> 00:19:31.520 align:middle line:84% and I'll come back to that momentarily. 00:19:31.520 --> 00:19:35.560 align:middle line:84% Well, of course, there's more than one curve. 00:19:35.560 --> 00:19:38.230 align:middle line:84% Because under a utility function, it's complete. 00:19:38.230 --> 00:19:39.730 align:middle line:90% So everything has a ranking. 00:19:39.730 --> 00:19:42.130 align:middle line:90% Let's jump up here-- 00:19:42.130 --> 00:19:43.770 align:middle line:90% utility levels are higher. 00:19:43.770 --> 00:19:46.410 align:middle line:90% Jump up here, higher still. 00:19:46.410 --> 00:19:49.620 align:middle line:84% So as you move northeast, we're moving in the direction 00:19:49.620 --> 00:19:53.260 align:middle line:90% of strictly increasing utility. 00:19:53.260 --> 00:19:57.340 align:middle line:84% In fact, stuff is going on between these curves, too. 00:19:57.340 --> 00:20:01.870 align:middle line:84% It's just utility levels that are between U1 and U2. 00:20:01.870 --> 00:20:03.760 align:middle line:84% So you typically graphically just 00:20:03.760 --> 00:20:06.730 align:middle line:90% draw one or a finite number. 00:20:06.730 --> 00:20:08.470 align:middle line:84% We don't try to fill the whole thing out 00:20:08.470 --> 00:20:11.760 align:middle line:84% because then it would just go black. 00:20:11.760 --> 00:20:14.130 align:middle line:90% Let's try this diagram. 00:20:14.130 --> 00:20:18.810 align:middle line:84% This is indifference curves that are crossing. 00:20:18.810 --> 00:20:26.310 align:middle line:84% Note that A is northeast of B, has more of both goods X and Y. 00:20:26.310 --> 00:20:30.420 align:middle line:84% So A must be strictly preferred to B. 00:20:30.420 --> 00:20:35.220 align:middle line:84% However, A is on the same indifference curve as E, 00:20:35.220 --> 00:20:37.800 align:middle line:84% the indifference curve labeled U1. 00:20:37.800 --> 00:20:41.730 align:middle line:84% And E is on the same indifference curve as B, 00:20:41.730 --> 00:20:45.280 align:middle line:90% the indifference curve U2. 00:20:45.280 --> 00:20:47.380 align:middle line:84% Under transitivity, if the consumer 00:20:47.380 --> 00:20:51.130 align:middle line:84% is indifferent between A and E and indifferent between E 00:20:51.130 --> 00:20:54.700 align:middle line:84% and B, then by transitivity, the consumer 00:20:54.700 --> 00:20:56.380 align:middle line:84% should be indifferent between A and B. 00:20:56.380 --> 00:20:59.110 align:middle line:84% But we started out by saying that the consumer strictly 00:20:59.110 --> 00:21:03.310 align:middle line:84% prefers A to B. So we end up with a contradiction. 00:21:03.310 --> 00:21:05.905 align:middle line:84% Hence, indifference curves can never cross. 00:21:05.905 --> 00:21:09.260 align:middle line:90% 00:21:09.260 --> 00:21:12.720 align:middle line:90% So let's come back to the slope. 00:21:12.720 --> 00:21:15.980 align:middle line:84% Let's define the marginal rate of substitution, 00:21:15.980 --> 00:21:18.440 align:middle line:84% the willingness to give up one good for another, 00:21:18.440 --> 00:21:23.090 align:middle line:84% as the negative slope of the indifference curve. 00:21:23.090 --> 00:21:29.260 align:middle line:84% So that is, the marginal rate of substitution, is how much of Y 00:21:29.260 --> 00:21:32.350 align:middle line:84% you have to get when you give up 1 unit of X 00:21:32.350 --> 00:21:35.470 align:middle line:84% while keeping utility constant at U1. 00:21:35.470 --> 00:21:39.430 align:middle line:84% So that curve and this number tell 00:21:39.430 --> 00:21:42.400 align:middle line:84% us something about trade-offs that a household would 00:21:42.400 --> 00:21:45.550 align:middle line:84% willingly do to be indifferent, or potentially, 00:21:45.550 --> 00:21:48.650 align:middle line:90% later, to even gain. 00:21:48.650 --> 00:21:52.510 align:middle line:84% So if you go back to this slide, this 00:21:52.510 --> 00:21:56.080 align:middle line:84% is a diminishing marginal rate of substitution. 00:21:56.080 --> 00:21:58.690 align:middle line:84% The slope is getting less and less. 00:21:58.690 --> 00:22:01.840 align:middle line:84% It's obviously negative because when you get more of one good, 00:22:01.840 --> 00:22:03.340 align:middle line:84% you're willing to give up the other. 00:22:03.340 --> 00:22:04.990 align:middle line:90% The issue is how much. 00:22:04.990 --> 00:22:08.200 align:middle line:84% If you start off with a modest amount of X1 00:22:08.200 --> 00:22:11.620 align:middle line:84% and a relatively large amount of Y1, 00:22:11.620 --> 00:22:15.310 align:middle line:84% and you move down this curve, the marginal rate 00:22:15.310 --> 00:22:17.950 align:middle line:84% of substitution would be this slope. 00:22:17.950 --> 00:22:25.450 align:middle line:84% But if you gave the household more of X and less of Y, 00:22:25.450 --> 00:22:28.450 align:middle line:84% the slope becomes less and less because here, 00:22:28.450 --> 00:22:32.530 align:middle line:84% as you move pointwise down, you have more and more X. 00:22:32.530 --> 00:22:35.290 align:middle line:84% But X is less and less valued because you 00:22:35.290 --> 00:22:36.700 align:middle line:90% have more and more of it. 00:22:36.700 --> 00:22:41.860 align:middle line:84% You have less and less Y, so Y is more and more valued 00:22:41.860 --> 00:22:44.240 align:middle line:90% as you give up more of it. 00:22:44.240 --> 00:22:47.560 align:middle line:84% And so to make this movement [INAUDIBLE],, 00:22:47.560 --> 00:22:52.660 align:middle line:84% you require more and more X as you surrender a given unit of Y 00:22:52.660 --> 00:22:54.520 align:middle line:84% in order to maintain the indifference 00:22:54.520 --> 00:22:56.630 align:middle line:90% on the indifference curve. 00:22:56.630 --> 00:22:59.460 align:middle line:84% I mean, down here, it's almost flat. 00:22:59.460 --> 00:23:03.630 align:middle line:84% You just have so much of good X that variation of it 00:23:03.630 --> 00:23:05.580 align:middle line:84% doesn't change utility very much. 00:23:05.580 --> 00:23:11.940 align:middle line:84% So you don't have to change Y. And here, it's almost vertical. 00:23:11.940 --> 00:23:14.490 align:middle line:84% So we normally assume the marginal rate of substitution 00:23:14.490 --> 00:23:16.950 align:middle line:84% is decreasing, just as in that diagram. 00:23:16.950 --> 00:23:21.190 align:middle line:84% And the intuition is they want things to be balanced. 00:23:21.190 --> 00:23:23.280 align:middle line:84% And then we're about to get to notions 00:23:23.280 --> 00:23:28.330 align:middle line:84% of convexity and concavity, but I'll wait for that slide. 00:23:28.330 --> 00:23:29.820 align:middle line:84% One thing I want to say right away 00:23:29.820 --> 00:23:32.010 align:middle line:84% is that the marginal rate of substitution 00:23:32.010 --> 00:23:34.560 align:middle line:90% doesn't have to be decreasing. 00:23:34.560 --> 00:23:38.370 align:middle line:84% If indifference curves are linear, as they are here, 00:23:38.370 --> 00:23:40.770 align:middle line:90% then the slope is the same. 00:23:40.770 --> 00:23:43.030 align:middle line:90% It's a line, after all. 00:23:43.030 --> 00:23:44.670 align:middle line:84% So the marginal rate of substitution 00:23:44.670 --> 00:23:47.580 align:middle line:84% is not diminishing, it's constant. 00:23:47.580 --> 00:23:52.670 align:middle line:84% And here, we have our classic case, 00:23:52.670 --> 00:23:54.970 align:middle line:84% which is the way they're typically drawn. 00:23:54.970 --> 00:23:58.390 align:middle line:84% Here, it's the opposite, which is 00:23:58.390 --> 00:24:02.020 align:middle line:84% if you start at this point of the L 00:24:02.020 --> 00:24:04.780 align:middle line:84% and move in the direction of having more of good X 00:24:04.780 --> 00:24:07.810 align:middle line:90% you would gain no extra utility. 00:24:07.810 --> 00:24:13.540 align:middle line:84% Likewise, if you started at this point and had more Y, 00:24:13.540 --> 00:24:15.970 align:middle line:90% you would gain no extra utility. 00:24:15.970 --> 00:24:19.450 align:middle line:84% The only way to achieve higher levels of utility going 00:24:19.450 --> 00:24:22.960 align:middle line:84% from U0 to U1, et cetera, is to increase 00:24:22.960 --> 00:24:27.190 align:middle line:84% both, and in the exact same proportions. 00:24:27.190 --> 00:24:29.620 align:middle line:84% So these are sometimes called Leontiefian indifference 00:24:29.620 --> 00:24:32.440 align:middle line:84% curves, although he really created them in production, 00:24:32.440 --> 00:24:35.330 align:middle line:84% and we'll get to that in two lectures. 00:24:35.330 --> 00:24:37.570 align:middle line:84% So the degree of substitutability-- 00:24:37.570 --> 00:24:42.580 align:middle line:84% these are very curved, these are linear, these are kinked-- 00:24:42.580 --> 00:24:45.460 align:middle line:84% is something called the "elasticity of substitution." 00:24:45.460 --> 00:24:49.480 align:middle line:84% So these are constant elasticity of substitution curves 00:24:49.480 --> 00:24:55.580 align:middle line:84% that are parameterized by the degree of curvature. 00:24:55.580 --> 00:24:58.270 align:middle line:84% And here are some particular functions. 00:24:58.270 --> 00:25:00.040 align:middle line:90% This one's called Cobb-Douglas. 00:25:00.040 --> 00:25:05.110 align:middle line:84% He was an Economist at Illinois, or Douglas was. 00:25:05.110 --> 00:25:10.250 align:middle line:84% Senator Douglas-- I think he was a Senator from Illinois when 00:25:10.250 --> 00:25:12.340 align:middle line:90% it was an agricultural state. 00:25:12.340 --> 00:25:15.260 align:middle line:84% I'd have to check that to be sure. 00:25:15.260 --> 00:25:18.830 align:middle line:84% So x to the power alpha, y to the power of beta-- 00:25:18.830 --> 00:25:25.220 align:middle line:84% perfect complements, which is this one is here, 00:25:25.220 --> 00:25:28.565 align:middle line:84% you can see the ratio I was referring to. 00:25:28.565 --> 00:25:33.050 align:middle line:84% To maintain utility constant, the ratio of y to x 00:25:33.050 --> 00:25:34.910 align:middle line:90% has to be alpha over beta. 00:25:34.910 --> 00:25:39.340 align:middle line:84% Perfect substitutes, linear, constant elasticity 00:25:39.340 --> 00:25:42.040 align:middle line:90% of substitution, this function. 00:25:42.040 --> 00:25:45.780 align:middle line:84% And we will use these for numerical examples, 00:25:45.780 --> 00:25:47.985 align:middle line:84% and p sets, and so on, from time to time. 00:25:47.985 --> 00:25:51.120 align:middle line:90% 00:25:51.120 --> 00:25:54.930 align:middle line:84% OK, now we get to convexity, and quasi-convexity, and all 00:25:54.930 --> 00:25:56.160 align:middle line:90% of that stuff. 00:25:56.160 --> 00:26:00.800 align:middle line:84% So a set X, like a consumption set, 00:26:00.800 --> 00:26:06.320 align:middle line:84% is convex if whenever you have two separate bundles x and x 00:26:06.320 --> 00:26:10.700 align:middle line:84% prime in the set, and you take a linear combination of those two 00:26:10.700 --> 00:26:13.310 align:middle line:90% points with weights alpha-- 00:26:13.310 --> 00:26:16.040 align:middle line:90% say alpha between 0 and 1-- 00:26:16.040 --> 00:26:18.770 align:middle line:84% then this new point is also in the set. 00:26:18.770 --> 00:26:22.450 align:middle line:84% So it's a standard definition of a convex set. 00:26:22.450 --> 00:26:26.530 align:middle line:84% A function is said to be convex if whenever 00:26:26.530 --> 00:26:30.940 align:middle line:84% you have two points x prime and y prime, and x and x prime 00:26:30.940 --> 00:26:36.100 align:middle line:84% and you evaluate the function f, then the weighted average 00:26:36.100 --> 00:26:40.090 align:middle line:84% of those two points over here evaluated under f 00:26:40.090 --> 00:26:43.000 align:middle line:84% is less than the weighted average 00:26:43.000 --> 00:26:46.060 align:middle line:90% in the range of the function. 00:26:46.060 --> 00:26:50.290 align:middle line:84% So that's a definition of a convex function. 00:26:50.290 --> 00:26:55.350 align:middle line:84% The set of points on or above the graph of a convex function 00:26:55.350 --> 00:26:56.680 align:middle line:90% is convex. 00:26:56.680 --> 00:26:59.710 align:middle line:84% So that's kind of linking these two definitions together. 00:26:59.710 --> 00:27:01.850 align:middle line:84% I'll show you the picture momentarily. 00:27:01.850 --> 00:27:05.460 align:middle line:84% A convex function and negative of a convex function 00:27:05.460 --> 00:27:06.900 align:middle line:90% is concave. 00:27:06.900 --> 00:27:10.140 align:middle line:84% To define quasi-convex-- a function 00:27:10.140 --> 00:27:15.600 align:middle line:84% f is not simply convex but weaker quasi-convex 00:27:15.600 --> 00:27:19.320 align:middle line:84% if whenever you evaluate the two end points, and x, 00:27:19.320 --> 00:27:22.200 align:middle line:84% and x prime, and take the max of the two, 00:27:22.200 --> 00:27:25.380 align:middle line:84% it's no less than the function evaluated 00:27:25.380 --> 00:27:26.700 align:middle line:90% at the weighted average. 00:27:26.700 --> 00:27:30.900 align:middle line:84% And likewise, the negative of a quasi-convex function 00:27:30.900 --> 00:27:33.540 align:middle line:90% is said to be quasi-concave. 00:27:33.540 --> 00:27:35.970 align:middle line:84% So what are we going to use here? 00:27:35.970 --> 00:27:37.650 align:middle line:84% I'll show you a picture where we'll 00:27:37.650 --> 00:27:41.970 align:middle line:84% display convexity of a set, convexity of a function, 00:27:41.970 --> 00:27:45.590 align:middle line:90% and the use of quasi-concavity. 00:27:45.590 --> 00:27:48.790 align:middle line:84% So this is the one that is consistent 00:27:48.790 --> 00:27:49.780 align:middle line:90% with the definitions. 00:27:49.780 --> 00:27:51.640 align:middle line:90% This is the one that is not. 00:27:51.640 --> 00:27:58.240 align:middle line:84% So this indifference curve is a curve in the space of x and y. 00:27:58.240 --> 00:28:02.200 align:middle line:84% And it's convex, actually strictly convex. 00:28:02.200 --> 00:28:05.410 align:middle line:84% The set of points with utility greater than 00:28:05.410 --> 00:28:08.380 align:middle line:84% or equal to the utility associated 00:28:08.380 --> 00:28:12.430 align:middle line:84% with this indifference curve is this hatched area. 00:28:12.430 --> 00:28:14.120 align:middle line:90% And this is a convex set. 00:28:14.120 --> 00:28:15.340 align:middle line:90% You pick any two points-- 00:28:15.340 --> 00:28:17.740 align:middle line:84% for example, this one and this one-- 00:28:17.740 --> 00:28:20.290 align:middle line:84% and take a weighted average, and you'll still 00:28:20.290 --> 00:28:23.980 align:middle line:84% have a utility level which is greater than the utility is 00:28:23.980 --> 00:28:25.720 align:middle line:90% associated with U1. 00:28:25.720 --> 00:28:29.830 align:middle line:84% If you looked at the definition of quasi-concave utility 00:28:29.830 --> 00:28:33.100 align:middle line:84% function, we take the utility function which 00:28:33.100 --> 00:28:35.830 align:middle line:84% is generating these indifference curves, and say, 00:28:35.830 --> 00:28:38.920 align:middle line:84% what's the utility at this point under that utility function? 00:28:38.920 --> 00:28:42.160 align:middle line:84% What's the utility at this point under that utility function? 00:28:42.160 --> 00:28:45.670 align:middle line:84% Take the max of those two, and then 00:28:45.670 --> 00:28:49.540 align:middle line:84% show that the weighted average on the line 00:28:49.540 --> 00:28:52.130 align:middle line:84% gives a strictly higher utility level. 00:28:52.130 --> 00:29:01.330 align:middle line:84% So quasi-concave function reverses the inequality here. 00:29:01.330 --> 00:29:04.770 align:middle line:84% So the max of the utility at each of the endpoints 00:29:04.770 --> 00:29:09.340 align:middle line:84% is less than the utility at the weighted average. 00:29:09.340 --> 00:29:13.240 align:middle line:84% So that is tedious, and it's a bit hard to absorb, 00:29:13.240 --> 00:29:17.682 align:middle line:84% especially if you're not used to using all of these things. 00:29:17.682 --> 00:29:19.390 align:middle line:84% Anyway, this is the good-looking picture, 00:29:19.390 --> 00:29:23.710 align:middle line:84% and this one is ugly in the sense that it violates every 00:29:23.710 --> 00:29:24.700 align:middle line:90% single-- 00:29:24.700 --> 00:29:28.270 align:middle line:84% this property, this utility indifference curve 00:29:28.270 --> 00:29:30.070 align:middle line:90% is not convex. 00:29:30.070 --> 00:29:32.860 align:middle line:90% It has some concave portion. 00:29:32.860 --> 00:29:36.310 align:middle line:84% The set of bundles on or above the indifference curve, 00:29:36.310 --> 00:29:38.170 align:middle line:84% when you take weighted averages of them, 00:29:38.170 --> 00:29:40.720 align:middle line:84% you would be below the indifference curve, 00:29:40.720 --> 00:29:41.960 align:middle line:90% violating the property. 00:29:41.960 --> 00:29:44.650 align:middle line:84% And similarly, if you take this point and this point, 00:29:44.650 --> 00:29:46.990 align:middle line:84% take the max of the two, and take a weighted average, 00:29:46.990 --> 00:29:51.030 align:middle line:84% the utility is actually now less than, not greater. 00:29:51.030 --> 00:29:54.660 align:middle line:84% That quasi-convexity or concavity of utility 00:29:54.660 --> 00:29:56.170 align:middle line:90% is a bit surprising. 00:29:56.170 --> 00:30:00.660 align:middle line:84% It has to do with the ordinal nature of the utility function. 00:30:00.660 --> 00:30:03.030 align:middle line:84% We don't want to say a function is necessarily 00:30:03.030 --> 00:30:07.080 align:middle line:84% strictly concave, although that would imply quasi-concavity. 00:30:07.080 --> 00:30:09.750 align:middle line:84% But we could take another transformation that 00:30:09.750 --> 00:30:12.450 align:middle line:84% removes the strict concavity, and it could still 00:30:12.450 --> 00:30:13.860 align:middle line:90% be quasi-concavity. 00:30:13.860 --> 00:30:17.700 align:middle line:84% It's clear by plotting in the choice space 00:30:17.700 --> 00:30:20.670 align:middle line:84% in this way, that quasi-concavity 00:30:20.670 --> 00:30:23.280 align:middle line:84% is what we want, although we haven't really 00:30:23.280 --> 00:30:24.205 align:middle line:90% talked about choices. 00:30:24.205 --> 00:30:25.830 align:middle line:84% We're just talking about rank ordering. 00:30:25.830 --> 00:30:28.730 align:middle line:90% 00:30:28.730 --> 00:30:33.020 align:middle line:84% So marginal rate of substitution defined a bit more 00:30:33.020 --> 00:30:35.000 align:middle line:84% mathematically-- we have a utility 00:30:35.000 --> 00:30:37.430 align:middle line:90% function over x1 and x2. 00:30:37.430 --> 00:30:40.760 align:middle line:84% We can call the marginal utility the derivative 00:30:40.760 --> 00:30:45.440 align:middle line:84% of the utility with respect to x1 or utility of x1, 00:30:45.440 --> 00:30:47.540 align:middle line:90% likewise for x2. 00:30:47.540 --> 00:30:51.500 align:middle line:84% Now, if we're moving along an indifference curve, 00:30:51.500 --> 00:30:55.220 align:middle line:84% we're changing the quantities of x and y, 00:30:55.220 --> 00:30:58.300 align:middle line:84% but we're keeping utility constant. 00:30:58.300 --> 00:31:01.920 align:middle line:84% So dU, or the total differentiable of utility, 00:31:01.920 --> 00:31:05.770 align:middle line:84% does not change as we move along an indifference curve. 00:31:05.770 --> 00:31:09.960 align:middle line:84% So dU here would be the derivative of utility 00:31:09.960 --> 00:31:14.010 align:middle line:84% with respect to X1 times dX1 plus the utility with respect 00:31:14.010 --> 00:31:18.090 align:middle line:84% to marginal utility with respect to X2 times dX2. 00:31:18.090 --> 00:31:19.390 align:middle line:90% That's this expression. 00:31:19.390 --> 00:31:22.350 align:middle line:84% This is going to be 0 by definition of moving along 00:31:22.350 --> 00:31:24.180 align:middle line:90% an indifference curve. 00:31:24.180 --> 00:31:27.110 align:middle line:84% Since it's equal to 0, we can solve for dX2. 00:31:27.110 --> 00:31:30.180 align:middle line:84% I think you can visualize what happens here. 00:31:30.180 --> 00:31:35.280 align:middle line:84% dX2 divided by dX1 is equal to the ratio 00:31:35.280 --> 00:31:37.870 align:middle line:90% of these marginal utilities. 00:31:37.870 --> 00:31:39.640 align:middle line:84% Rather than derive this every time, 00:31:39.640 --> 00:31:42.250 align:middle line:84% although it's a bit dangerous to memorize things, 00:31:42.250 --> 00:31:44.410 align:middle line:84% basically in that diagram, we have 00:31:44.410 --> 00:31:47.410 align:middle line:84% x1 on the x-axis, x2 on the y-axis. 00:31:47.410 --> 00:31:49.240 align:middle line:84% And the marginal rate of substitution 00:31:49.240 --> 00:31:52.570 align:middle line:84% is the negative of the marginal utility 00:31:52.570 --> 00:31:55.030 align:middle line:84% on the x-axis and marginal utility on the y-axis. 00:31:55.030 --> 00:31:59.000 align:middle line:90% 00:31:59.000 --> 00:32:02.880 align:middle line:84% Another definition-- homothetic preferences-- 00:32:02.880 --> 00:32:06.340 align:middle line:84% so preferences are homothetic means 00:32:06.340 --> 00:32:08.770 align:middle line:84% that the marginal rate of substitution 00:32:08.770 --> 00:32:11.590 align:middle line:84% depends only on the ratio of the goods 00:32:11.590 --> 00:32:13.490 align:middle line:90% and not on the total quantity. 00:32:13.490 --> 00:32:16.070 align:middle line:84% So a picture is worth a thousand words. 00:32:16.070 --> 00:32:19.560 align:middle line:84% Here's an indifference curve, here's a point. 00:32:19.560 --> 00:32:22.530 align:middle line:84% What is the marginal rate of substitution? 00:32:22.530 --> 00:32:26.350 align:middle line:84% The slope of that point, of the curve at that point. 00:32:26.350 --> 00:32:31.680 align:middle line:84% Here's the ratio of the goods, X2 to X1 on this line. 00:32:31.680 --> 00:32:35.220 align:middle line:84% As you move on this line, that ratio stays constant. 00:32:35.220 --> 00:32:38.460 align:middle line:84% We go up here to this higher level of indifference, 00:32:38.460 --> 00:32:39.940 align:middle line:90% the slope is the same. 00:32:39.940 --> 00:32:42.450 align:middle line:84% So essentially, this indifference curve 00:32:42.450 --> 00:32:44.230 align:middle line:90% is a blowup of this one. 00:32:44.230 --> 00:32:47.160 align:middle line:84% And you could draw one out here, which is 00:32:47.160 --> 00:32:50.140 align:middle line:90% a radial blowup of these two. 00:32:50.140 --> 00:32:52.110 align:middle line:84% So you can scale up and down, and just 00:32:52.110 --> 00:32:55.890 align:middle line:84% keep the slopes the same as you move, for example, 00:32:55.890 --> 00:32:58.810 align:middle line:84% along either one of these two lines. 00:32:58.810 --> 00:33:00.030 align:middle line:90% So who cares? 00:33:00.030 --> 00:33:04.050 align:middle line:84% Well it's going to turn out that a household with these kinds 00:33:04.050 --> 00:33:07.530 align:middle line:84% of indifference curves is going to respond in a certain way 00:33:07.530 --> 00:33:11.010 align:middle line:84% to income, respond in a certain linear way, that's 00:33:11.010 --> 00:33:13.530 align:middle line:84% going to allow us to, for some purposes, 00:33:13.530 --> 00:33:15.990 align:middle line:84% aggregate up over many diverse households, 00:33:15.990 --> 00:33:18.840 align:middle line:84% and act like there's only one representative 00:33:18.840 --> 00:33:22.590 align:middle line:84% Robinson Crusoe macro household in the economy. 00:33:22.590 --> 00:33:24.480 align:middle line:84% You probably don't remember, but when 00:33:24.480 --> 00:33:27.540 align:middle line:84% I went through the great, big, long syllabus last time, 00:33:27.540 --> 00:33:31.870 align:middle line:84% the next to last page talked about aggregation. 00:33:31.870 --> 00:33:34.630 align:middle line:84% And the lecture on Tuesday will be 00:33:34.630 --> 00:33:38.560 align:middle line:84% an example that violates this kind of homotheticity. 00:33:38.560 --> 00:33:41.560 align:middle line:90% 00:33:41.560 --> 00:33:43.300 align:middle line:90% Here's a violation. 00:33:43.300 --> 00:33:44.920 align:middle line:84% We have a utility function which is 00:33:44.920 --> 00:33:49.780 align:middle line:84% linear in one good, x1, which we'll call money. 00:33:49.780 --> 00:33:55.640 align:middle line:84% And the log of x2 is not linear, but it is concave. 00:33:55.640 --> 00:34:00.470 align:middle line:84% So we call this the money good because as you 00:34:00.470 --> 00:34:03.920 align:middle line:84% inject or take away good 1, utility 00:34:03.920 --> 00:34:06.680 align:middle line:90% goes up or down linearly. 00:34:06.680 --> 00:34:10.370 align:middle line:84% And it is also true that if everyone in the economy 00:34:10.370 --> 00:34:15.320 align:middle line:84% has linear utility functions in x1, then the amount of utility 00:34:15.320 --> 00:34:19.340 align:middle line:84% goes down when you tax somebody is exactly the same amount 00:34:19.340 --> 00:34:20.989 align:middle line:84% that utility goes up for somebody 00:34:20.989 --> 00:34:22.310 align:middle line:90% else who got the subsidy. 00:34:22.310 --> 00:34:26.460 align:middle line:84% So again, that's a useful property to have, 00:34:26.460 --> 00:34:27.679 align:middle line:90% but it's very special. 00:34:27.679 --> 00:34:31.230 align:middle line:84% Anyway, in this case, the marginal rate of substitution, 00:34:31.230 --> 00:34:34.760 align:middle line:84% which is not constant on any rate from the origin, 00:34:34.760 --> 00:34:36.260 align:middle line:84% the marginal rate of substitution 00:34:36.260 --> 00:34:40.460 align:middle line:84% is constant as we fix x2 and vary x1, 00:34:40.460 --> 00:34:43.040 align:middle line:90% so going along horizontally. 00:34:43.040 --> 00:34:45.409 align:middle line:84% And you can convince yourself that has to be true. 00:34:45.409 --> 00:34:47.449 align:middle line:84% Because if you took the definition, which 00:34:47.449 --> 00:34:49.730 align:middle line:84% is the marginal utility of x1 divided 00:34:49.730 --> 00:34:53.000 align:middle line:84% by the marginal utility of x2, the marginal x1 00:34:53.000 --> 00:34:54.440 align:middle line:90% is constant because it's linear. 00:34:54.440 --> 00:34:56.780 align:middle line:84% And this thing is varying with x2. 00:34:56.780 --> 00:34:59.120 align:middle line:84% So if you fix x2 and vary x1, you're 00:34:59.120 --> 00:35:03.530 align:middle line:84% not going to change the marginal rate of substitution. 00:35:03.530 --> 00:35:06.700 align:middle line:84% So we have now successfully reviewed 00:35:06.700 --> 00:35:09.850 align:middle line:84% basic properties of consumption sets, 00:35:09.850 --> 00:35:12.500 align:middle line:90% consumer preferences, and so on. 00:35:12.500 --> 00:35:15.010 align:middle line:84% You might want to squirrel away some of those properties 00:35:15.010 --> 00:35:18.470 align:middle line:84% because we'll come back to them later throughout the course, 00:35:18.470 --> 00:35:19.780 align:middle line:90% in the subsequent lectures. 00:35:19.780 --> 00:35:22.510 align:middle line:84% But here they are, essentially all listed now. 00:35:22.510 --> 00:35:24.500 align:middle line:84% It doesn't mean they all have to be true. 00:35:24.500 --> 00:35:26.590 align:middle line:84% Some of them contradict each other. 00:35:26.590 --> 00:35:29.710 align:middle line:84% Theorems are if we have local non-satiation 00:35:29.710 --> 00:35:31.870 align:middle line:84% then a competitive equilibrium is optimal. 00:35:31.870 --> 00:35:33.640 align:middle line:84% But we may not have it, in which case 00:35:33.640 --> 00:35:36.200 align:middle line:84% competitive markets don't work, and things like that. 00:35:36.200 --> 00:35:38.000 align:middle line:90% So these are not requirements. 00:35:38.000 --> 00:35:39.610 align:middle line:90% These are options. 00:35:39.610 --> 00:35:42.970 align:middle line:84% All right, but now we get to an application, 00:35:42.970 --> 00:35:45.470 align:middle line:84% this partial equilibrium problem, 00:35:45.470 --> 00:35:48.670 align:middle line:84% which is to maximize utility subject to a constraint 00:35:48.670 --> 00:35:49.640 align:middle line:90% for the first time. 00:35:49.640 --> 00:35:52.360 align:middle line:84% So this is the budget constraint. 00:35:52.360 --> 00:35:54.280 align:middle line:90% Income on the right-hand side-- 00:35:54.280 --> 00:35:56.920 align:middle line:84% I for income-- is taken as given. 00:35:56.920 --> 00:35:59.560 align:middle line:90% The prices are of each good. 00:35:59.560 --> 00:36:04.430 align:middle line:84% There are n of them, p1 for good 1, p2 for good 2, and so on. 00:36:04.430 --> 00:36:09.820 align:middle line:84% So the sum of the expenditures at prices p over choices x 00:36:09.820 --> 00:36:14.950 align:middle line:84% cannot exceed the income I. So this is your budget. 00:36:14.950 --> 00:36:16.330 align:middle line:90% We all have budgets. 00:36:16.330 --> 00:36:19.630 align:middle line:84% I sometimes wonder about the US Treasury, 00:36:19.630 --> 00:36:22.570 align:middle line:84% its budget deficit, debt outstanding, 00:36:22.570 --> 00:36:26.410 align:middle line:84% is now larger than any time since World War II. 00:36:26.410 --> 00:36:29.440 align:middle line:90% But households have budgets. 00:36:29.440 --> 00:36:31.010 align:middle line:84% And I'm worried about the government, 00:36:31.010 --> 00:36:34.510 align:middle line:90% but we'll get to that later. 00:36:34.510 --> 00:36:37.870 align:middle line:84% So suppose you were to double all the prices. 00:36:37.870 --> 00:36:42.140 align:middle line:84% We start with this, and we double income as well. 00:36:42.140 --> 00:36:47.590 align:middle line:84% So we pre-multiply every term by t, little t, both on the left 00:36:47.590 --> 00:36:49.030 align:middle line:90% and on the right. 00:36:49.030 --> 00:36:51.260 align:middle line:84% It's a constraint in the problem, 00:36:51.260 --> 00:36:53.440 align:middle line:84% but it doesn't change the constraint 00:36:53.440 --> 00:36:55.420 align:middle line:90% by multiplying through by t. 00:36:55.420 --> 00:36:59.650 align:middle line:84% Or if you started with t, in there, just divide everything 00:36:59.650 --> 00:37:00.940 align:middle line:90% by t and you take it out. 00:37:00.940 --> 00:37:04.390 align:middle line:84% So it can't change the math of anything. 00:37:04.390 --> 00:37:05.800 align:middle line:90% So that's one way to look at it. 00:37:05.800 --> 00:37:07.810 align:middle line:84% I'll show you in the picture in a minute 00:37:07.810 --> 00:37:12.700 align:middle line:84% that no price illusion-- if prices double as in inflation, 00:37:12.700 --> 00:37:15.160 align:middle line:84% but your nominal income also doubles, 00:37:15.160 --> 00:37:19.210 align:middle line:84% no harm, no foul, at least not for this household. 00:37:19.210 --> 00:37:21.250 align:middle line:84% Now, maybe consumers aren't rational 00:37:21.250 --> 00:37:24.820 align:middle line:84% and maybe they're sensitive to income movement and not 00:37:24.820 --> 00:37:26.103 align:middle line:90% price or vice versa. 00:37:26.103 --> 00:37:27.520 align:middle line:84% But for the most part, we're going 00:37:27.520 --> 00:37:29.583 align:middle line:84% to assume that these are equivalent 00:37:29.583 --> 00:37:31.750 align:middle line:84% constraints from the point of view of the household. 00:37:31.750 --> 00:37:35.380 align:middle line:84% And this is a definition of homogeneity of degree 00:37:35.380 --> 00:37:37.390 align:middle line:90% 0 of individual demand. 00:37:37.390 --> 00:37:40.760 align:middle line:84% The demand can't change because the constraint doesn't change. 00:37:40.760 --> 00:37:42.140 align:middle line:90% And here's the picture. 00:37:42.140 --> 00:37:46.180 align:middle line:84% So this is the set of points which completely exhaust income 00:37:46.180 --> 00:37:50.050 align:middle line:84% when there are two goods, x and y, P x and P y. 00:37:50.050 --> 00:37:52.300 align:middle line:84% You could move over to this extreme point, 00:37:52.300 --> 00:37:54.820 align:middle line:84% take all of your income, and buy good y. 00:37:54.820 --> 00:37:57.220 align:middle line:90% How many units can you get? 00:37:57.220 --> 00:38:00.310 align:middle line:84% Well, it's $10 divided by 5, or 2. 00:38:00.310 --> 00:38:02.740 align:middle line:84% So we have a maximum amount that would 00:38:02.740 --> 00:38:04.630 align:middle line:84% be possible to purchase of good y, 00:38:04.630 --> 00:38:07.120 align:middle line:84% and maximum amount you could purchase of good x. 00:38:07.120 --> 00:38:10.030 align:middle line:84% We have exhaustion of the budget on this line. 00:38:10.030 --> 00:38:14.140 align:middle line:84% Actually in the interior, we're spending less than income. 00:38:14.140 --> 00:38:16.840 align:middle line:84% Consumer might not want to end up in the interior, 00:38:16.840 --> 00:38:20.020 align:middle line:84% but that has to do with satiation. 00:38:20.020 --> 00:38:22.720 align:middle line:84% What is the slope of this budget line? 00:38:22.720 --> 00:38:24.640 align:middle line:84% Well, you can derive this the same way 00:38:24.640 --> 00:38:27.490 align:middle line:84% we kept utility constant-- keep income constant, 00:38:27.490 --> 00:38:30.310 align:middle line:84% take dI, set it to 0, and then take 00:38:30.310 --> 00:38:34.630 align:middle line:84% the derivative of P X times dX plus P Y times dY, 00:38:34.630 --> 00:38:38.440 align:middle line:84% and you get dYdX is equal to P X over P Y. 00:38:38.440 --> 00:38:44.380 align:middle line:84% So the ratio of this line is exactly P X over P Y. 00:38:44.380 --> 00:38:48.670 align:middle line:84% It is kind of interesting that only the ratio matters. 00:38:48.670 --> 00:38:50.620 align:middle line:84% That is to say, setting aside what's 00:38:50.620 --> 00:38:52.210 align:middle line:84% going to happen to income, if you 00:38:52.210 --> 00:38:54.370 align:middle line:84% double all the prices of both goods, 00:38:54.370 --> 00:38:57.110 align:middle line:84% you don't change the slope of this line. 00:38:57.110 --> 00:38:59.830 align:middle line:84% In fact, more than that, we could pick one of these goods-- 00:38:59.830 --> 00:39:02.050 align:middle line:90% say Y as the numeraire-- 00:39:02.050 --> 00:39:04.330 align:middle line:84% set P Y equal to 1, and then we'd 00:39:04.330 --> 00:39:08.840 align:middle line:84% only be fussing about the price of good X. 00:39:08.840 --> 00:39:09.890 align:middle line:90% Here it's more general. 00:39:09.890 --> 00:39:13.350 align:middle line:84% Everything's in dollars or some unit of account, 00:39:13.350 --> 00:39:14.960 align:middle line:84% depending on what country you're in. 00:39:14.960 --> 00:39:17.330 align:middle line:84% Could be measured in wheat, goods. 00:39:17.330 --> 00:39:20.810 align:middle line:84% Price is measured in terms of how many bushels of wheat 00:39:20.810 --> 00:39:21.920 align:middle line:90% you can buy. 00:39:21.920 --> 00:39:23.270 align:middle line:90% So it's arbitrary. 00:39:23.270 --> 00:39:27.200 align:middle line:84% You have to specify what it is, but only the ratio matters. 00:39:27.200 --> 00:39:30.210 align:middle line:90% That thing about inflation-- 00:39:30.210 --> 00:39:32.540 align:middle line:84% it's like doubling prices and doubling income 00:39:32.540 --> 00:39:34.580 align:middle line:90% doesn't change this line. 00:39:34.580 --> 00:39:37.880 align:middle line:84% And where does the consumer choose to be? 00:39:37.880 --> 00:39:40.870 align:middle line:90% Here, C for choice, maybe. 00:39:40.870 --> 00:39:43.790 align:middle line:90% So why? 00:39:43.790 --> 00:39:46.460 align:middle line:90% Well, one way to view it-- 00:39:46.460 --> 00:39:50.530 align:middle line:84% the household wants maximal utility. 00:39:50.530 --> 00:39:55.870 align:middle line:84% So in the set of choices inside or on the budget line, 00:39:55.870 --> 00:39:58.300 align:middle line:84% what point has the highest utility level? 00:39:58.300 --> 00:40:01.370 align:middle line:84% It's point C. If you go this way, it's less. 00:40:01.370 --> 00:40:03.750 align:middle line:90% If you go this way, it's less. 00:40:03.750 --> 00:40:05.670 align:middle line:84% There's another somewhat interesting way 00:40:05.670 --> 00:40:08.940 align:middle line:84% to view this marginal analysis, which 00:40:08.940 --> 00:40:12.810 align:middle line:84% is if you start it up here, the slope of this indifference 00:40:12.810 --> 00:40:15.390 align:middle line:90% curve U1 is pretty steep. 00:40:15.390 --> 00:40:18.360 align:middle line:84% It's negative, but I mean steep in absolute value. 00:40:18.360 --> 00:40:21.150 align:middle line:84% And the slope of the budget line is less. 00:40:21.150 --> 00:40:23.520 align:middle line:84% That means this household's willing to give up 00:40:23.520 --> 00:40:27.840 align:middle line:84% a substantial amount of good Y to get one more unit of good X 00:40:27.840 --> 00:40:31.650 align:middle line:84% as he or she moves away from B. But in fact, given 00:40:31.650 --> 00:40:34.920 align:middle line:84% the prices in the market, you don't have to give up that 00:40:34.920 --> 00:40:38.400 align:middle line:84% much to gain value in order to be able to purchase 00:40:38.400 --> 00:40:40.440 align:middle line:90% X. You give up less Y. 00:40:40.440 --> 00:40:43.230 align:middle line:84% So this is what you're willing to give up and be indifferent, 00:40:43.230 --> 00:40:46.510 align:middle line:84% but you are allowed to give up less on the budget line. 00:40:46.510 --> 00:40:48.570 align:middle line:84% So you're definitely moving in the direction here 00:40:48.570 --> 00:40:50.340 align:middle line:90% of higher and higher utility. 00:40:50.340 --> 00:40:53.880 align:middle line:84% And you can conduct the same reverse experiment down here. 00:40:53.880 --> 00:40:58.180 align:middle line:84% So next, let's do a utility maximization problem. 00:40:58.180 --> 00:41:03.270 align:middle line:84% And as I promised last time, I'm going to introduce a cool tool. 00:41:03.270 --> 00:41:07.190 align:middle line:90% And it's called the Lagrangian. 00:41:07.190 --> 00:41:11.330 align:middle line:84% So if we want to maximize utility subject to a budget, 00:41:11.330 --> 00:41:13.580 align:middle line:90% set up the problem in this way-- 00:41:13.580 --> 00:41:18.890 align:middle line:84% Lagrangian, which is the French guy that figured all this out, 00:41:18.890 --> 00:41:23.120 align:middle line:84% Lagrange, is to write down the utility function 00:41:23.120 --> 00:41:30.080 align:middle line:84% plus a shadow price lambda times a repetition of the constraint. 00:41:30.080 --> 00:41:33.770 align:middle line:84% So we turn the direct maximization problem, 00:41:33.770 --> 00:41:39.400 align:middle line:84% which was this, into a pseudo problem where 00:41:39.400 --> 00:41:42.700 align:middle line:84% the constraint is actually in the objective function. 00:41:42.700 --> 00:41:44.620 align:middle line:84% Anyway, if you start taking derivatives 00:41:44.620 --> 00:41:47.530 align:middle line:84% of this looking for the first-order conditions 00:41:47.530 --> 00:41:51.520 align:middle line:84% for a maximum, the derivative of the entire expression 00:41:51.520 --> 00:41:55.570 align:middle line:84% with respect to x k is the way that k will enter and generate 00:41:55.570 --> 00:41:58.810 align:middle line:84% marginal utility under U. That's this term. 00:41:58.810 --> 00:42:00.610 align:middle line:90% Plus, we'll pick up lambda. 00:42:00.610 --> 00:42:03.520 align:middle line:84% And where else does little x enter? 00:42:03.520 --> 00:42:05.620 align:middle line:90% Over here somewhere, P k. 00:42:05.620 --> 00:42:10.270 align:middle line:84% It's linear in x k coefficient P k. 00:42:10.270 --> 00:42:12.340 align:middle line:90% So we get this thing. 00:42:12.340 --> 00:42:14.950 align:middle line:84% And again, we're looking for an interior solution. 00:42:14.950 --> 00:42:18.240 align:middle line:84% We set this thing equal to 0 as a necessary condition 00:42:18.240 --> 00:42:21.000 align:middle line:90% for an optimum. 00:42:21.000 --> 00:42:24.630 align:middle line:84% Lambda is actually a control variable, too, 00:42:24.630 --> 00:42:27.970 align:middle line:84% the way the x's are, just looks a little weird. 00:42:27.970 --> 00:42:30.810 align:middle line:84% We take the derivative of the Lagrangian 00:42:30.810 --> 00:42:33.180 align:middle line:84% with respect to lambda, and obviously, we're 00:42:33.180 --> 00:42:36.100 align:middle line:84% going to get the constraint right back again. 00:42:36.100 --> 00:42:37.770 align:middle line:84% And we're going to set it to 0, which 00:42:37.770 --> 00:42:40.050 align:middle line:84% we assumed before that they wanted 00:42:40.050 --> 00:42:41.430 align:middle line:90% to spend all their income. 00:42:41.430 --> 00:42:43.590 align:middle line:84% We were implicitly assuming before 00:42:43.590 --> 00:42:46.230 align:middle line:84% that the marginal utility of income was positive, 00:42:46.230 --> 00:42:48.940 align:middle line:84% or in this case, that lambda is positive. 00:42:48.940 --> 00:42:52.210 align:middle line:84% So that's why we get the constraint right back. 00:42:52.210 --> 00:42:54.930 align:middle line:84% I'm not sure if you've seen Lagrangians before, 00:42:54.930 --> 00:42:58.500 align:middle line:84% but it's just a version of multivariate calculus 00:42:58.500 --> 00:43:00.570 align:middle line:84% where you find the first-order conditions 00:43:00.570 --> 00:43:01.990 align:middle line:90% to a maximum problem. 00:43:01.990 --> 00:43:07.680 align:middle line:84% Note that we can take any two pair of bundles k and j, i 00:43:07.680 --> 00:43:09.360 align:middle line:90% and j, or whatever. 00:43:09.360 --> 00:43:11.130 align:middle line:84% When you take them pairwise, we're 00:43:11.130 --> 00:43:14.070 align:middle line:84% going to have the derivative with respect to good I here 00:43:14.070 --> 00:43:17.100 align:middle line:84% and the price of good I here and the lambda. 00:43:17.100 --> 00:43:20.400 align:middle line:84% That lambda is the same when we go J, J. 00:43:20.400 --> 00:43:24.630 align:middle line:84% So we can divide the left by the right, 00:43:24.630 --> 00:43:27.700 align:middle line:90% and the lambda will disappear. 00:43:27.700 --> 00:43:31.200 align:middle line:84% So we're going to get these ratios of marginal utilities, 00:43:31.200 --> 00:43:34.570 align:middle line:84% which are, of course, the marginal rate of substitution 00:43:34.570 --> 00:43:35.820 align:middle line:90% equal to the price ratio. 00:43:35.820 --> 00:43:39.110 align:middle line:90% 00:43:39.110 --> 00:43:40.880 align:middle line:90% And that's what we were saying-- 00:43:40.880 --> 00:43:43.310 align:middle line:90% maybe I didn't say it-- 00:43:43.310 --> 00:43:45.830 align:middle line:84% when we have this tangency over here, 00:43:45.830 --> 00:43:48.230 align:middle line:84% the marginal rate of substitution in utility 00:43:48.230 --> 00:43:50.520 align:middle line:84% is the marginal rate of substitution in the budget, 00:43:50.520 --> 00:43:52.170 align:middle line:90% which is the price ratio. 00:43:52.170 --> 00:43:55.700 align:middle line:84% So we're getting that result back now 00:43:55.700 --> 00:43:59.667 align:middle line:84% as a consequence of this optimization. 00:43:59.667 --> 00:44:01.500 align:middle line:84% Another danger-- I feel like I ought to have 00:44:01.500 --> 00:44:02.850 align:middle line:90% this slide that says, stop. 00:44:02.850 --> 00:44:03.720 align:middle line:90% Pause. 00:44:03.720 --> 00:44:06.840 align:middle line:90% Do not memorize these diagrams. 00:44:06.840 --> 00:44:10.050 align:middle line:84% Just like the notation becomes mind numbing, 00:44:10.050 --> 00:44:13.260 align:middle line:84% you get used to drawing a picture in a certain way. 00:44:13.260 --> 00:44:18.480 align:middle line:84% And then when someone nasty puts another specification 00:44:18.480 --> 00:44:23.090 align:middle line:84% on the table, you don't know what to do with it. 00:44:23.090 --> 00:44:25.840 align:middle line:90% So here I am being nasty. 00:44:25.840 --> 00:44:29.950 align:middle line:84% I'll show you n incorporated it into the Lagrangian. 00:44:29.950 --> 00:44:32.350 align:middle line:90% So this is a corner solution. 00:44:32.350 --> 00:44:35.840 align:middle line:90% Where is the best utility level? 00:44:35.840 --> 00:44:39.310 align:middle line:84% The highest level is here because as you move down 00:44:39.310 --> 00:44:41.890 align:middle line:84% this budget line on the line, you 00:44:41.890 --> 00:44:44.140 align:middle line:84% get higher and higher levels of utility. 00:44:44.140 --> 00:44:46.970 align:middle line:84% But if the consumption set is all 00:44:46.970 --> 00:44:50.080 align:middle line:84% of this non-negative [INAUDIBLE] you can't go negative in y. 00:44:50.080 --> 00:44:51.730 align:middle line:90% So you stop. 00:44:51.730 --> 00:44:53.380 align:middle line:90% This is a corner solution. 00:44:53.380 --> 00:44:56.890 align:middle line:84% By the way, you would get this as well if the utility 00:44:56.890 --> 00:44:58.580 align:middle line:90% functions were linear. 00:44:58.580 --> 00:45:02.320 align:middle line:84% They're almost linear here, and you have a corner solution. 00:45:02.320 --> 00:45:04.720 align:middle line:84% Of course, they could be linear and coincident 00:45:04.720 --> 00:45:07.690 align:middle line:84% with the budget, in which case everything on the budget line 00:45:07.690 --> 00:45:08.740 align:middle line:90% is an optimum. 00:45:08.740 --> 00:45:11.110 align:middle line:84% Or those linear indifference curves 00:45:11.110 --> 00:45:13.570 align:middle line:84% could be really flat, in which case 00:45:13.570 --> 00:45:15.700 align:middle line:84% you'll go over here to a different corner. 00:45:15.700 --> 00:45:18.130 align:middle line:84% It doesn't have to be linear to generate corners. 00:45:18.130 --> 00:45:19.900 align:middle line:90% This is an example. 00:45:19.900 --> 00:45:23.290 align:middle line:84% But we need to incorporate it into our maximum techniques. 00:45:23.290 --> 00:45:26.500 align:middle line:90% So what's the general recipe? 00:45:26.500 --> 00:45:30.050 align:middle line:84% Before we translate it into the consumer optimization problem, 00:45:30.050 --> 00:45:31.600 align:middle line:90% we'll get the technique down. 00:45:31.600 --> 00:45:35.660 align:middle line:84% As I promised, you're being gifted with a powerful tool. 00:45:35.660 --> 00:45:39.550 align:middle line:84% So we have a function f over a range x. 00:45:39.550 --> 00:45:41.170 align:middle line:90% We want to maximize it. 00:45:41.170 --> 00:45:44.140 align:middle line:90% And we have various constraints. 00:45:44.140 --> 00:45:46.240 align:middle line:90% We have n of them. 00:45:46.240 --> 00:45:50.620 align:middle line:90% g i of x-- g is a function. 00:45:50.620 --> 00:45:55.000 align:middle line:84% There are i functions, each of which, when evaluated at x, 00:45:55.000 --> 00:45:56.650 align:middle line:90% cannot be negative. 00:45:56.650 --> 00:46:01.660 align:middle line:84% Properties f and g i are real-valued, continuously 00:46:01.660 --> 00:46:02.590 align:middle line:90% differentiable. 00:46:02.590 --> 00:46:04.960 align:middle line:84% In other words, they're very smooth, 00:46:04.960 --> 00:46:08.060 align:middle line:84% and not only continuous, the derivatives are continuous. 00:46:08.060 --> 00:46:09.610 align:middle line:90% How do we solve this problem? 00:46:09.610 --> 00:46:13.210 align:middle line:84% Look it up in the cookbook, or memorize it, even better. 00:46:13.210 --> 00:46:16.060 align:middle line:84% There's a trick here I'll show you. 00:46:16.060 --> 00:46:20.090 align:middle line:84% The Lagrangian put the objective function, just like we had U. 00:46:20.090 --> 00:46:23.410 align:middle line:84% And now write down each and every one of these constraints 00:46:23.410 --> 00:46:28.890 align:middle line:84% with these Lagrange multipliers lambda i for constraint i, 00:46:28.890 --> 00:46:30.310 align:middle line:90% add them all up. 00:46:30.310 --> 00:46:31.750 align:middle line:90% What's the little trick? 00:46:31.750 --> 00:46:33.430 align:middle line:90% The g i is non-negative. 00:46:33.430 --> 00:46:36.310 align:middle line:84% We enter it with a plus sign, and then things 00:46:36.310 --> 00:46:38.410 align:middle line:84% are going to come out right with the math. 00:46:38.410 --> 00:46:39.940 align:middle line:90% It's not necessary. 00:46:39.940 --> 00:46:42.280 align:middle line:84% Things work even if you write down 00:46:42.280 --> 00:46:44.290 align:middle line:84% the g's as less than or equal to 0, 00:46:44.290 --> 00:46:46.930 align:middle line:84% but then you have to struggle a little bit more 00:46:46.930 --> 00:46:48.790 align:middle line:84% to make sure you've got the math right, 00:46:48.790 --> 00:46:50.680 align:middle line:90% and you didn't reverse the sign. 00:46:50.680 --> 00:46:53.590 align:middle line:84% So anyway, try to do it this way-- g i greater than 00:46:53.590 --> 00:46:55.390 align:middle line:90% 0 lambda g i. 00:46:55.390 --> 00:46:57.040 align:middle line:84% Write out the first-order conditions. 00:46:57.040 --> 00:46:58.730 align:middle line:90% That's just a derivative. 00:46:58.730 --> 00:47:01.420 align:middle line:84% So we're taking the derivative of the Lagrangian 00:47:01.420 --> 00:47:04.210 align:middle line:84% with respect to x k is the derivative of the function 00:47:04.210 --> 00:47:08.470 align:middle line:84% f with respect to k plus the way in which x k 00:47:08.470 --> 00:47:11.740 align:middle line:84% is entering in each of these constraints g 00:47:11.740 --> 00:47:15.520 align:middle line:84% i, each pre-multiplied by lambda i. 00:47:15.520 --> 00:47:20.560 align:middle line:84% Set it equal to 0, and do that for every argument x k. 00:47:20.560 --> 00:47:21.340 align:middle line:90% With me? 00:47:21.340 --> 00:47:24.240 align:middle line:90% 00:47:24.240 --> 00:47:29.020 align:middle line:84% So next, write down each constraint again. 00:47:29.020 --> 00:47:33.780 align:middle line:84% Write down these multipliers which are non-negative. 00:47:33.780 --> 00:47:36.540 align:middle line:84% And this is cool-- this is called 00:47:36.540 --> 00:47:39.750 align:middle line:84% a complementary slackness condition, which 00:47:39.750 --> 00:47:44.520 align:middle line:84% is the multiplication-- oh, it should be a lambda i on that. 00:47:44.520 --> 00:47:45.960 align:middle line:90% That's a typo. 00:47:45.960 --> 00:47:49.890 align:middle line:84% Lambda i times g i equal to 0, and then we're 00:47:49.890 --> 00:47:53.160 align:middle line:84% going to look for a solution to all these equations. 00:47:53.160 --> 00:47:57.270 align:middle line:84% Now, if lambda i were 0 then it doesn't matter 00:47:57.270 --> 00:47:59.280 align:middle line:90% what g is evaluated at x. 00:47:59.280 --> 00:48:00.420 align:middle line:90% It's still 0. 00:48:00.420 --> 00:48:05.640 align:middle line:84% Well, if lambda is 0, then effectively, it's 00:48:05.640 --> 00:48:09.390 align:middle line:84% not entering into the problem, and you can ignore it, and drop 00:48:09.390 --> 00:48:11.220 align:middle line:90% it by setting it equal to 0. 00:48:11.220 --> 00:48:14.010 align:middle line:84% If lambda i were strictly positive, 00:48:14.010 --> 00:48:17.670 align:middle line:84% though, then the only way you can get this product to be 0 00:48:17.670 --> 00:48:20.520 align:middle line:90% is to have this g i be 0. 00:48:20.520 --> 00:48:21.580 align:middle line:90% And that's here. 00:48:21.580 --> 00:48:24.330 align:middle line:84% And this is what happened with the income maximization 00:48:24.330 --> 00:48:25.350 align:middle line:90% problem. 00:48:25.350 --> 00:48:28.200 align:middle line:84% I asserted-- if we go back now and look-- 00:48:28.200 --> 00:48:31.560 align:middle line:84% that the marginal utility of income lambda on the budget 00:48:31.560 --> 00:48:33.460 align:middle line:90% constraint was positive. 00:48:33.460 --> 00:48:35.040 align:middle line:84% And if it's positive, then we have 00:48:35.040 --> 00:48:37.890 align:middle line:84% to set the budget constraint equal to 0, income 00:48:37.890 --> 00:48:40.000 align:middle line:84% equal to the sum of expenditures, 00:48:40.000 --> 00:48:43.120 align:middle line:84% which is the way we solved that simpler problem to begin with. 00:48:43.120 --> 00:48:45.690 align:middle line:84% So what we did earlier generalizes 00:48:45.690 --> 00:48:47.850 align:middle line:90% to this more complex. 00:48:47.850 --> 00:48:49.080 align:middle line:90% Now, why are we doing it? 00:48:49.080 --> 00:48:51.720 align:middle line:84% Well, let's come back to this problem. 00:48:51.720 --> 00:48:52.800 align:middle line:90% Are they necessary? 00:48:52.800 --> 00:48:54.330 align:middle line:90% They could also be sufficient. 00:48:54.330 --> 00:48:57.060 align:middle line:84% If that function f is quasi-concave, 00:48:57.060 --> 00:49:00.510 align:middle line:84% and the constraint sets, these guys, are convex, 00:49:00.510 --> 00:49:03.510 align:middle line:84% then finding a solution to the first-order condition is 00:49:03.510 --> 00:49:06.330 align:middle line:84% equivalent with finding a global maximizer-- 00:49:06.330 --> 00:49:10.920 align:middle line:84% very powerful, and we will use it subsequently, again, 00:49:10.920 --> 00:49:15.010 align:middle line:84% to address the issue of whether competitive equilibria are 00:49:15.010 --> 00:49:15.510 align:middle line:90% optimal. 00:49:15.510 --> 00:49:18.160 align:middle line:90% 00:49:18.160 --> 00:49:23.080 align:middle line:84% So going back to the motivation, which was this picture-- 00:49:23.080 --> 00:49:25.510 align:middle line:84% we kind of know that we're going to hit a corner. 00:49:25.510 --> 00:49:26.860 align:middle line:90% What comes up with the-- 00:49:26.860 --> 00:49:30.820 align:middle line:84% if these were linear indifference curves, 00:49:30.820 --> 00:49:33.310 align:middle line:84% and you had this linear budget set, then 00:49:33.310 --> 00:49:34.840 align:middle line:84% when you take first-order conditions 00:49:34.840 --> 00:49:37.828 align:middle line:84% you're going to be in trouble because everything 00:49:37.828 --> 00:49:39.370 align:middle line:84% in the first-order condition is going 00:49:39.370 --> 00:49:42.730 align:middle line:84% to be parameters, not control variables 00:49:42.730 --> 00:49:44.300 align:middle line:90% that you want to find. 00:49:44.300 --> 00:49:47.410 align:middle line:84% So that's a version of being nasty. 00:49:47.410 --> 00:49:49.550 align:middle line:84% But like I said, we're working it all out. 00:49:49.550 --> 00:49:52.120 align:middle line:84% So how do we deal with this hitting the corner? 00:49:52.120 --> 00:49:54.610 align:middle line:90% What does the Lagrangian find? 00:49:54.610 --> 00:49:57.520 align:middle line:84% So we maximize utility subject to the budget, 00:49:57.520 --> 00:50:00.400 align:middle line:84% subject to non-negativity constraints. 00:50:00.400 --> 00:50:04.040 align:middle line:84% As in that picture, consumption cannot go negative. 00:50:04.040 --> 00:50:09.790 align:middle line:84% We form the Lagrangian mechanically, which 00:50:09.790 --> 00:50:12.460 align:middle line:84% says write down the function, write down 00:50:12.460 --> 00:50:17.620 align:middle line:84% each of the constraints, lambda on the budget constraint, mu-- 00:50:17.620 --> 00:50:21.430 align:middle line:84% I changed notation-- mu k for constraint, 00:50:21.430 --> 00:50:26.320 align:middle line:84% this one, the non-negativity constraint on x k, 00:50:26.320 --> 00:50:28.460 align:middle line:90% take the first-order conditions. 00:50:28.460 --> 00:50:31.580 align:middle line:84% And now we get something similar, 00:50:31.580 --> 00:50:35.050 align:middle line:84% except that you pick up this shadow price or Lagrange 00:50:35.050 --> 00:50:39.640 align:middle line:84% multiplier on the non-negativity constraint of good k. 00:50:39.640 --> 00:50:41.920 align:middle line:84% Whereas before, that mu wasn't there. 00:50:41.920 --> 00:50:44.200 align:middle line:84% So then applying the recipe, we write down 00:50:44.200 --> 00:50:45.340 align:middle line:90% the constraints again. 00:50:45.340 --> 00:50:48.970 align:middle line:84% We write down the non-negativity of the Lagrange multipliers. 00:50:48.970 --> 00:50:51.830 align:middle line:84% We have this complementary slackness condition, 00:50:51.830 --> 00:50:53.950 align:middle line:84% which is the product of the Lagrange multipliers 00:50:53.950 --> 00:50:58.640 align:middle line:84% with the budgets, and we look for a solution. 00:50:58.640 --> 00:51:03.200 align:middle line:84% If we were at a corner where good 1-- 00:51:03.200 --> 00:51:05.840 align:middle line:84% or good x, whatever was in that figure-- 00:51:05.840 --> 00:51:13.570 align:middle line:84% were 0, then if we go back here, this would be positive. 00:51:13.570 --> 00:51:16.270 align:middle line:84% This mu k would be positive, and we take it out. 00:51:16.270 --> 00:51:18.700 align:middle line:90% This would then be less than 0. 00:51:18.700 --> 00:51:21.790 align:middle line:84% Something less than 0 plus something positive makes it 0. 00:51:21.790 --> 00:51:24.640 align:middle line:84% We leave the positive thing out, and we 00:51:24.640 --> 00:51:28.820 align:middle line:84% get the thing that's less than 0 looks like this. 00:51:28.820 --> 00:51:30.820 align:middle line:84% And what this says, essentially, is-- 00:51:30.820 --> 00:51:32.843 align:middle line:84% there's a number of ways to read it. 00:51:32.843 --> 00:51:33.760 align:middle line:90% Let me do it this way. 00:51:33.760 --> 00:51:35.170 align:middle line:90% Put lambda over here. 00:51:35.170 --> 00:51:38.270 align:middle line:84% Lambda is the marginal utility of income. 00:51:38.270 --> 00:51:41.860 align:middle line:84% So how much income does it take to get good k? 00:51:41.860 --> 00:51:45.320 align:middle line:84% And what are the utility consequences of that? 00:51:45.320 --> 00:51:49.930 align:middle line:84% p k is the number of dollars you have to give up to get good k. 00:51:49.930 --> 00:51:52.700 align:middle line:84% The utility of those dollars is lambda. 00:51:52.700 --> 00:51:58.300 align:middle line:84% So lambda times p k is the marginal cost 00:51:58.300 --> 00:52:00.610 align:middle line:90% of getting a unit of good k. 00:52:00.610 --> 00:52:02.960 align:middle line:90% And what's the marginal benefit? 00:52:02.960 --> 00:52:05.080 align:middle line:84% The marginal utility is the benefit. 00:52:05.080 --> 00:52:08.120 align:middle line:84% In this case, the marginal cost exceeds the marginal benefit, 00:52:08.120 --> 00:52:09.290 align:middle line:90% so you don't do it. 00:52:09.290 --> 00:52:11.120 align:middle line:84% In fact, the solution, obviously, 00:52:11.120 --> 00:52:14.990 align:middle line:84% is to set x k equal to 0 because at any positive level, 00:52:14.990 --> 00:52:17.590 align:middle line:84% the marginal costs are higher than the marginal benefits. 00:52:17.590 --> 00:52:20.542 align:middle line:84% The best you can do is to set it to 0. 00:52:20.542 --> 00:52:23.960 align:middle line:90% 00:52:23.960 --> 00:52:32.030 align:middle line:84% So Cobb-Douglass utility-- this was an example we had before. 00:52:32.030 --> 00:52:35.420 align:middle line:84% Good X and Y to power alpha and 1 minus alpha 00:52:35.420 --> 00:52:36.980 align:middle line:90% subject to a budget-- 00:52:36.980 --> 00:52:38.870 align:middle line:90% you can solve this. 00:52:38.870 --> 00:52:42.800 align:middle line:84% It's really not necessary to go through in class. 00:52:42.800 --> 00:52:44.060 align:middle line:90% It's a good example. 00:52:44.060 --> 00:52:44.960 align:middle line:90% You maximize. 00:52:44.960 --> 00:52:50.130 align:middle line:84% Using the Lagrangian, you get the first-order conditions. 00:52:50.130 --> 00:52:52.440 align:middle line:84% Do some algebra with that first-order condition 00:52:52.440 --> 00:52:54.360 align:middle line:90% comparing ratios. 00:52:54.360 --> 00:52:56.520 align:middle line:84% This would be the marginal rate of substitution. 00:52:56.520 --> 00:52:58.500 align:middle line:84% This is going to be the price ratio. 00:52:58.500 --> 00:53:00.990 align:middle line:84% They have to be equal for an interior solution. 00:53:00.990 --> 00:53:03.840 align:middle line:84% And the solution has to be interior 00:53:03.840 --> 00:53:06.180 align:middle line:84% because the marginal utility is going 00:53:06.180 --> 00:53:09.690 align:middle line:84% to infinity when one or the other of these goods goes to 0. 00:53:09.690 --> 00:53:13.230 align:middle line:84% So you never want to go down that far. 00:53:13.230 --> 00:53:15.150 align:middle line:90% We can rearrange these terms. 00:53:15.150 --> 00:53:18.180 align:middle line:84% And the bottom line is, p X times X 00:53:18.180 --> 00:53:22.290 align:middle line:84% is the expenditure on X, and it's equal to alpha times 00:53:22.290 --> 00:53:25.680 align:middle line:84% if I. If you go back to the utility function, 00:53:25.680 --> 00:53:29.250 align:middle line:90% alpha is the power on X. 00:53:29.250 --> 00:53:31.470 align:middle line:84% So whenever you see this Cobb-Douglas, 00:53:31.470 --> 00:53:34.950 align:middle line:84% you can immediately jump to something you should know, 00:53:34.950 --> 00:53:38.640 align:middle line:84% which is the expenditure shares have to do with these powers. 00:53:38.640 --> 00:53:41.220 align:middle line:84% The expenditure shares on X would be alpha. 00:53:41.220 --> 00:53:44.940 align:middle line:84% The expenditure share on Y would be 1 minus alpha. 00:53:44.940 --> 00:53:47.190 align:middle line:90% We want to see that in notation. 00:53:47.190 --> 00:53:51.450 align:middle line:84% p X times X is the expenditure share on X at the optimum. 00:53:51.450 --> 00:53:55.680 align:middle line:84% It's alpha times I. Likewise, p Y times Y 00:53:55.680 --> 00:53:58.560 align:middle line:90% is 1 minus alpha times I. 00:53:58.560 --> 00:54:00.135 align:middle line:90% So you have your whole budget. 00:54:00.135 --> 00:54:01.260 align:middle line:90% You spend the whole budget. 00:54:01.260 --> 00:54:03.660 align:middle line:84% And the share of the budget you're spending 00:54:03.660 --> 00:54:09.040 align:middle line:84% is dictated by the alpha, and 1 minus alpha, and the prices. 00:54:09.040 --> 00:54:11.360 align:middle line:84% So that's constant expenditure shares. 00:54:11.360 --> 00:54:13.960 align:middle line:84% And again, when you go to data, you go take a look. 00:54:13.960 --> 00:54:16.435 align:middle line:84% And we'll do that in the lecture on next Tuesday. 00:54:16.435 --> 00:54:19.710 align:middle line:90% 00:54:19.710 --> 00:54:23.530 align:middle line:84% Now again, we got through all that stuff-- 00:54:23.530 --> 00:54:27.690 align:middle line:84% hopefully useful, maybe even interesting to you. 00:54:27.690 --> 00:54:30.270 align:middle line:84% And now you've forgotten about geography. 00:54:30.270 --> 00:54:32.220 align:middle line:90% You've forgotten about time. 00:54:32.220 --> 00:54:34.510 align:middle line:84% You've forgotten about states of the world. 00:54:34.510 --> 00:54:38.220 align:middle line:90% So let me end with two slides-- 00:54:38.220 --> 00:54:41.190 align:middle line:84% the lecture about consumer choice, 00:54:41.190 --> 00:54:44.340 align:middle line:84% when the consumer decides how much leisure 00:54:44.340 --> 00:54:47.670 align:middle line:84% to provide or negatively, how much to work 00:54:47.670 --> 00:54:49.830 align:middle line:90% and how much consumption to buy. 00:54:49.830 --> 00:54:53.100 align:middle line:84% So it's going to be one thing to say right away-- 00:54:53.100 --> 00:54:55.620 align:middle line:84% leisure is going to be a good thing. 00:54:55.620 --> 00:54:57.490 align:middle line:90% Consumption is a good thing. 00:54:57.490 --> 00:54:59.940 align:middle line:84% So those axes that were labeled x and y 00:54:59.940 --> 00:55:02.310 align:middle line:84% are now labeled c and l, where c is 00:55:02.310 --> 00:55:05.160 align:middle line:90% consumption and l is leisure. 00:55:05.160 --> 00:55:07.980 align:middle line:84% So it looks standard now, but it's a much more interesting 00:55:07.980 --> 00:55:08.710 align:middle line:90% problem. 00:55:08.710 --> 00:55:09.960 align:middle line:90% It's how much to work problem. 00:55:09.960 --> 00:55:12.180 align:middle line:84% It's like the bicycle messengers. 00:55:12.180 --> 00:55:15.900 align:middle line:84% The other thing is maximize utility subject to the budget. 00:55:15.900 --> 00:55:18.330 align:middle line:84% So this is kind of a twist because instead 00:55:18.330 --> 00:55:20.610 align:middle line:84% of having income over here, we've 00:55:20.610 --> 00:55:25.530 align:middle line:84% got the wage, w, per unit time times the time endowment 00:55:25.530 --> 00:55:27.720 align:middle line:84% that the household is endowed with. 00:55:27.720 --> 00:55:32.310 align:middle line:84% So this is as if you took your entire time endowment, 00:55:32.310 --> 00:55:36.810 align:middle line:84% worked all of it, and then went back and bought some leisure 00:55:36.810 --> 00:55:38.580 align:middle line:90% and bought consumption. 00:55:38.580 --> 00:55:41.250 align:middle line:84% Probably a more obvious way to put this 00:55:41.250 --> 00:55:45.420 align:middle line:84% is to take wl, subtract it from both sides. 00:55:45.420 --> 00:55:47.640 align:middle line:90% Then you have T minus l. 00:55:47.640 --> 00:55:51.370 align:middle line:84% So time minus leisure is the amount spent working. 00:55:51.370 --> 00:55:52.830 align:middle line:84% You surrendered it to the market. 00:55:52.830 --> 00:55:56.100 align:middle line:84% You don't enjoy it, but you do get the wage for it. 00:55:56.100 --> 00:55:58.680 align:middle line:84% And then this becomes income, and income 00:55:58.680 --> 00:56:00.263 align:middle line:90% is spent on consumption. 00:56:00.263 --> 00:56:01.680 align:middle line:84% The other thing about this problem 00:56:01.680 --> 00:56:04.350 align:middle line:84% is like the choice of a numeraire, 00:56:04.350 --> 00:56:07.140 align:middle line:84% everything is measured in terms of consumption. 00:56:07.140 --> 00:56:08.190 align:middle line:90% There's no price. 00:56:08.190 --> 00:56:09.780 align:middle line:90% You could have done it. 00:56:09.780 --> 00:56:13.440 align:middle line:84% You could have price of consumption times consumption, 00:56:13.440 --> 00:56:17.280 align:middle line:84% price of leisure, the wage times the leisure. 00:56:17.280 --> 00:56:19.600 align:middle line:90% But we just normalized wages. 00:56:19.600 --> 00:56:22.320 align:middle line:84% So this is what's called the real wage. 00:56:22.320 --> 00:56:24.540 align:middle line:84% It's how much consumption bundle you can 00:56:24.540 --> 00:56:27.840 align:middle line:90% get for a given unit of labor. 00:56:27.840 --> 00:56:29.250 align:middle line:90% And here's the picture. 00:56:29.250 --> 00:56:35.610 align:middle line:84% So you're effectively starting here, with 16 hours a day 00:56:35.610 --> 00:56:37.650 align:middle line:90% of leisure. 00:56:37.650 --> 00:56:42.060 align:middle line:84% I don't know, some sleep is allowed, eight hour bedtime. 00:56:42.060 --> 00:56:43.710 align:middle line:90% The remaining is 16 hours. 00:56:43.710 --> 00:56:46.900 align:middle line:84% But you may want to earn some money. 00:56:46.900 --> 00:56:48.780 align:middle line:84% So if you move in this direction, 00:56:48.780 --> 00:56:53.100 align:middle line:84% you're giving up a unit of leisure, working that unit, 00:56:53.100 --> 00:56:55.500 align:middle line:84% and getting the wage for it, which 00:56:55.500 --> 00:56:57.340 align:middle line:90% allows you to buy consumption. 00:56:57.340 --> 00:57:01.290 align:middle line:84% So the slope of this line is the wage. 00:57:01.290 --> 00:57:03.100 align:middle line:84% In this case, the wage rate is 10. 00:57:03.100 --> 00:57:06.360 align:middle line:84% So every unit of work that you supply 00:57:06.360 --> 00:57:08.970 align:middle line:84% gives you 10 units worth of consumption. 00:57:08.970 --> 00:57:14.895 align:middle line:84% And you could work your entire 16-hour day at a wage of 10, 00:57:14.895 --> 00:57:17.730 align:middle line:84% and have 160 units of consumption. 00:57:17.730 --> 00:57:20.040 align:middle line:84% But this guy is stopping short of that, 00:57:20.040 --> 00:57:23.440 align:middle line:84% where the indifference curve is tangent to the budget line. 00:57:23.440 --> 00:57:26.140 align:middle line:84% So this is the choice of how hard to work. 00:57:26.140 --> 00:57:30.600 align:middle line:84% And again, I could have put in a dynamic choice problem. 00:57:30.600 --> 00:57:32.040 align:middle line:90% I could have put time-- 00:57:32.040 --> 00:57:34.453 align:middle line:84% consumption today versus consumption tomorrow. 00:57:34.453 --> 00:57:36.870 align:middle line:84% We're going to have one of those at the end of the lecture 00:57:36.870 --> 00:57:37.530 align:middle line:90% next time. 00:57:37.530 --> 00:57:39.240 align:middle line:84% As a constant reminder, we could have 00:57:39.240 --> 00:57:41.580 align:middle line:90% had choice about locations. 00:57:41.580 --> 00:57:45.820 align:middle line:84% And we'll do some of that pretty shortly, as well. 00:57:45.820 --> 00:57:47.920 align:middle line:84% All right, so that's the prepared slides. 00:57:47.920 --> 00:57:53.450 align:middle line:84% And we have one or two minutes left for questions. 00:57:53.450 --> 00:57:55.970 align:middle line:84% All right, so you have all the slides. 00:57:55.970 --> 00:57:57.650 align:middle line:90% Take a look at them, study them. 00:57:57.650 --> 00:57:59.797 align:middle line:84% You also have the review sheet questions. 00:57:59.797 --> 00:58:01.630 align:middle line:84% Go back and take a look and make sure you're 00:58:01.630 --> 00:58:02.740 align:middle line:90% able to answer them. 00:58:02.740 --> 00:58:05.720 align:middle line:84% We'll have a little review next time of what we did today. 00:58:05.720 --> 00:58:10.330 align:middle line:84% And then we're going to go on to explore these expenditure 00:58:10.330 --> 00:58:13.300 align:middle line:84% shares, and income expansion paths, 00:58:13.300 --> 00:58:16.930 align:middle line:84% and whether poor people consume luxury goods 00:58:16.930 --> 00:58:18.520 align:middle line:90% or the opposite, et cetera. 00:58:18.520 --> 00:58:22.120 align:middle line:90% So that's for next time. 00:58:22.120 --> 00:58:25.000 align:middle line:84% I've already mentioned last time something drawing 00:58:25.000 --> 00:58:27.490 align:middle line:84% on a real application from the material we're 00:58:27.490 --> 00:58:30.760 align:middle line:84% going to do in the lecture next Tuesday on China, 00:58:30.760 --> 00:58:37.270 align:middle line:84% and kind of varying these price ratios of these budget lines 00:58:37.270 --> 00:58:41.030 align:middle line:84% by subsidizing consumer purchases. 00:58:41.030 --> 00:58:43.690 align:middle line:84% It's a wonderful problem set, worked out 00:58:43.690 --> 00:58:45.970 align:middle line:84% by one of your former classmates, actually, 00:58:45.970 --> 00:58:48.460 align:middle line:90% who I hired as a UROP. 00:58:48.460 --> 00:58:51.460 align:middle line:84% So you're almost in the position to start 00:58:51.460 --> 00:58:54.640 align:middle line:84% to take the look at that, especially after Friday, 00:58:54.640 --> 00:58:56.350 align:middle line:90% and certainly after Tuesday. 00:58:56.350 --> 00:58:59.400 align:middle line:90% So that's all I have for today. 00:58:59.400 --> 00:59:06.000 align:middle line:90%