WEBVTT 00:00:00.000 --> 00:00:01.924 align:middle line:90% [SQUEAKING] 00:00:01.924 --> 00:00:02.886 align:middle line:90% [RUSTLING] 00:00:02.886 --> 00:00:05.772 align:middle line:90% [CLICKING] 00:00:05.772 --> 00:00:11.063 align:middle line:90% 00:00:11.063 --> 00:00:13.690 align:middle line:84% AUDIENCE: Hi, Professor Townsend. 00:00:13.690 --> 00:00:18.610 align:middle line:84% Can I ask a question about the income factor 00:00:18.610 --> 00:00:21.700 align:middle line:90% on the Slutsky equation? 00:00:21.700 --> 00:00:23.350 align:middle line:90% ROBERT TOWNSEND: Yes. 00:00:23.350 --> 00:00:28.690 align:middle line:84% AUDIENCE: Yeah, I noticed that in the last lecture slides, 00:00:28.690 --> 00:00:35.230 align:middle line:84% the income factor, you'll note that it was x multiply data x 00:00:35.230 --> 00:00:37.660 align:middle line:90% over [INAUDIBLE]-- 00:00:37.660 --> 00:00:44.920 align:middle line:84% why it is not minus x modify data x over data y. 00:00:44.920 --> 00:00:48.190 align:middle line:84% ROBERT TOWNSEND: You're wondering about the sign. 00:00:48.190 --> 00:00:50.020 align:middle line:90% AUDIENCE: Yeah. 00:00:50.020 --> 00:00:54.690 align:middle line:84% ROBERT TOWNSEND: Yeah, it is confusing because the way that 00:00:54.690 --> 00:00:56.970 align:middle line:84% equation is written, it looks as if we're-- 00:00:56.970 --> 00:00:59.652 align:middle line:90% 00:00:59.652 --> 00:01:00.360 align:middle line:90% I can't remember. 00:01:00.360 --> 00:01:01.380 align:middle line:90% Which way is it written? 00:01:01.380 --> 00:01:03.602 align:middle line:90% Minus, right? 00:01:03.602 --> 00:01:05.540 align:middle line:90% AUDIENCE: Yeah, minus x. 00:01:05.540 --> 00:01:07.820 align:middle line:84% ROBERT TOWNSEND: Subtracting the income effect, 00:01:07.820 --> 00:01:12.830 align:middle line:84% but the graph on the previous slide 00:01:12.830 --> 00:01:15.290 align:middle line:84% says we're adding the income effect. 00:01:15.290 --> 00:01:19.970 align:middle line:84% If prices go down and income effectively increases, 00:01:19.970 --> 00:01:25.370 align:middle line:84% then you buy more than you would if you held income constant. 00:01:25.370 --> 00:01:31.040 align:middle line:84% So the answer to the puzzle is the demand 00:01:31.040 --> 00:01:36.830 align:middle line:84% curve figures have reversed the dependent 00:01:36.830 --> 00:01:39.320 align:middle line:90% and the independent variables. 00:01:39.320 --> 00:01:41.960 align:middle line:84% That is to say, we usually write dy dx, 00:01:41.960 --> 00:01:43.820 align:middle line:84% where y is the dependent variable, 00:01:43.820 --> 00:01:47.180 align:middle line:84% x is the right-hand side independent variable, 00:01:47.180 --> 00:01:53.240 align:middle line:84% but the demand curves actually put price on the y-axis. 00:01:53.240 --> 00:01:55.910 align:middle line:90% 00:01:55.910 --> 00:02:01.740 align:middle line:84% So it's the inverse slope, if that makes sense. 00:02:01.740 --> 00:02:07.320 align:middle line:84% So the slope dy dx, as written in that Slutsky equation, 00:02:07.320 --> 00:02:11.830 align:middle line:84% is the inverse of the slope in the diagram. 00:02:11.830 --> 00:02:14.710 align:middle line:90% AUDIENCE: OK, thank you. 00:02:14.710 --> 00:02:17.020 align:middle line:84% ROBERT TOWNSEND: Yeah, don't worry. 00:02:17.020 --> 00:02:19.810 align:middle line:90% I stumble over that myself. 00:02:19.810 --> 00:02:22.240 align:middle line:84% I was pondering it, and I actually 00:02:22.240 --> 00:02:23.650 align:middle line:90% asked Michael to look at it. 00:02:23.650 --> 00:02:27.580 align:middle line:84% And I'd forgotten that strange labeling that we use, 00:02:27.580 --> 00:02:30.825 align:middle line:90% so it is a little bit unusual. 00:02:30.825 --> 00:02:33.550 align:middle line:90% 00:02:33.550 --> 00:02:36.520 align:middle line:84% Thank you for asking the question. 00:02:36.520 --> 00:02:38.480 align:middle line:90% I do always start the class. 00:02:38.480 --> 00:02:41.950 align:middle line:84% So you've got us off slightly early, but that's fine. 00:02:41.950 --> 00:02:45.150 align:middle line:90% Are there other questions? 00:02:45.150 --> 00:02:50.370 align:middle line:84% The reading list for this lecture, it's just one starred 00:02:50.370 --> 00:02:53.100 align:middle line:90% reading here, Kreps chapter 7. 00:02:53.100 --> 00:02:56.640 align:middle line:84% So again, a lot but not all of the material today 00:02:56.640 --> 00:02:59.910 align:middle line:84% is in that text, and it's starred for a reason. 00:02:59.910 --> 00:03:01.980 align:middle line:84% I think you will find it to be very 00:03:01.980 --> 00:03:03.730 align:middle line:90% helpful to read through it. 00:03:03.730 --> 00:03:04.230 align:middle line:90% Let's see. 00:03:04.230 --> 00:03:05.970 align:middle line:90% So that's that. 00:03:05.970 --> 00:03:10.800 align:middle line:84% And then the study guide on what we did last time. 00:03:10.800 --> 00:03:14.310 align:middle line:84% OK, income and substitution effects-- 00:03:14.310 --> 00:03:18.780 align:middle line:84% can someone volunteer to answer this request 00:03:18.780 --> 00:03:21.300 align:middle line:90% to discuss Engel's law? 00:03:21.300 --> 00:03:25.950 align:middle line:84% Be as clear as possible about the data used, 00:03:25.950 --> 00:03:31.730 align:middle line:84% and what you would find if you look at the data properly? 00:03:31.730 --> 00:03:36.970 align:middle line:84% AUDIENCE: Engel's law is the law it 00:03:36.970 --> 00:03:42.370 align:middle line:84% says that the higher proportion of your income 00:03:42.370 --> 00:03:43.555 align:middle line:90% is spent on food, right? 00:03:43.555 --> 00:03:48.510 align:middle line:90% 00:03:48.510 --> 00:03:51.220 align:middle line:84% ROBERT TOWNSEND: The higher is your income, 00:03:51.220 --> 00:03:54.670 align:middle line:84% some good expenditure shares go up, 00:03:54.670 --> 00:03:57.950 align:middle line:84% and some expenditure shares go down. 00:03:57.950 --> 00:04:02.380 align:middle line:84% So the exact statement of it was, I guess, in this figure 00:04:02.380 --> 00:04:03.130 align:middle line:90% with the data. 00:04:03.130 --> 00:04:06.720 align:middle line:90% 00:04:06.720 --> 00:04:11.370 align:middle line:84% And food is a necessity, so the consumption actually 00:04:11.370 --> 00:04:19.089 align:middle line:84% goes down or rises less rapidly than income does. 00:04:19.089 --> 00:04:21.790 align:middle line:90% So thank you. 00:04:21.790 --> 00:04:24.930 align:middle line:84% So that's where it is in the lecture. 00:04:24.930 --> 00:04:32.760 align:middle line:84% One comment I have, this is about expenditure shares. 00:04:32.760 --> 00:04:39.410 align:middle line:84% It doesn't say the quantity of food. 00:04:39.410 --> 00:04:43.040 align:middle line:84% It says 3 times the quantity of food is going up. 00:04:43.040 --> 00:04:45.227 align:middle line:84% So expenditures for food are going up. 00:04:45.227 --> 00:04:46.310 align:middle line:90% It could be a normal good. 00:04:46.310 --> 00:04:51.380 align:middle line:84% But it's not going up as fast as total income is going up. 00:04:51.380 --> 00:04:56.370 align:middle line:90% 00:04:56.370 --> 00:05:00.870 align:middle line:84% So last time I drew what I called an income expansion 00:05:00.870 --> 00:05:04.110 align:middle line:84% path, which traces out the locus of tangencies 00:05:04.110 --> 00:05:07.300 align:middle line:84% as income rises, holding prices fixed. 00:05:07.300 --> 00:05:14.260 align:middle line:84% And if they were linear, then the expenditure shares 00:05:14.260 --> 00:05:16.060 align:middle line:90% are constant. 00:05:16.060 --> 00:05:18.770 align:middle line:90% That is to say, income goes up. 00:05:18.770 --> 00:05:21.010 align:middle line:84% The amount spent on the first good goes-- 00:05:21.010 --> 00:05:23.560 align:middle line:84% the amount spent on the second good goes up. 00:05:23.560 --> 00:05:25.870 align:middle line:84% Everything is rising proportionally. 00:05:25.870 --> 00:05:28.790 align:middle line:90% 00:05:28.790 --> 00:05:32.450 align:middle line:84% In order to get the expenditure share 00:05:32.450 --> 00:05:35.960 align:middle line:84% to be not rising as fast for food, 00:05:35.960 --> 00:05:38.840 align:middle line:84% you have to have those income expansion 00:05:38.840 --> 00:05:46.860 align:middle line:84% paths be somewhat curved so that you don't get 00:05:46.860 --> 00:05:52.068 align:middle line:90% a proportional rise in food. 00:05:52.068 --> 00:05:53.610 align:middle line:84% And really, answering a question that 00:05:53.610 --> 00:05:57.990 align:middle line:84% was asked in class last time, or at least 00:05:57.990 --> 00:06:03.160 align:middle line:84% what I thought about afterwards might be helpful. 00:06:03.160 --> 00:06:05.190 align:middle line:90% Another way to say this. 00:06:05.190 --> 00:06:06.910 align:middle line:90% We talked about Giffen goods. 00:06:06.910 --> 00:06:08.730 align:middle line:84% Oh, well, maybe that's a question. 00:06:08.730 --> 00:06:11.280 align:middle line:84% So someone volunteer to tell me, what is a Giffen good? 00:06:11.280 --> 00:06:18.470 align:middle line:90% 00:06:18.470 --> 00:06:20.630 align:middle line:84% AUDIENCE: It's when the price and the quantity 00:06:20.630 --> 00:06:23.310 align:middle line:84% demanded move in the same direction. 00:06:23.310 --> 00:06:26.600 align:middle line:84% So if the price of something goes up, then you want-- 00:06:26.600 --> 00:06:29.160 align:middle line:90% then you buy more of it. 00:06:29.160 --> 00:06:30.410 align:middle line:90% ROBERT TOWNSEND: That's right. 00:06:30.410 --> 00:06:32.720 align:middle line:90% OK, perfect. 00:06:32.720 --> 00:06:33.810 align:middle line:90% Perfect answer. 00:06:33.810 --> 00:06:36.350 align:middle line:84% So the point is, the quantity is what 00:06:36.350 --> 00:06:38.750 align:middle line:84% we have in mind when we talk about Giffen goods, 00:06:38.750 --> 00:06:44.150 align:middle line:84% not price times quantity, not expenditure shares. 00:06:44.150 --> 00:06:47.630 align:middle line:84% What creates a Giffen good is a good 00:06:47.630 --> 00:06:51.500 align:middle line:84% which is not normal, that is to say, 00:06:51.500 --> 00:06:56.470 align:middle line:84% where expenditures are falling, relatively speaking, 00:06:56.470 --> 00:06:58.890 align:middle line:90% as we increase income. 00:06:58.890 --> 00:07:05.440 align:middle line:84% And the point is there's two effects of price changes. 00:07:05.440 --> 00:07:08.830 align:middle line:84% Engel's law is about income changes. 00:07:08.830 --> 00:07:12.160 align:middle line:84% Giffen's paradox is about a price change. 00:07:12.160 --> 00:07:13.900 align:middle line:84% But the price change can be broken 00:07:13.900 --> 00:07:16.540 align:middle line:84% into two pieces, the substitution 00:07:16.540 --> 00:07:20.470 align:middle line:84% effect and the consequent income effect. 00:07:20.470 --> 00:07:26.330 align:middle line:84% And the substitution effect does mean that as price goes up, 00:07:26.330 --> 00:07:28.910 align:middle line:90% you would demand less. 00:07:28.910 --> 00:07:32.310 align:middle line:84% But the income effect means that as price goes up, 00:07:32.310 --> 00:07:35.040 align:middle line:84% you have effectively less income. 00:07:35.040 --> 00:07:37.550 align:middle line:84% And if it's a necessity, then you're 00:07:37.550 --> 00:07:40.880 align:middle line:84% going to spend relatively less on that good. 00:07:40.880 --> 00:07:46.250 align:middle line:84% So that's the-- tends to make the quantity go down, 00:07:46.250 --> 00:07:49.060 align:middle line:84% so it's about income effect dominating 00:07:49.060 --> 00:07:56.740 align:middle line:84% the price effect with implications for quantities. 00:07:56.740 --> 00:07:58.340 align:middle line:90% I hope that's helpful. 00:07:58.340 --> 00:08:01.420 align:middle line:84% Another way to say this is when we look at income expansion 00:08:01.420 --> 00:08:04.300 align:middle line:84% paths just changing income, we're not 00:08:04.300 --> 00:08:08.580 align:middle line:90% moving the price around at all. 00:08:08.580 --> 00:08:11.990 align:middle line:84% And we are assuming somehow you can make comparisons 00:08:11.990 --> 00:08:15.110 align:middle line:84% across to different income groups, 00:08:15.110 --> 00:08:17.370 align:middle line:84% that they're all facing the same prices. 00:08:17.370 --> 00:08:20.870 align:middle line:84% It's just that some people have more income than others, which 00:08:20.870 --> 00:08:24.620 align:middle line:84% is effectively what would happen for a given individual 00:08:24.620 --> 00:08:30.580 align:middle line:84% if somehow a poor individual were given more income. 00:08:30.580 --> 00:08:34.630 align:middle line:84% But it's all about moving income around in the cross-section 00:08:34.630 --> 00:08:37.900 align:middle line:84% or as if, over time for a given individual, 00:08:37.900 --> 00:08:40.419 align:middle line:84% holding the prices fixed, whereas the Giffen 00:08:40.419 --> 00:08:45.250 align:middle line:84% good has to do entirely with just changing prices. 00:08:45.250 --> 00:08:48.130 align:middle line:84% Income changes only as a consequence of the price 00:08:48.130 --> 00:08:52.488 align:middle line:84% change, and the Giffen good is about quantity. 00:08:52.488 --> 00:08:53.530 align:middle line:90% So I hope that's helpful. 00:08:53.530 --> 00:08:56.280 align:middle line:90% 00:08:56.280 --> 00:08:59.720 align:middle line:84% Can someone do the last one here? 00:08:59.720 --> 00:09:03.290 align:middle line:84% Compare and contrast the utility maximization problem 00:09:03.290 --> 00:09:08.550 align:middle line:84% with the expenditure minimization problem. 00:09:08.550 --> 00:09:10.770 align:middle line:84% Essentially, give a statement of duality. 00:09:10.770 --> 00:09:11.880 align:middle line:90% What does duality mean? 00:09:11.880 --> 00:09:17.110 align:middle line:90% 00:09:17.110 --> 00:09:18.840 align:middle line:84% AUDIENCE: Yep, so the duality just 00:09:18.840 --> 00:09:22.410 align:middle line:84% means that these two ways of formulating the consumer's 00:09:22.410 --> 00:09:25.440 align:middle line:84% choice problem are, in fact, just identical. 00:09:25.440 --> 00:09:27.810 align:middle line:84% Utility maximization says given a budget, 00:09:27.810 --> 00:09:29.760 align:middle line:84% what's the highest utility I can achieve? 00:09:29.760 --> 00:09:35.160 align:middle line:84% And expenditure minimization is given a level of utility 00:09:35.160 --> 00:09:36.990 align:middle line:84% I want to hit, what's the minimum budget I 00:09:36.990 --> 00:09:37.830 align:middle line:90% need to hit that? 00:09:37.830 --> 00:09:42.270 align:middle line:84% And these two will, they're basically identical problems 00:09:42.270 --> 00:09:43.660 align:middle line:90% of each other. 00:09:43.660 --> 00:09:46.400 align:middle line:90% ROBERT TOWNSEND: Perfect. 00:09:46.400 --> 00:09:47.725 align:middle line:90% That's a perfect answer. 00:09:47.725 --> 00:09:51.250 align:middle line:90% 00:09:51.250 --> 00:09:52.000 align:middle line:90% Thank you. 00:09:52.000 --> 00:09:54.700 align:middle line:90% I can't add anything. 00:09:54.700 --> 00:10:00.450 align:middle line:84% OK, so as always, it's quite apparent 00:10:00.450 --> 00:10:05.640 align:middle line:84% these are review questions that you should go over after class 00:10:05.640 --> 00:10:07.500 align:middle line:84% and before the next class to make sure 00:10:07.500 --> 00:10:09.270 align:middle line:84% you're really following the material 00:10:09.270 --> 00:10:15.120 align:middle line:84% to go with the production lecture, lecture 4. 00:10:15.120 --> 00:10:17.620 align:middle line:84% To get reoriented, we have an economy, 00:10:17.620 --> 00:10:22.590 align:middle line:84% which consists of preferences, endowments, and technology. 00:10:22.590 --> 00:10:26.040 align:middle line:84% Last two lectures, we focused on preferences. 00:10:26.040 --> 00:10:30.550 align:middle line:84% Oh, I should have also said, the application at the end 00:10:30.550 --> 00:10:32.310 align:middle line:84% having to do with borrowing and lending, 00:10:32.310 --> 00:10:37.890 align:middle line:84% in addition to labeling the goods by the date, date 00:10:37.890 --> 00:10:40.950 align:middle line:84% 1 and day 2, it was quite apparent 00:10:40.950 --> 00:10:43.960 align:middle line:84% that we had also actually plotted the endowment. 00:10:43.960 --> 00:10:46.020 align:middle line:84% So that was a bit different than what 00:10:46.020 --> 00:10:49.950 align:middle line:84% we had been doing with good x and good y, where we just 00:10:49.950 --> 00:10:51.550 align:middle line:90% talked about income. 00:10:51.550 --> 00:10:57.270 align:middle line:84% So in addition to having income i as exogenous in that diagram, 00:10:57.270 --> 00:10:59.880 align:middle line:84% we had the endowments, the amount 00:10:59.880 --> 00:11:01.980 align:middle line:84% of money they had in day 1, the amount of money 00:11:01.980 --> 00:11:04.470 align:middle line:84% they had in day 2, and plotted the endowment 00:11:04.470 --> 00:11:05.830 align:middle line:90% point in the diagram. 00:11:05.830 --> 00:11:07.920 align:middle line:84% Which is actually, either way is fine. 00:11:07.920 --> 00:11:10.140 align:middle line:90% But we did add more information. 00:11:10.140 --> 00:11:13.440 align:middle line:84% Anyway, you've seen endowments and preferences. 00:11:13.440 --> 00:11:16.110 align:middle line:84% Today, we're going to do technology. 00:11:16.110 --> 00:11:19.500 align:middle line:84% In choice of overproduction plans, 00:11:19.500 --> 00:11:23.190 align:middle line:84% we'll explore some possible properties 00:11:23.190 --> 00:11:28.600 align:middle line:84% that technology sets can have, as we did for consumer 00:11:28.600 --> 00:11:30.200 align:middle line:90% preferences, so on. 00:11:30.200 --> 00:11:32.660 align:middle line:84% And then we'll go to the application, 00:11:32.660 --> 00:11:35.890 align:middle line:84% which is going to be similar to what we did for households. 00:11:35.890 --> 00:11:40.000 align:middle line:84% We're going to have firms maximizing profits given 00:11:40.000 --> 00:11:42.560 align:middle line:90% a production possibility set. 00:11:42.560 --> 00:11:45.910 align:middle line:84% And we'll take that into an application. 00:11:45.910 --> 00:11:48.320 align:middle line:84% And actually, by the end of this lecture, 00:11:48.320 --> 00:11:52.690 align:middle line:84% we will actually look at our first entirely specified 00:11:52.690 --> 00:11:55.420 align:middle line:84% economy, Robinson Crusoe economy, 00:11:55.420 --> 00:11:58.370 align:middle line:84% and begin to talk about international trade. 00:11:58.370 --> 00:12:02.770 align:middle line:84% So this slide is meant to portray technology. 00:12:02.770 --> 00:12:06.160 align:middle line:84% It's a picture I took, actually, in Northern Thailand, where 00:12:06.160 --> 00:12:08.750 align:middle line:90% I was doing my field research. 00:12:08.750 --> 00:12:11.350 align:middle line:84% And if you're really into economics, 00:12:11.350 --> 00:12:15.460 align:middle line:84% you'll start talking about land, labor, and capital as inputs 00:12:15.460 --> 00:12:16.520 align:middle line:90% into production. 00:12:16.520 --> 00:12:18.970 align:middle line:90% And here you see all three. 00:12:18.970 --> 00:12:21.850 align:middle line:90% The land is kind of obvious. 00:12:21.850 --> 00:12:24.220 align:middle line:90% He's getting ready to plant. 00:12:24.220 --> 00:12:26.740 align:middle line:84% Labor, this guy, who's showing off, 00:12:26.740 --> 00:12:29.860 align:middle line:84% by the way, because he's not using any hands. 00:12:29.860 --> 00:12:32.650 align:middle line:84% He's not grabbing the handlebars on what you might 00:12:32.650 --> 00:12:35.080 align:middle line:90% think is like a lawnmower. 00:12:35.080 --> 00:12:40.060 align:middle line:84% And then this tiller, which they call an iron buffalo, 00:12:40.060 --> 00:12:43.810 align:middle line:84% replaced the buffalo that used to do the plowing, 00:12:43.810 --> 00:12:47.470 align:middle line:84% and it's just this small horsepower motor driving 00:12:47.470 --> 00:12:50.440 align:middle line:90% the treads on the wheels. 00:12:50.440 --> 00:12:52.810 align:middle line:90% But anyway, that's the capital-- 00:12:52.810 --> 00:12:54.790 align:middle line:90% land, labor, and capital. 00:12:54.790 --> 00:12:58.600 align:middle line:90% So a picture is worth 1,000. 00:12:58.600 --> 00:13:02.410 align:middle line:84% Anyway, production, OK-- so firms 00:13:02.410 --> 00:13:06.310 align:middle line:84% possess production capabilities, meaning 00:13:06.310 --> 00:13:08.710 align:middle line:90% some commodities are inputs. 00:13:08.710 --> 00:13:10.150 align:middle line:90% Others are outputs. 00:13:10.150 --> 00:13:16.520 align:middle line:84% And the capability is described by a set of possible values, 00:13:16.520 --> 00:13:18.190 align:middle line:84% which you can think of as vectors 00:13:18.190 --> 00:13:21.130 align:middle line:84% with inputs and outputs, or some people say netput. 00:13:21.130 --> 00:13:25.000 align:middle line:84% If we have the number of goods, including inputs and outputs, 00:13:25.000 --> 00:13:28.570 align:middle line:84% being of dimension capital K, then a vector 00:13:28.570 --> 00:13:33.790 align:middle line:84% has dimension capital K, and the set of all feasible production 00:13:33.790 --> 00:13:37.180 align:middle line:84% vectors is called the production possibility set. 00:13:37.180 --> 00:13:40.930 align:middle line:84% Now, we do have a convention, namely, 00:13:40.930 --> 00:13:44.530 align:middle line:84% inputs are negative, or at least not positive, 00:13:44.530 --> 00:13:47.860 align:middle line:84% and outputs or positive, or at least not negative. 00:13:47.860 --> 00:13:51.880 align:middle line:84% And with a couple of exceptions, I'm going to continue that. 00:13:51.880 --> 00:13:56.920 align:middle line:84% It is easier to interpret if we designate the inputs 00:13:56.920 --> 00:14:00.820 align:middle line:84% and talk about labor and capital and land 00:14:00.820 --> 00:14:03.700 align:middle line:84% as being positive into a production function 00:14:03.700 --> 00:14:06.760 align:middle line:90% to get rice as the output. 00:14:06.760 --> 00:14:09.970 align:middle line:84% But for reasons of very succinct accounting, 00:14:09.970 --> 00:14:13.000 align:middle line:84% it's quite useful to have the inputs as negative 00:14:13.000 --> 00:14:16.270 align:middle line:90% and the outputs as positive. 00:14:16.270 --> 00:14:21.040 align:middle line:84% You'll see more of that powerful notation a little bit later. 00:14:21.040 --> 00:14:27.100 align:middle line:84% Anyway, so for example, if we have three commodities, two 00:14:27.100 --> 00:14:29.950 align:middle line:84% of which are inputs, say, labor and steel, 00:14:29.950 --> 00:14:32.830 align:middle line:84% numbered 1 and 2 in the vector, and the output 00:14:32.830 --> 00:14:35.800 align:middle line:84% being safety pins, then, for example, 00:14:35.800 --> 00:14:41.700 align:middle line:84% we have a netput vector of minus 5, minus 8, and 10. 00:14:41.700 --> 00:14:44.160 align:middle line:84% And the inputs are each negative, 00:14:44.160 --> 00:14:48.940 align:middle line:84% and the output, the steel safety pins, is positive. 00:14:48.940 --> 00:14:53.800 align:middle line:90% So not too hard. 00:14:53.800 --> 00:14:58.510 align:middle line:84% More generally, Z could be a production possibility set. 00:14:58.510 --> 00:15:01.900 align:middle line:84% Sometimes we call it Y. I apologize. 00:15:01.900 --> 00:15:05.830 align:middle line:90% We used different sources here. 00:15:05.830 --> 00:15:10.690 align:middle line:84% We can write Z from 1 to K as this vector, 00:15:10.690 --> 00:15:13.800 align:middle line:84% where, again, K is the number of commodities. 00:15:13.800 --> 00:15:19.770 align:middle line:84% And if we wish, we had a firm J, we could index everything by J 00:15:19.770 --> 00:15:23.100 align:middle line:84% and talk about the production possibilities set of firm J 00:15:23.100 --> 00:15:29.310 align:middle line:84% and the inputs j1, j2, through jK. 00:15:29.310 --> 00:15:33.810 align:middle line:84% And then we drop the j for most of this talk. 00:15:33.810 --> 00:15:37.830 align:middle line:84% Now, this is very similar to what we did for consumption. 00:15:37.830 --> 00:15:44.010 align:middle line:84% We had x1 through xL, I think it was. 00:15:44.010 --> 00:15:51.730 align:middle line:84% And then we put an i for [INAUDIBLE] needed. 00:15:51.730 --> 00:15:53.670 align:middle line:90% So here, we're dropping the j. 00:15:53.670 --> 00:15:58.410 align:middle line:84% Here's a picture of a production possibility set, Z, 00:15:58.410 --> 00:16:03.870 align:middle line:84% where we have one input only, and it's negative. 00:16:03.870 --> 00:16:07.620 align:middle line:84% The more negative it is, the larger the quantity 00:16:07.620 --> 00:16:10.440 align:middle line:90% being utilized is. 00:16:10.440 --> 00:16:14.970 align:middle line:84% And likewise, if you use more of it, you get more output. 00:16:14.970 --> 00:16:20.130 align:middle line:84% This entire speckled region is illustrative 00:16:20.130 --> 00:16:24.040 align:middle line:84% of the entire production possibility set. 00:16:24.040 --> 00:16:26.820 align:middle line:84% You don't have to be on the boundary, which 00:16:26.820 --> 00:16:29.430 align:middle line:84% is the maximum output you can get for a given input 00:16:29.430 --> 00:16:30.600 align:middle line:90% in order to be in the set. 00:16:30.600 --> 00:16:33.150 align:middle line:84% That boundary is typically included in the set, 00:16:33.150 --> 00:16:37.810 align:middle line:84% but so are these interior points. 00:16:37.810 --> 00:16:40.000 align:middle line:90% So here are possible properties. 00:16:40.000 --> 00:16:41.790 align:middle line:84% And again, just like for the household, 00:16:41.790 --> 00:16:45.990 align:middle line:84% we didn't insist that every class of every production 00:16:45.990 --> 00:16:48.000 align:middle line:84% possibility set contain each of these. 00:16:48.000 --> 00:16:51.660 align:middle line:84% In fact, some of them contradict each other, 00:16:51.660 --> 00:16:53.230 align:middle line:90% as you'll see in a minute. 00:16:53.230 --> 00:16:55.140 align:middle line:84% So we usually assume it's not empty. 00:16:55.140 --> 00:16:58.950 align:middle line:84% I mean, why are we writing it down if there's nothing to do? 00:16:58.950 --> 00:17:02.430 align:middle line:84% Closed means it contains its boundary. 00:17:02.430 --> 00:17:04.480 align:middle line:90% That's one way to say it. 00:17:04.480 --> 00:17:07.740 align:middle line:84% Another is just that every limit point in-- 00:17:07.740 --> 00:17:10.380 align:middle line:84% every sequence that consists of points 00:17:10.380 --> 00:17:15.599 align:middle line:84% in the set that converges has a point that is also in the set. 00:17:15.599 --> 00:17:17.970 align:middle line:90% That's a technical condition. 00:17:17.970 --> 00:17:23.790 align:middle line:84% I'm sure you've noticed I'm not overdoing the continuity 00:17:23.790 --> 00:17:25.890 align:middle line:90% part and the closure part. 00:17:25.890 --> 00:17:28.830 align:middle line:84% It's a bit more technical than we need, 00:17:28.830 --> 00:17:32.580 align:middle line:84% and these properties are pretty intuitive. 00:17:32.580 --> 00:17:36.720 align:middle line:84% No free lunch means that whenever Z is in the set 00:17:36.720 --> 00:17:41.320 align:middle line:84% and has no negative components, you can't get any output. 00:17:41.320 --> 00:17:42.660 align:middle line:90% So Z is a vector. 00:17:42.660 --> 00:17:46.380 align:middle line:84% Z can have positive and negative components. 00:17:46.380 --> 00:17:51.330 align:middle line:84% Here, writing this Z vector being non-negative 00:17:51.330 --> 00:17:53.940 align:middle line:84% means that no component in the vector 00:17:53.940 --> 00:17:57.720 align:middle line:84% is negative, which means they're either 0 or positive. 00:17:57.720 --> 00:18:00.900 align:middle line:84% And if they were, say, all 0, then you have no inputs, 00:18:00.900 --> 00:18:02.370 align:middle line:90% and you can't get any output. 00:18:02.370 --> 00:18:04.350 align:middle line:90% You can't get lunch out of that. 00:18:04.350 --> 00:18:06.600 align:middle line:84% And that's the expression "no free lunch." 00:18:06.600 --> 00:18:08.970 align:middle line:84% This is used by cynical economists 00:18:08.970 --> 00:18:11.910 align:middle line:84% to mean something else, which means if someone offered you 00:18:11.910 --> 00:18:15.480 align:middle line:84% something for free, be careful because there may be 00:18:15.480 --> 00:18:17.070 align:middle line:90% [INAUDIBLE]. 00:18:17.070 --> 00:18:20.760 align:middle line:84% Anyway, I won't share that too much. 00:18:20.760 --> 00:18:24.750 align:middle line:84% Convexity-- if you have two elements, Z and Z prime, 00:18:24.750 --> 00:18:28.860 align:middle line:84% in the set, then the alpha-weighted combination 00:18:28.860 --> 00:18:30.430 align:middle line:90% is also in the set. 00:18:30.430 --> 00:18:32.980 align:middle line:84% We did this with consumption sets, 00:18:32.980 --> 00:18:35.610 align:middle line:90% so this is not surprising. 00:18:35.610 --> 00:18:38.670 align:middle line:84% Free disposal means that if you have 00:18:38.670 --> 00:18:45.390 align:middle line:84% a vector Z in the production set and Z prime is not higher-- 00:18:45.390 --> 00:18:47.220 align:middle line:84% say it's less than or equal to Z-- 00:18:47.220 --> 00:18:50.640 align:middle line:84% then that Z prime is also in the set. 00:18:50.640 --> 00:18:53.200 align:middle line:90% This is called free disposal. 00:18:53.200 --> 00:18:54.640 align:middle line:84% There are two ways to look at it. 00:18:54.640 --> 00:18:55.770 align:middle line:90% It's kind of the same. 00:18:55.770 --> 00:18:58.560 align:middle line:84% One-- suppose Z prime had the same amount 00:18:58.560 --> 00:19:01.410 align:middle line:84% of all the negative inputs, quantities 00:19:01.410 --> 00:19:05.230 align:middle line:84% of all the inputs, which are negative, but at less output. 00:19:05.230 --> 00:19:09.410 align:middle line:84% So the positive components are not greater and maybe 00:19:09.410 --> 00:19:10.860 align:middle line:90% extremely less. 00:19:10.860 --> 00:19:13.740 align:middle line:84% Then you're perfectly free to throw away that extra output 00:19:13.740 --> 00:19:17.760 align:middle line:84% and maintain yourself in the production possibilities set. 00:19:17.760 --> 00:19:20.640 align:middle line:84% A little less obvious, if Z prime 00:19:20.640 --> 00:19:24.780 align:middle line:84% were say strictly less than Z because the input quantities 00:19:24.780 --> 00:19:26.800 align:middle line:90% are actually higher-- 00:19:26.800 --> 00:19:30.250 align:middle line:84% and again, the negative makes it look like it's less. 00:19:30.250 --> 00:19:33.310 align:middle line:84% Higher inputs means a bigger negative number. 00:19:33.310 --> 00:19:34.860 align:middle line:90% Then that's also in the set. 00:19:34.860 --> 00:19:35.640 align:middle line:90% Why? 00:19:35.640 --> 00:19:37.680 align:middle line:84% Because if you've got more inputs-- 00:19:37.680 --> 00:19:39.990 align:middle line:84% say you're maintaining the same output. 00:19:39.990 --> 00:19:42.210 align:middle line:84% Well, if the inputs were at all productive, 00:19:42.210 --> 00:19:43.590 align:middle line:90% you would have had more output. 00:19:43.590 --> 00:19:45.160 align:middle line:90% But you throw away that output. 00:19:45.160 --> 00:19:47.640 align:middle line:90% That's the free disposal again. 00:19:47.640 --> 00:19:50.160 align:middle line:84% And then, finally, you can always shut down. 00:19:50.160 --> 00:19:53.220 align:middle line:84% Most, I think, production functions, production sets have 00:19:53.220 --> 00:19:55.890 align:middle line:84% that property, that 0 is feasible, 00:19:55.890 --> 00:19:59.880 align:middle line:84% which is a bit different from saying it's empty. 00:19:59.880 --> 00:20:02.250 align:middle line:84% It means 0 is in there, but other things are usually 00:20:02.250 --> 00:20:04.360 align:middle line:90% typically also in there. 00:20:04.360 --> 00:20:09.750 align:middle line:84% So three other properties-- decreasing returns to scale. 00:20:09.750 --> 00:20:11.760 align:middle line:90% And I'll show you some pictures. 00:20:11.760 --> 00:20:15.120 align:middle line:84% If Z is in there and alpha is less than 1, 00:20:15.120 --> 00:20:17.170 align:middle line:84% then alpha times Z is also in there. 00:20:17.170 --> 00:20:19.680 align:middle line:84% So decreasing returns actually means 00:20:19.680 --> 00:20:24.360 align:middle line:84% you can scale back something in the production function, 00:20:24.360 --> 00:20:27.690 align:middle line:90% like US Steel blast furnaces. 00:20:27.690 --> 00:20:32.310 align:middle line:84% And somehow they can pare that down and make a smaller one. 00:20:32.310 --> 00:20:35.580 align:middle line:84% So it may not be true, but that's what the property is. 00:20:35.580 --> 00:20:37.440 align:middle line:84% Constant returns means that you can 00:20:37.440 --> 00:20:41.220 align:middle line:84% scale up or down willy-nilly and still 00:20:41.220 --> 00:20:45.390 align:middle line:84% have the vector alpha Z in the production set when Z is. 00:20:45.390 --> 00:20:48.520 align:middle line:84% And increasing returns goes the other way. 00:20:48.520 --> 00:20:51.990 align:middle line:84% If you have Z in the set and alpha is greater than 1, 00:20:51.990 --> 00:20:55.270 align:middle line:84% then alpha times Z is also in the production set, 00:20:55.270 --> 00:20:57.720 align:middle line:84% which means, in this case, to continue 00:20:57.720 --> 00:20:59.820 align:middle line:90% the example of US Steel. 00:20:59.820 --> 00:21:02.250 align:middle line:84% If you still has one big blast furnace, 00:21:02.250 --> 00:21:05.580 align:middle line:84% they can just as easily have two or three. 00:21:05.580 --> 00:21:06.960 align:middle line:90% They can run them. 00:21:06.960 --> 00:21:09.760 align:middle line:84% OK, so let me draw you some pictures. 00:21:09.760 --> 00:21:14.550 align:middle line:84% This is decreasing returns, which intuitively 00:21:14.550 --> 00:21:20.010 align:middle line:84% comes from the curvature declining 00:21:20.010 --> 00:21:24.990 align:middle line:84% of the boundary of the production possibility set. 00:21:24.990 --> 00:21:28.320 align:middle line:84% The property says if you're given Y in the set 00:21:28.320 --> 00:21:32.760 align:middle line:84% and it's not 0, then alpha between 0 and 1 00:21:32.760 --> 00:21:38.050 align:middle line:84% times Y is also in the set, which these things are. 00:21:38.050 --> 00:21:40.930 align:middle line:84% This is increasing returns, which is, 00:21:40.930 --> 00:21:44.680 align:middle line:84% whoa, getting steeper and steeper, 00:21:44.680 --> 00:21:48.210 align:middle line:84% and then there's [INAUDIBLE],, but it's instructive. 00:21:48.210 --> 00:21:51.990 align:middle line:84% So this is the same alpha Y, yes. 00:21:51.990 --> 00:21:54.090 align:middle line:84% But the property of increasing returns 00:21:54.090 --> 00:22:01.230 align:middle line:84% says if you have Y, say, here and alpha is greater than 1, 00:22:01.230 --> 00:22:02.940 align:middle line:90% then alpha lies in the set. 00:22:02.940 --> 00:22:05.310 align:middle line:90% So, actually, the error-- 00:22:05.310 --> 00:22:06.810 align:middle line:90% we'll fix this, I guess-- 00:22:06.810 --> 00:22:10.950 align:middle line:84% is that the alpha Y should have been depicted out here 00:22:10.950 --> 00:22:12.090 align:middle line:90% in the production set. 00:22:12.090 --> 00:22:16.840 align:middle line:84% All this area is in the production set. 00:22:16.840 --> 00:22:21.430 align:middle line:84% And then constant returns is neither of those extremes. 00:22:21.430 --> 00:22:23.840 align:middle line:90% It's just a linear boundary. 00:22:23.840 --> 00:22:25.900 align:middle line:84% And it says if you pick anything in the set, 00:22:25.900 --> 00:22:31.330 align:middle line:84% say, here, and then draw a line, a ray, from the 0 00:22:31.330 --> 00:22:34.450 align:middle line:84% through that picked point, everything 00:22:34.450 --> 00:22:39.190 align:middle line:84% on that ray towards 0 or out to infinity is also in the set. 00:22:39.190 --> 00:22:41.710 align:middle line:84% And I guess we should have put alpha down here instead 00:22:41.710 --> 00:22:44.110 align:middle line:84% of lambda, but it's kind of obvious. 00:22:44.110 --> 00:22:46.420 align:middle line:84% So constant returns to scale is going 00:22:46.420 --> 00:22:51.160 align:middle line:84% to have some strong properties, which we'll come to. 00:22:51.160 --> 00:22:56.240 align:middle line:84% One word about aggregating up production sets-- 00:22:56.240 --> 00:23:00.820 align:middle line:84% so this is one of the exceptions, where what-- 00:23:00.820 --> 00:23:03.110 align:middle line:84% well, this is the classic picture. 00:23:03.110 --> 00:23:05.360 align:middle line:90% Inputs on the x-axis. 00:23:05.360 --> 00:23:06.865 align:middle line:90% Output on the y-axis. 00:23:06.865 --> 00:23:08.240 align:middle line:84% This would be the production set. 00:23:08.240 --> 00:23:10.460 align:middle line:84% Here's a point on the boundary of one 00:23:10.460 --> 00:23:14.460 align:middle line:84% firm's, labeled number 1 firm here. 00:23:14.460 --> 00:23:18.050 align:middle line:84% Also, the exception is we're labeling firms 1 and 2. 00:23:18.050 --> 00:23:21.800 align:middle line:84% This is the production set of another one 00:23:21.800 --> 00:23:26.090 align:middle line:84% that actually uses as an input the output that this produces. 00:23:26.090 --> 00:23:27.890 align:middle line:84% And I'll come back to that later. 00:23:27.890 --> 00:23:33.230 align:middle line:84% In this case, the inputs and outputs are reversed. 00:23:33.230 --> 00:23:36.740 align:middle line:84% And finally, we have an endowment point. 00:23:36.740 --> 00:23:38.690 align:middle line:84% So we wanted to add all these things up 00:23:38.690 --> 00:23:42.140 align:middle line:84% and get the aggregate set of possibilities. 00:23:42.140 --> 00:23:43.970 align:middle line:84% You got to do a lot of vector algebra. 00:23:43.970 --> 00:23:47.750 align:middle line:84% But essentially, what's going on here is start at 0. 00:23:47.750 --> 00:23:48.860 align:middle line:90% That's feasible. 00:23:48.860 --> 00:23:50.480 align:middle line:90% It uses less than endowment. 00:23:50.480 --> 00:23:51.800 align:middle line:90% That's fine. 00:23:51.800 --> 00:23:56.690 align:middle line:84% Then find y1 on the first production set-- y1. 00:23:56.690 --> 00:23:59.060 align:middle line:90% Then add that endowment back in. 00:23:59.060 --> 00:24:02.210 align:middle line:84% Notice this, the parallel line shift. 00:24:02.210 --> 00:24:06.290 align:middle line:84% And then, so far, we've been keeping y2 at 0, 00:24:06.290 --> 00:24:10.250 align:middle line:84% but let's move y2 to something non-trivial production. 00:24:10.250 --> 00:24:13.230 align:middle line:84% That movement here is the same as this movement here. 00:24:13.230 --> 00:24:18.410 align:middle line:84% So at this point, which is, of course, the sum of y1, y2, 00:24:18.410 --> 00:24:22.370 align:middle line:84% and z1, is feasible because it comes 00:24:22.370 --> 00:24:25.020 align:middle line:90% from these three possibilities. 00:24:25.020 --> 00:24:29.720 align:middle line:84% So this larger area out here is the aggregated or aggregate 00:24:29.720 --> 00:24:31.190 align:middle line:90% production set. 00:24:31.190 --> 00:24:33.750 align:middle line:84% Now let's talk about the firm's problem. 00:24:33.750 --> 00:24:36.170 align:middle line:84% So we're already getting into an application now. 00:24:36.170 --> 00:24:40.580 align:middle line:84% There are these prices, p, which the firm takes as given, 00:24:40.580 --> 00:24:43.230 align:middle line:84% just like the household took prices as given. 00:24:43.230 --> 00:24:48.440 align:middle line:84% And the firm wants to come up with a plan, input-output plan, 00:24:48.440 --> 00:24:50.790 align:middle line:90% to maximize its profit. 00:24:50.790 --> 00:24:56.090 align:middle line:84% So this capital pi here is just a statement of prices time pk 00:24:56.090 --> 00:25:00.170 align:middle line:84% times zk, summing up over all the k's. 00:25:00.170 --> 00:25:03.530 align:middle line:84% And here, again, it looks like everything's positive. 00:25:03.530 --> 00:25:04.610 align:middle line:90% What a wonderful world. 00:25:04.610 --> 00:25:08.720 align:middle line:84% But some of these z's are required to be negative. 00:25:08.720 --> 00:25:12.960 align:middle line:84% And hence, [INAUDIBLE] times a negative object. 00:25:12.960 --> 00:25:16.712 align:middle line:84% So there's expenses in here and revenues in here. 00:25:16.712 --> 00:25:18.920 align:middle line:84% And then when we talk about the firm's problem, which 00:25:18.920 --> 00:25:23.900 align:middle line:84% is FP, for Firm's Problem, is to maximize the profits 00:25:23.900 --> 00:25:27.410 align:middle line:84% by choosing feasible points vectors in the production 00:25:27.410 --> 00:25:29.210 align:middle line:90% possibilities set. 00:25:29.210 --> 00:25:34.160 align:middle line:84% So this very cryptic abbreviated notation, max p of z, 00:25:34.160 --> 00:25:37.820 align:middle line:90% [? proves ?] z and z. 00:25:37.820 --> 00:25:39.660 align:middle line:90% And here's our picture. 00:25:39.660 --> 00:25:43.070 align:middle line:84% So inputs, which are negative in sign, 00:25:43.070 --> 00:25:45.140 align:middle line:90% outputs, which are positive. 00:25:45.140 --> 00:25:48.860 align:middle line:84% These guys here, these linear hyperplanes, 00:25:48.860 --> 00:25:51.740 align:middle line:84% represent possible profit levels. 00:25:51.740 --> 00:25:54.350 align:middle line:90% Call them isoprofit lines. 00:25:54.350 --> 00:25:56.150 align:middle line:84% When we introduced indifference curves, 00:25:56.150 --> 00:25:59.810 align:middle line:84% I said it might be better to call them isoutility lines, 00:25:59.810 --> 00:26:02.750 align:middle line:84% and I was anticipating this slide 00:26:02.750 --> 00:26:04.370 align:middle line:90% and another one that's coming. 00:26:04.370 --> 00:26:09.440 align:middle line:84% So an isoprofit line means as you, say, use more inputs 00:26:09.440 --> 00:26:13.670 align:middle line:84% but get more output, you have expenses balancing revenues, 00:26:13.670 --> 00:26:15.450 align:middle line:90% and you stay on this line. 00:26:15.450 --> 00:26:21.380 align:middle line:84% If you go put in lower inputs, so for the same prices, 00:26:21.380 --> 00:26:23.840 align:middle line:90% you would have higher profits. 00:26:23.840 --> 00:26:28.400 align:middle line:84% The slope of any isoprofit line is just 00:26:28.400 --> 00:26:31.250 align:middle line:84% price of output divided by the price of input, 00:26:31.250 --> 00:26:33.320 align:middle line:90% like p2 over p1. 00:26:33.320 --> 00:26:35.990 align:middle line:84% And of course, we're depicting this tangency 00:26:35.990 --> 00:26:37.760 align:middle line:90% as the maximum profit line. 00:26:37.760 --> 00:26:40.400 align:middle line:84% You could be in the interior, but then you 00:26:40.400 --> 00:26:42.560 align:middle line:90% would not be maximizing. 00:26:42.560 --> 00:26:51.380 align:middle line:84% So let's call little pi of p to be the maximized profit, not 00:26:51.380 --> 00:26:55.100 align:middle line:84% just any old profit, which was capital pi a minute ago. 00:26:55.100 --> 00:27:00.260 align:middle line:84% This is the best choice of production vector 00:27:00.260 --> 00:27:02.690 align:middle line:84% z in the production set, given price 00:27:02.690 --> 00:27:05.450 align:middle line:90% is p. p is a parameter here. 00:27:05.450 --> 00:27:07.970 align:middle line:84% This is, quote, partial equilibrium, so 00:27:07.970 --> 00:27:11.330 align:middle line:90% we index the solution by p. 00:27:11.330 --> 00:27:14.360 align:middle line:84% Of course, we'll do experiments where we vary p. 00:27:14.360 --> 00:27:15.980 align:middle line:90% There are some profits-- 00:27:15.980 --> 00:27:19.530 align:middle line:84% properties of the profit function, 00:27:19.530 --> 00:27:23.340 align:middle line:84% namely, this profit function pi is 00:27:23.340 --> 00:27:28.290 align:middle line:84% homogeneous of degree 1 in the price vector p. 00:27:28.290 --> 00:27:32.610 align:middle line:84% Homogeneous of degree 1 means that if you scale prices 00:27:32.610 --> 00:27:36.090 align:middle line:84% up by some proportion, like doubling them, 00:27:36.090 --> 00:27:40.680 align:middle line:84% then maximized profit should also double. 00:27:40.680 --> 00:27:44.280 align:middle line:84% The reason that should be obvious, at least when 00:27:44.280 --> 00:27:47.280 align:middle line:84% you think about it, if we're doubling prices, 00:27:47.280 --> 00:27:49.920 align:middle line:90% we're not changing the slope. 00:27:49.920 --> 00:27:53.630 align:middle line:84% So we're not changing the maximizing choice. 00:27:53.630 --> 00:27:57.320 align:middle line:84% But we have changed the levels of the prices, 00:27:57.320 --> 00:28:01.130 align:middle line:84% and the levels means even though the z's aren't moving, 00:28:01.130 --> 00:28:02.150 align:middle line:90% the prices are moving. 00:28:02.150 --> 00:28:04.340 align:middle line:84% If prices have doubled, then it must 00:28:04.340 --> 00:28:10.760 align:middle line:84% be that profits have doubled and constant returns to scale. 00:28:10.760 --> 00:28:13.040 align:middle line:84% Homogeneous to degree one, sorry-- 00:28:13.040 --> 00:28:15.170 align:middle line:90% homogeneous to degree 1. 00:28:15.170 --> 00:28:19.160 align:middle line:90% Profit function is continuous. 00:28:19.160 --> 00:28:21.940 align:middle line:84% It's actually true much more generally than you might think, 00:28:21.940 --> 00:28:27.790 align:middle line:84% but I'm not into emphasizing continuity too much these days. 00:28:27.790 --> 00:28:29.690 align:middle line:90% Profit function is convex. 00:28:29.690 --> 00:28:31.390 align:middle line:90% That's an np. 00:28:31.390 --> 00:28:33.140 align:middle line:84% So that's an interesting property. 00:28:33.140 --> 00:28:34.990 align:middle line:90% So let's look at that one. 00:28:34.990 --> 00:28:38.740 align:middle line:90% The profit function is convex. 00:28:38.740 --> 00:28:41.530 align:middle line:90% OK, what do we want to show? 00:28:41.530 --> 00:28:43.030 align:middle line:90% If we have two p's-- 00:28:43.030 --> 00:28:50.790 align:middle line:84% two prices p and [INAUDIBLE] the associated profits, pi of p 00:28:50.790 --> 00:28:54.240 align:middle line:84% and pi of p prime, then we take the linear combination 00:28:54.240 --> 00:28:58.410 align:middle line:84% of the prices, say, p double prime, this alpha weighted sum, 00:28:58.410 --> 00:29:02.628 align:middle line:84% the associated profits should be not greater 00:29:02.628 --> 00:29:04.170 align:middle line:84% than the weighted sum of the profits. 00:29:04.170 --> 00:29:05.790 align:middle line:90% So I said that in words. 00:29:05.790 --> 00:29:08.430 align:middle line:84% It's probably just a whole lot easier to do the mechanics. 00:29:08.430 --> 00:29:11.430 align:middle line:84% So we start with two prices, p and p prime. 00:29:11.430 --> 00:29:12.960 align:middle line:90% We pick an alpha. 00:29:12.960 --> 00:29:16.650 align:middle line:84% We consider p double prime to be the alpha-weighted sum. 00:29:16.650 --> 00:29:23.760 align:middle line:84% We let z be the solution to the firm's profit maximization 00:29:23.760 --> 00:29:27.070 align:middle line:90% problem at p double prime. 00:29:27.070 --> 00:29:31.500 align:middle line:84% So in other words, that special maximized profit pi 00:29:31.500 --> 00:29:34.170 align:middle line:84% of p double prime is just p double prime times z 00:29:34.170 --> 00:29:36.310 align:middle line:90% because z is the solution. 00:29:36.310 --> 00:29:39.240 align:middle line:90% Now, z, therefore, was feasible. 00:29:39.240 --> 00:29:41.940 align:middle line:84% It was in the production possibility set. 00:29:41.940 --> 00:29:44.250 align:middle line:84% So even though we changed the prices, 00:29:44.250 --> 00:29:49.260 align:middle line:84% that z remains feasible, which means whatever solves 00:29:49.260 --> 00:29:52.980 align:middle line:84% the problem at p, it cannot deliver a lower number than 00:29:52.980 --> 00:29:56.520 align:middle line:84% this and may deliver a higher one. 00:29:56.520 --> 00:29:58.950 align:middle line:84% The profit-maximizing solution at p 00:29:58.950 --> 00:30:03.150 align:middle line:84% might involve a different z different from z. 00:30:03.150 --> 00:30:06.270 align:middle line:84% And likewise, the profit-maximizing solution 00:30:06.270 --> 00:30:10.680 align:middle line:84% at p prime can only be not lower and maybe higher 00:30:10.680 --> 00:30:14.370 align:middle line:84% than p prime evaluated at this z. 00:30:14.370 --> 00:30:17.020 align:middle line:90% OK, so now we do the math. 00:30:17.020 --> 00:30:20.110 align:middle line:84% We want to show that pi at p double prime, 00:30:20.110 --> 00:30:23.880 align:middle line:84% which is, by definition, p double prime at z. 00:30:23.880 --> 00:30:25.050 align:middle line:90% Remember the starting point. 00:30:25.050 --> 00:30:26.520 align:middle line:84% We're starting at p double prime, 00:30:26.520 --> 00:30:29.510 align:middle line:84% and we're told that z is the solution. 00:30:29.510 --> 00:30:31.840 align:middle line:90% So this is p double prime z. 00:30:31.840 --> 00:30:34.720 align:middle line:84% But p double prime written out is just alpha 00:30:34.720 --> 00:30:36.820 align:middle line:90% p, 1 minus alpha p prime-- 00:30:36.820 --> 00:30:39.580 align:middle line:84% that was the definition-- times z. 00:30:39.580 --> 00:30:43.610 align:middle line:84% And now we invoke each of these two weak inequalities and say, 00:30:43.610 --> 00:30:50.320 align:middle line:84% well, alpha p times z is alpha pz less than pi of p. 00:30:50.320 --> 00:30:52.780 align:middle line:84% So the inequality is less than or equal to, 00:30:52.780 --> 00:30:55.900 align:middle line:84% and we get the same thing for p prime. 00:30:55.900 --> 00:30:59.800 align:middle line:84% Sorry, I think the ratio of my words to what's on the slide-- 00:30:59.800 --> 00:31:01.360 align:middle line:84% I must have talked for three minutes. 00:31:01.360 --> 00:31:06.150 align:middle line:84% But there's only four lines here, so I apologize. 00:31:06.150 --> 00:31:11.415 align:middle line:84% Alpha function is convex, maybe weakly, but it's not concave, 00:31:11.415 --> 00:31:13.650 align:middle line:90% or not strictly concave. 00:31:13.650 --> 00:31:17.020 align:middle line:84% It may be convex or strictly convex. 00:31:17.020 --> 00:31:20.800 align:middle line:84% OK, so we'll come back to that momentarily. 00:31:20.800 --> 00:31:23.140 align:middle line:84% Homogeneity-- two more properties, 00:31:23.140 --> 00:31:24.590 align:middle line:90% and back to convexity. 00:31:24.590 --> 00:31:27.940 align:middle line:84% So if f is a real valued function, 00:31:27.940 --> 00:31:31.330 align:middle line:84% mapping rn into r and k is an integer, 00:31:31.330 --> 00:31:34.390 align:middle line:84% we say f is homogeneous of degree k 00:31:34.390 --> 00:31:37.900 align:middle line:84% if whenever you scale the arguments of the function up 00:31:37.900 --> 00:31:38.590 align:middle line:90% by-- 00:31:38.590 --> 00:31:42.670 align:middle line:84% the arguments of the function v by alpha, the value 00:31:42.670 --> 00:31:44.680 align:middle line:90% in the domain goes up by-- 00:31:44.680 --> 00:31:47.800 align:middle line:84% in the range goes up by alpha to the power k. 00:31:47.800 --> 00:31:54.490 align:middle line:84% So homogeneous of degree 0 would mean nothing changes. 00:31:54.490 --> 00:31:57.310 align:middle line:84% Homogeneous of degree 1 we just did 00:31:57.310 --> 00:32:00.640 align:middle line:84% for profits, where k is equal to 1, 00:32:00.640 --> 00:32:03.295 align:middle line:84% and this is the most general concept. 00:32:03.295 --> 00:32:06.200 align:middle line:90% 00:32:06.200 --> 00:32:09.950 align:middle line:84% Another property of the profit function-- 00:32:09.950 --> 00:32:13.280 align:middle line:84% it's continuously differentiable. 00:32:13.280 --> 00:32:15.590 align:middle line:84% So I'm going to show you an expression 00:32:15.590 --> 00:32:17.180 align:middle line:90% with the derivatives. 00:32:17.180 --> 00:32:22.370 align:middle line:84% If you take that function, which has as a parameter the price 00:32:22.370 --> 00:32:27.200 align:middle line:84% vector p, and differentiate the maximized profit with respect 00:32:27.200 --> 00:32:33.410 align:middle line:84% to, say, the k-th item, pk, holding all the prices-- 00:32:33.410 --> 00:32:39.110 align:middle line:84% doing it locally around the maximizing value p star, 00:32:39.110 --> 00:32:41.420 align:middle line:90% then the solution is the input. 00:32:41.420 --> 00:32:43.970 align:middle line:90% You get the input back. 00:32:43.970 --> 00:32:49.570 align:middle line:84% Derivative with respect to pk, you get the input zk. 00:32:49.570 --> 00:32:52.690 align:middle line:84% A quick, maybe misleading, intuition 00:32:52.690 --> 00:32:56.620 align:middle line:84% for that is you write down profits-- 00:32:56.620 --> 00:33:00.490 align:middle line:84% p1, z1, p2, z2, et cetera, et cetera. 00:33:00.490 --> 00:33:03.730 align:middle line:84% Take a derivative at the maximizing solution 00:33:03.730 --> 00:33:07.090 align:middle line:84% The derivative is with respective pk. 00:33:07.090 --> 00:33:08.810 align:middle line:90% Where does that show up? 00:33:08.810 --> 00:33:12.310 align:middle line:84% It shows up in the pk, zk term, so it seems 00:33:12.310 --> 00:33:15.140 align:middle line:90% like this ought to be true. 00:33:15.140 --> 00:33:19.230 align:middle line:84% Actually, what it's using is the convexity. 00:33:19.230 --> 00:33:22.310 align:middle line:84% So this thing is called Hotelling's lemma. 00:33:22.310 --> 00:33:25.910 align:middle line:84% Here is the actual profit function, but now 00:33:25.910 --> 00:33:27.740 align:middle line:90% with all the prices. 00:33:27.740 --> 00:33:31.640 align:middle line:84% It's actually got a star on everything but the k-th price 00:33:31.640 --> 00:33:33.440 align:middle line:90% here, no star. 00:33:33.440 --> 00:33:37.580 align:middle line:84% But we're going to evaluate profit at star k. 00:33:37.580 --> 00:33:40.910 align:middle line:84% So we're right here on this profit function. 00:33:40.910 --> 00:33:44.900 align:middle line:84% This would be the maximized profit in general, 00:33:44.900 --> 00:33:47.900 align:middle line:84% and also the maximized profit in particular 00:33:47.900 --> 00:33:51.890 align:middle line:84% when the price of the k-th input or output is pk. 00:33:51.890 --> 00:33:53.660 align:middle line:90% What is this n? 00:33:53.660 --> 00:33:55.520 align:middle line:90% And this is like a tangency? 00:33:55.520 --> 00:33:56.940 align:middle line:90% It's a straight line. 00:33:56.940 --> 00:33:58.670 align:middle line:90% And what is this straight line? 00:33:58.670 --> 00:34:02.600 align:middle line:84% That's just what I was saying in words. 00:34:02.600 --> 00:34:06.350 align:middle line:84% That's just the evaluation of inputs and outputs 00:34:06.350 --> 00:34:09.820 align:middle line:84% at this price vector star, and it starred everywhere. 00:34:09.820 --> 00:34:13.560 align:middle line:84% OK, so pk doesn't have a star here. 00:34:13.560 --> 00:34:17.480 align:middle line:84% But because we're plotting everything is a function of pq, 00:34:17.480 --> 00:34:20.929 align:middle line:84% but there is a particular pk, namely pk star, 00:34:20.929 --> 00:34:23.659 align:middle line:84% where we can evaluate that linear function. 00:34:23.659 --> 00:34:25.670 align:middle line:84% Now, the point of the tangency is 00:34:25.670 --> 00:34:29.100 align:middle line:84% that the curved function, the profit margin, 00:34:29.100 --> 00:34:32.280 align:middle line:84% and this linear function have the same slope. 00:34:32.280 --> 00:34:35.330 align:middle line:84% So if we want to know how the profit function is changing 00:34:35.330 --> 00:34:39.679 align:middle line:84% as we change pk locally, we can just as easily 00:34:39.679 --> 00:34:42.230 align:middle line:84% show how this straight line is changing 00:34:42.230 --> 00:34:46.340 align:middle line:84% as we change pk locally because they have the same slope. 00:34:46.340 --> 00:34:50.870 align:middle line:84% And how is the linear function changing as we change pk? 00:34:50.870 --> 00:34:52.580 align:middle line:84% Well, it's just changing linearly. 00:34:52.580 --> 00:34:55.190 align:middle line:84% So that's the derivative, namely zk. 00:34:55.190 --> 00:34:57.900 align:middle line:90% So that's the proof of this. 00:34:57.900 --> 00:35:00.500 align:middle line:84% So we did actually use the convexity of the profit 00:35:00.500 --> 00:35:03.475 align:middle line:84% function, and the picture is strict convexity. 00:35:03.475 --> 00:35:04.850 align:middle line:84% It actually works more generally, 00:35:04.850 --> 00:35:08.710 align:middle line:84% but it would be a pretty hard picture to interpret. 00:35:08.710 --> 00:35:12.050 align:middle line:84% Now, this is also illustrative of something else called 00:35:12.050 --> 00:35:16.190 align:middle line:84% the envelope theorem, and I'll just show you the definition. 00:35:16.190 --> 00:35:18.890 align:middle line:84% Oh, and remember the strategy here. 00:35:18.890 --> 00:35:23.440 align:middle line:84% Tools-- so every so often, we bump into a concept 00:35:23.440 --> 00:35:24.790 align:middle line:90% that generalizes. 00:35:24.790 --> 00:35:30.100 align:middle line:84% In this case, we just went over some example of something 00:35:30.100 --> 00:35:31.300 align:middle line:90% called the envelope theorem. 00:35:31.300 --> 00:35:34.540 align:middle line:84% So let me give you a statement of the general theorem. 00:35:34.540 --> 00:35:38.180 align:middle line:84% If fact, [INAUDIBLE] [? set ?] and let 00:35:38.180 --> 00:35:41.000 align:middle line:84% t be some parameter of the problem. 00:35:41.000 --> 00:35:43.390 align:middle line:84% We have a function f mapping elements 00:35:43.390 --> 00:35:45.880 align:middle line:84% in the choice set in that parameter 00:35:45.880 --> 00:35:52.250 align:middle line:84% into some real valued objective function, namely like this. 00:35:52.250 --> 00:35:55.270 align:middle line:84% So put max here if you're more comfortable. 00:35:55.270 --> 00:35:58.030 align:middle line:84% So f is the thing we want to max. 00:35:58.030 --> 00:36:01.230 align:middle line:84% We want to max it by choice of x. 00:36:01.230 --> 00:36:04.830 align:middle line:84% But we have a parameter in the problem called t, 00:36:04.830 --> 00:36:10.170 align:middle line:84% so we denote the solution, the maximized value of f, at t 00:36:10.170 --> 00:36:13.470 align:middle line:84% to be V of t, the value function at t. 00:36:13.470 --> 00:36:15.330 align:middle line:90% And what is X star? 00:36:15.330 --> 00:36:18.870 align:middle line:84% X star is just the set of all maximizers X. 00:36:18.870 --> 00:36:20.830 align:middle line:90% It could be unique. 00:36:20.830 --> 00:36:22.410 align:middle line:90% There could be many. 00:36:22.410 --> 00:36:25.020 align:middle line:84% But each one of them, if there are many, 00:36:25.020 --> 00:36:28.380 align:middle line:84% has to have the property that evaluating 00:36:28.380 --> 00:36:33.210 align:middle line:84% the function f at that maximizer x gives you V back. 00:36:33.210 --> 00:36:36.510 align:middle line:84% And again, everything is indexed by t. 00:36:36.510 --> 00:36:40.410 align:middle line:84% Now for the big E. If you took the derivative of the value 00:36:40.410 --> 00:36:44.940 align:middle line:84% function with respect to t, even though t is a parameter, 00:36:44.940 --> 00:36:50.100 align:middle line:84% it will be the derivative of the underlying 00:36:50.100 --> 00:36:52.530 align:middle line:84% function you're trying to maximize 00:36:52.530 --> 00:36:56.370 align:middle line:90% at the optimized solution. 00:36:56.370 --> 00:37:02.070 align:middle line:84% That is for an x in the set of maximizers at t. 00:37:02.070 --> 00:37:07.080 align:middle line:84% So one thing to say this is an example of that, right? 00:37:07.080 --> 00:37:12.100 align:middle line:84% We wanted the profit function, which 00:37:12.100 --> 00:37:15.430 align:middle line:84% is parameterized by not t, but p, 00:37:15.430 --> 00:37:18.910 align:middle line:84% and we wanted to know how it would behave at the optimum 00:37:18.910 --> 00:37:22.780 align:middle line:84% if we were to slightly change the parameter p. 00:37:22.780 --> 00:37:26.380 align:middle line:84% And we just took the derivative of the objective function 00:37:26.380 --> 00:37:30.010 align:middle line:84% at the optimizing solution, which 00:37:30.010 --> 00:37:35.320 align:middle line:84% here is what this line is at, the optimizing solution. 00:37:35.320 --> 00:37:38.740 align:middle line:90% And the derivative was just z. 00:37:38.740 --> 00:37:40.960 align:middle line:84% Well, at the risk of confusing the issue, 00:37:40.960 --> 00:37:44.170 align:middle line:84% there was this chat going on about lambda 00:37:44.170 --> 00:37:47.770 align:middle line:84% as the margin utility of income in the consumers problem. 00:37:47.770 --> 00:37:51.830 align:middle line:84% And, again, income was a parameter of the problem. 00:37:51.830 --> 00:37:55.810 align:middle line:84% And nevertheless, we wanted to know how utility would change-- 00:37:55.810 --> 00:37:58.870 align:middle line:90% in this case, utility f-- 00:37:58.870 --> 00:38:04.060 align:middle line:84% how utility would change as we change that parameter income i. 00:38:04.060 --> 00:38:07.840 align:middle line:84% And the answer from the envelope theorem is it 00:38:07.840 --> 00:38:12.250 align:middle line:84% is the derivative of a maximized objective function-- 00:38:12.250 --> 00:38:15.100 align:middle line:84% that is to say, of utility substituting in the budget 00:38:15.100 --> 00:38:16.330 align:middle line:90% constraint-- 00:38:16.330 --> 00:38:18.520 align:middle line:90% at parameter i. 00:38:18.520 --> 00:38:20.680 align:middle line:84% So that's consistent with this language 00:38:20.680 --> 00:38:22.900 align:middle line:84% of the marginal utility of income. 00:38:22.900 --> 00:38:25.690 align:middle line:84% In the chat, there is a great discussion 00:38:25.690 --> 00:38:30.760 align:middle line:84% about that Lagrangian, which is so powerful it's mysterious, 00:38:30.760 --> 00:38:35.230 align:middle line:84% because the Lagrangian has those Lagrange multipliers types 00:38:35.230 --> 00:38:38.800 align:middle line:84% constraints, not just the objective function. 00:38:38.800 --> 00:38:42.550 align:middle line:84% So the chat was, it's not really marginal utility because it's 00:38:42.550 --> 00:38:43.510 align:middle line:90% a Lagrangian. 00:38:43.510 --> 00:38:46.850 align:middle line:84% It didn't just write down the utility term, et cetera, 00:38:46.850 --> 00:38:47.350 align:middle line:90% et cetera. 00:38:47.350 --> 00:38:49.960 align:middle line:90% But you can see how it-- 00:38:49.960 --> 00:38:53.320 align:middle line:84% hopefully get a little more intuition about that problem 00:38:53.320 --> 00:38:56.020 align:middle line:90% from the envelope theorem. 00:38:56.020 --> 00:39:00.950 align:middle line:84% OK, so, as I said before, in many applications that we're 00:39:00.950 --> 00:39:03.800 align:middle line:84% clear about what's an input and what's an output, 00:39:03.800 --> 00:39:06.710 align:middle line:84% we could have, for example, n possible inputs 00:39:06.710 --> 00:39:09.800 align:middle line:84% and m possible outputs, and then we'd 00:39:09.800 --> 00:39:14.300 align:middle line:84% be writing these vectors as negatives for the n 00:39:14.300 --> 00:39:18.590 align:middle line:84% inputs and positive [INAUDIBLE] m outputs [INAUDIBLE] 00:39:18.590 --> 00:39:21.290 align:middle line:84% some degrees of freedom, depending on the production 00:39:21.290 --> 00:39:21.870 align:middle line:90% function. 00:39:21.870 --> 00:39:23.720 align:middle line:90% The other things being 0. 00:39:23.720 --> 00:39:26.810 align:middle line:84% There are things other firms can do this firm cannot. 00:39:26.810 --> 00:39:29.870 align:middle line:90% So we could maximize if-- 00:39:29.870 --> 00:39:30.960 align:middle line:90% even easier. 00:39:30.960 --> 00:39:36.740 align:middle line:84% There's a single output and capital N possible inputs. 00:39:36.740 --> 00:39:41.380 align:middle line:84% We can talk about a production function. 00:39:41.380 --> 00:39:46.450 align:middle line:84% Rice is a function of labor, holding capital and land fixed, 00:39:46.450 --> 00:39:48.260 align:middle line:90% for example. 00:39:48.260 --> 00:39:51.920 align:middle line:84% Or we could have multiple vectors x-- 00:39:51.920 --> 00:39:54.390 align:middle line:90% land, labor, and capital. 00:39:54.390 --> 00:39:57.530 align:middle line:84% And we write down this output function, this production 00:39:57.530 --> 00:39:59.530 align:middle line:90% function. 00:39:59.530 --> 00:40:05.380 align:middle line:84% We will have prices of the output p, say, taken as given. 00:40:05.380 --> 00:40:09.573 align:middle line:84% We'll have the prices of the N inputs, W1 00:40:09.573 --> 00:40:11.610 align:middle line:90% through wn taken as given. 00:40:11.610 --> 00:40:13.790 align:middle line:84% And we can write down the profit maximization 00:40:13.790 --> 00:40:17.060 align:middle line:84% problem, now with a bit more intuitive signs. 00:40:17.060 --> 00:40:21.530 align:middle line:84% It's revenue, price times output, minus the cost 00:40:21.530 --> 00:40:26.585 align:middle line:84% of the inputs that were used to create that output at the price 00:40:26.585 --> 00:40:27.740 align:middle line:90% is wn. 00:40:27.740 --> 00:40:32.060 align:middle line:84% If f is differentiable, then we can look for the first order 00:40:32.060 --> 00:40:32.710 align:middle line:90% conditions. 00:40:32.710 --> 00:40:35.120 align:middle line:90% That derivative's equal to 0. 00:40:35.120 --> 00:40:38.990 align:middle line:84% Say, in particular with respect to the nth input, 00:40:38.990 --> 00:40:44.810 align:middle line:84% we'll get p times the derivative of [? f of ?] xn over here, 00:40:44.810 --> 00:40:47.580 align:middle line:90% and we'll get wn here. 00:40:47.580 --> 00:40:49.030 align:middle line:90% That's just expression. 00:40:49.030 --> 00:40:51.990 align:middle line:84% So the shorthand way to say this is 00:40:51.990 --> 00:40:56.800 align:middle line:84% the value of the marginal product or the marginal revenue 00:40:56.800 --> 00:41:00.670 align:middle line:90% product is equal to the price. 00:41:00.670 --> 00:41:03.670 align:middle line:90% That's economist speak. 00:41:03.670 --> 00:41:05.800 align:middle line:84% If you think about what would happen when this is, 00:41:05.800 --> 00:41:11.410 align:middle line:84% say, greater than the right-hand side, that means one more 00:41:11.410 --> 00:41:14.890 align:middle line:90% unit of input costs you wn. 00:41:14.890 --> 00:41:18.730 align:middle line:84% But that input applied to the production function 00:41:18.730 --> 00:41:23.020 align:middle line:84% gives you a marginal output, and that marginal output 00:41:23.020 --> 00:41:27.730 align:middle line:84% evaluated at the price gives you the marginal revenue. 00:41:27.730 --> 00:41:29.830 align:middle line:84% So if the left-hand side is greater 00:41:29.830 --> 00:41:31.300 align:middle line:84% than the right-hand side, you can't 00:41:31.300 --> 00:41:35.140 align:middle line:84% be done because you get more out of increasing your costs. 00:41:35.140 --> 00:41:38.650 align:middle line:84% You more balance your costs by increasing your revenue. 00:41:38.650 --> 00:41:42.190 align:middle line:84% And likewise, when the inequality goes the other way, 00:41:42.190 --> 00:41:46.600 align:middle line:84% you've gone too far and used up too much of the inputs. 00:41:46.600 --> 00:41:49.830 align:middle line:84% So this is a very intuitive expression. 00:41:49.830 --> 00:41:53.700 align:middle line:84% So now let's talk about cost minimization. 00:41:53.700 --> 00:41:57.030 align:middle line:84% Instead of solving the global problem 00:41:57.030 --> 00:41:59.900 align:middle line:84% and finding the maximizing profit, 00:41:59.900 --> 00:42:03.180 align:middle line:84% maximizing inputs and outputs, we're 00:42:03.180 --> 00:42:05.190 align:middle line:84% going to break the problem into pieces 00:42:05.190 --> 00:42:09.315 align:middle line:84% and take the output y as a given, 00:42:09.315 --> 00:42:11.280 align:middle line:90% like something exogenous. 00:42:11.280 --> 00:42:13.280 align:middle line:90% Got to do it. 00:42:13.280 --> 00:42:14.780 align:middle line:90% Got to achieve it. 00:42:14.780 --> 00:42:18.120 align:middle line:84% But let's achieve it in the cost minimizing way. 00:42:18.120 --> 00:42:21.580 align:middle line:84% So here's a statement of the problem. 00:42:21.580 --> 00:42:28.600 align:middle line:84% If this V of y is the input set that delivers y as the output, 00:42:28.600 --> 00:42:32.050 align:middle line:84% taking y as given, we can choose any input vector, as long 00:42:32.050 --> 00:42:34.570 align:middle line:90% as it gives us the output of y. 00:42:34.570 --> 00:42:38.200 align:middle line:84% So let's minimize the cost, the dot product w 00:42:38.200 --> 00:42:42.940 align:middle line:84% times x, which is the cost for achieving that output y. 00:42:42.940 --> 00:42:47.960 align:middle line:84% And let's call that the cost function. 00:42:47.960 --> 00:42:52.810 align:middle line:84% It's the cost of producing y in the least possible cost way, 00:42:52.810 --> 00:42:56.890 align:middle line:90% given input, price, vector w. 00:42:56.890 --> 00:42:59.110 align:middle line:84% The price of output doesn't matter here 00:42:59.110 --> 00:43:01.750 align:middle line:90% because that's a given anyway. 00:43:01.750 --> 00:43:03.820 align:middle line:84% That is to say, price is not only given. 00:43:03.820 --> 00:43:05.020 align:middle line:90% Output is given. 00:43:05.020 --> 00:43:07.450 align:middle line:84% So revenue is not in question here. 00:43:07.450 --> 00:43:11.690 align:middle line:84% It's just the cost of financing it through the inputs. 00:43:11.690 --> 00:43:15.940 align:middle line:84% So we'll call the solution conditional factor demands. 00:43:15.940 --> 00:43:18.850 align:middle line:84% And in principle, they're a function 00:43:18.850 --> 00:43:25.000 align:middle line:90% of these parameters W and y. 00:43:25.000 --> 00:43:26.830 align:middle line:90% So let me draw you a picture. 00:43:26.830 --> 00:43:30.220 align:middle line:84% In this case, there's going to be an output level. 00:43:30.220 --> 00:43:32.920 align:middle line:84% That's to think of it as a single good. 00:43:32.920 --> 00:43:35.020 align:middle line:90% q is the level. 00:43:35.020 --> 00:43:36.700 align:middle line:90% And we're going to plot-- 00:43:36.700 --> 00:43:39.310 align:middle line:84% and there's two inputs, x1 and x2. 00:43:39.310 --> 00:43:44.140 align:middle line:84% And we're going to plot here the set of all inputs that give you 00:43:44.140 --> 00:43:45.910 align:middle line:90% the same output. 00:43:45.910 --> 00:43:47.260 align:middle line:90% That's that curve. 00:43:47.260 --> 00:43:49.990 align:middle line:84% Since the quantity is the same, namely q, 00:43:49.990 --> 00:43:51.490 align:middle line:90% we've got a name for this guy. 00:43:51.490 --> 00:43:56.230 align:middle line:84% It's isoquant, which is the final instance of iso 00:43:56.230 --> 00:43:57.250 align:middle line:90% this and that. 00:43:57.250 --> 00:44:04.270 align:middle line:84% So isoquant for same quantity as we move along this isoquant. 00:44:04.270 --> 00:44:06.700 align:middle line:84% And what are these straight lines? 00:44:06.700 --> 00:44:09.850 align:middle line:84% Our expenditure curves-- they're down here. 00:44:09.850 --> 00:44:13.330 align:middle line:84% w times w1 x1 plus w2 x2 equals c, 00:44:13.330 --> 00:44:15.730 align:middle line:90% where c is some level of cost. 00:44:15.730 --> 00:44:19.210 align:middle line:84% The higher ones are associated with higher costs. 00:44:19.210 --> 00:44:23.500 align:middle line:84% To minimize [INAUDIBLE] one, move towards 0. 00:44:23.500 --> 00:44:26.740 align:middle line:90% So the solution is here. 00:44:26.740 --> 00:44:29.710 align:middle line:84% It's the minimized cost that allows 00:44:29.710 --> 00:44:31.780 align:middle line:90% you to be on the isoquant. 00:44:31.780 --> 00:44:35.650 align:middle line:84% In fact, the way it's drawn, and the other point on the isoquant 00:44:35.650 --> 00:44:38.770 align:middle line:84% would give strictly higher costs. 00:44:38.770 --> 00:44:40.860 align:middle line:90% You with me? 00:44:40.860 --> 00:44:45.240 align:middle line:84% So you can also characterize this tangency. 00:44:45.240 --> 00:44:50.610 align:middle line:84% The slope of this cost line is w1 over w2. 00:44:50.610 --> 00:44:55.020 align:middle line:84% The slope of the isoquant is like a marginal rate 00:44:55.020 --> 00:44:58.980 align:middle line:84% of substitution, except we call it the marginal rate of product 00:44:58.980 --> 00:44:59.970 align:middle line:90% transformation. 00:44:59.970 --> 00:45:05.010 align:middle line:84% It's the rate at which you can give up the use of input 2 00:45:05.010 --> 00:45:10.210 align:middle line:84% and use input 1 in such a way as to maintain quantity constant. 00:45:10.210 --> 00:45:13.950 align:middle line:84% So this is very similar to the indifference curve 00:45:13.950 --> 00:45:16.200 align:middle line:90% that we had in consumer theory. 00:45:16.200 --> 00:45:19.530 align:middle line:84% And this characterization of the tangency 00:45:19.530 --> 00:45:24.270 align:middle line:84% is similar in terms of the ratios of the two tangent lines 00:45:24.270 --> 00:45:26.200 align:middle line:90% being the same. 00:45:26.200 --> 00:45:28.860 align:middle line:84% AUDIENCE: So the thing that's getting toggled in this case 00:45:28.860 --> 00:45:32.637 align:middle line:90% is the budget line, right? 00:45:32.637 --> 00:45:34.970 align:middle line:84% ROBERT TOWNSEND: The thing that we're trying to minimize 00:45:34.970 --> 00:45:35.640 align:middle line:90% is cost. 00:45:35.640 --> 00:45:38.510 align:middle line:84% So this, instead of being a budget line, 00:45:38.510 --> 00:45:40.425 align:middle line:90% it's an isocost line. 00:45:40.425 --> 00:45:41.300 align:middle line:90% AUDIENCE: Yeah, yeah. 00:45:41.300 --> 00:45:44.690 align:middle line:84% ROBERT TOWNSEND: Yeah, and we're trying to move this way 00:45:44.690 --> 00:45:48.770 align:middle line:90% to make it as low as possible. 00:45:48.770 --> 00:45:50.510 align:middle line:84% Yeah, so that's another difference. 00:45:50.510 --> 00:45:55.308 align:middle line:84% When we did utility max, we had a budget, 00:45:55.308 --> 00:45:56.600 align:middle line:90% and we were trying to maximize. 00:45:56.600 --> 00:46:00.650 align:middle line:84% Although, we did in the duality lecture 00:46:00.650 --> 00:46:03.440 align:middle line:84% talk about minimizing the cost of achieving 00:46:03.440 --> 00:46:06.960 align:middle line:84% a certain level of utility, and this is very close to that. 00:46:06.960 --> 00:46:09.110 align:middle line:84% So here, we're minimizing the cost 00:46:09.110 --> 00:46:11.270 align:middle line:84% of achieving this level of output. 00:46:11.270 --> 00:46:14.290 align:middle line:90% 00:46:14.290 --> 00:46:20.410 align:middle line:84% OK, let me just say in words, this is a particular q. 00:46:20.410 --> 00:46:23.020 align:middle line:84% I could have picked another one, which would minimize 00:46:23.020 --> 00:46:25.170 align:middle line:90% the cost of achieving that q. 00:46:25.170 --> 00:46:27.760 align:middle line:84% And then we would get cost curves, 00:46:27.760 --> 00:46:30.680 align:middle line:84% which are going to show up momentarily. 00:46:30.680 --> 00:46:33.320 align:middle line:84% But that was too much clutter to put on the diagram, 00:46:33.320 --> 00:46:36.120 align:middle line:90% so we didn't do it. 00:46:36.120 --> 00:46:40.020 align:middle line:84% Instead, we put on this, the idea 00:46:40.020 --> 00:46:43.180 align:middle line:90% of changing the ratio of costs. 00:46:43.180 --> 00:46:51.120 align:middle line:84% So say Disney Company producing movies uses humans, animators, 00:46:51.120 --> 00:46:52.300 align:middle line:90% and computers. 00:46:52.300 --> 00:46:55.140 align:middle line:90% 00:46:55.140 --> 00:47:00.870 align:middle line:84% [INAUDIBLE] the year, computers got cheaper relative 00:47:00.870 --> 00:47:03.370 align:middle line:90% to the wages of humans. 00:47:03.370 --> 00:47:10.530 align:middle line:84% So in the early 1900s, 1990s, the slope of this line 00:47:10.530 --> 00:47:14.400 align:middle line:84% would be the cost of computers divided 00:47:14.400 --> 00:47:16.920 align:middle line:90% by the cost of animators-- 00:47:16.920 --> 00:47:18.810 align:middle line:90% relatively flat. 00:47:18.810 --> 00:47:22.470 align:middle line:84% By today's standards, the line is steeper. 00:47:22.470 --> 00:47:27.780 align:middle line:84% So the cost of computers, well-- it's-- 00:47:27.780 --> 00:47:30.930 align:middle line:84% always have to remember, it's like p1 over p2. 00:47:30.930 --> 00:47:34.470 align:middle line:84% So the cost of animators relative to the cost 00:47:34.470 --> 00:47:38.310 align:middle line:84% of computers has become a lot steeper. 00:47:38.310 --> 00:47:41.520 align:middle line:84% And of course, you're seeing different tangencies A and B. 00:47:41.520 --> 00:47:44.730 align:middle line:84% Naturally, Walt Disney Company would 00:47:44.730 --> 00:47:48.570 align:middle line:84% switch in the direction of using the cheaper 00:47:48.570 --> 00:47:53.640 align:middle line:84% input, namely, computers at the expense of using more humans. 00:47:53.640 --> 00:47:56.260 align:middle line:84% So later, when we get into international trade, 00:47:56.260 --> 00:47:57.720 align:middle line:90% we're going to talk about-- 00:47:57.720 --> 00:48:00.060 align:middle line:84% we're going to talk about changing factor prices 00:48:00.060 --> 00:48:04.050 align:middle line:84% and wages and so on, so this is motivation 00:48:04.050 --> 00:48:06.660 align:middle line:90% for that when we get there. 00:48:06.660 --> 00:48:11.320 align:middle line:84% Cost function is homogeneous of degree 1. 00:48:11.320 --> 00:48:14.500 align:middle line:84% If you double the inputs, you double the cost. 00:48:14.500 --> 00:48:16.780 align:middle line:90% Same argument as before. 00:48:16.780 --> 00:48:18.520 align:middle line:84% We're not going to change the optimum. 00:48:18.520 --> 00:48:20.770 align:middle line:84% We're just changing the prices of the inputs, 00:48:20.770 --> 00:48:22.030 align:middle line:90% so the cost doubles. 00:48:22.030 --> 00:48:25.030 align:middle line:84% The cost function is continuous, and the cost function 00:48:25.030 --> 00:48:26.500 align:middle line:90% is concave. 00:48:26.500 --> 00:48:30.190 align:middle line:84% Just like we did that the profit function was convex, 00:48:30.190 --> 00:48:33.190 align:middle line:84% we're getting that this cost function is concave. 00:48:33.190 --> 00:48:35.480 align:middle line:84% And the argument is very similar. 00:48:35.480 --> 00:48:37.570 align:middle line:84% And instead of chewing up time, I'm 00:48:37.570 --> 00:48:40.720 align:middle line:84% actually going to break precedent and leave 00:48:40.720 --> 00:48:43.300 align:middle line:84% this for you to do, but it's exactly 00:48:43.300 --> 00:48:45.850 align:middle line:84% the same argument that we went through 00:48:45.850 --> 00:48:47.620 align:middle line:84% with the convexity of the profit function, 00:48:47.620 --> 00:48:49.660 align:middle line:90% except the signs are-- 00:48:49.660 --> 00:48:52.310 align:middle line:84% the inequalities go the other way. 00:48:52.310 --> 00:48:56.620 align:middle line:84% OK, and then we get Shepard's lemma, 00:48:56.620 --> 00:49:00.720 align:middle line:84% which is a version of Hotelling's lemma focusing 00:49:00.720 --> 00:49:05.310 align:middle line:84% on costs, and hence only on the inputs prices. 00:49:05.310 --> 00:49:10.260 align:middle line:84% If we differentiate the minimized cost 00:49:10.260 --> 00:49:19.310 align:middle line:84% of producing output level y with respect to an input price w-- 00:49:19.310 --> 00:49:22.070 align:middle line:84% I pride myself on the ability to go through these slides 00:49:22.070 --> 00:49:23.990 align:middle line:90% and remove all the typos. 00:49:23.990 --> 00:49:26.870 align:middle line:84% But here, I'm kind of wishing that I had had 00:49:26.870 --> 00:49:31.340 align:middle line:90% a wn on here because it's xn. 00:49:31.340 --> 00:49:33.680 align:middle line:84% So if there's only one input, it's fine. 00:49:33.680 --> 00:49:35.840 align:middle line:84% If there's multiple inputs, then there's a vector, 00:49:35.840 --> 00:49:38.000 align:middle line:84% and we should be differentiating with respect 00:49:38.000 --> 00:49:43.070 align:middle line:84% to a particular item in the vector, namely, the nth input. 00:49:43.070 --> 00:49:46.960 align:middle line:84% So probably better to have written this as wn 00:49:46.960 --> 00:49:51.860 align:middle line:90% and have an xn over here. 00:49:51.860 --> 00:49:56.050 align:middle line:84% Anyway, hopefully, we'll take notes and correct that. 00:49:56.050 --> 00:49:57.360 align:middle line:90% It's not wrong. 00:49:57.360 --> 00:49:59.320 align:middle line:84% It could have been written better. 00:49:59.320 --> 00:50:02.450 align:middle line:84% But the real point is you've seen this before. 00:50:02.450 --> 00:50:04.390 align:middle line:84% This is a version of Hotelling's lemma. 00:50:04.390 --> 00:50:06.310 align:middle line:90% It's called Shepard's lemma. 00:50:06.310 --> 00:50:08.560 align:middle line:84% But Shepard's only dealing with cost function. 00:50:08.560 --> 00:50:11.970 align:middle line:84% Hotelling was dealing with the whole profit function. 00:50:11.970 --> 00:50:13.910 align:middle line:84% So all that stuff about the envelope theorem, 00:50:13.910 --> 00:50:16.010 align:middle line:84% blah, blah, blah, that all applies here. 00:50:16.010 --> 00:50:19.770 align:middle line:90% 00:50:19.770 --> 00:50:27.120 align:middle line:84% Cost curves-- so we've got the total cost of producing q 00:50:27.120 --> 00:50:29.640 align:middle line:84% as the minimized cost of producing q. 00:50:29.640 --> 00:50:31.950 align:middle line:84% The marginal cost is the derivative of that. 00:50:31.950 --> 00:50:37.680 align:middle line:84% And the average cost is total cost divided by the quantity q. 00:50:37.680 --> 00:50:44.130 align:middle line:84% That's the cost on average of the various units of q. 00:50:44.130 --> 00:50:50.220 align:middle line:84% Of course, profits would be just the revenue minus the cost. 00:50:50.220 --> 00:50:53.250 align:middle line:84% This is the minimized cost function 00:50:53.250 --> 00:50:57.420 align:middle line:84% suppressing the reference to w as if it were fixed. 00:50:57.420 --> 00:51:01.410 align:middle line:84% And take the derivative of this, of course with respect to q, 00:51:01.410 --> 00:51:03.450 align:middle line:84% we're getting a price on the left-hand side 00:51:03.450 --> 00:51:06.600 align:middle line:84% and the marginal cost on the right. 00:51:06.600 --> 00:51:08.930 align:middle line:84% So again, this has an intuitive-- 00:51:08.930 --> 00:51:12.230 align:middle line:84% you've probably seen this in your principles of economics 00:51:12.230 --> 00:51:14.660 align:middle line:84% course, that when the firm is optimizing, 00:51:14.660 --> 00:51:17.390 align:middle line:84% it should pick a quantity q, where price 00:51:17.390 --> 00:51:19.610 align:middle line:90% is equal to marginal cost. 00:51:19.610 --> 00:51:22.550 align:middle line:84% And the reason is, of course, if price were greater 00:51:22.550 --> 00:51:25.490 align:middle line:84% than marginal cost, you get more in revenue 00:51:25.490 --> 00:51:29.060 align:middle line:84% for one more unit than you sacrifice in terms of costs 00:51:29.060 --> 00:51:32.720 align:middle line:84% and vice versa when it goes the other way. 00:51:32.720 --> 00:51:37.500 align:middle line:84% In particular-- we're going to get a strong property here-- 00:51:37.500 --> 00:51:40.560 align:middle line:84% when we have constant returns to scale, 00:51:40.560 --> 00:51:45.840 align:middle line:84% this production function you've seen before, production set. 00:51:45.840 --> 00:51:50.680 align:middle line:84% And the cost is going to be linear. 00:51:50.680 --> 00:51:55.600 align:middle line:84% It just scales up and down as you vary q. 00:51:55.600 --> 00:51:58.790 align:middle line:84% And in other words, the marginal cost 00:51:58.790 --> 00:52:01.220 align:middle line:84% is equal to the average cost, and we 00:52:01.220 --> 00:52:04.250 align:middle line:84% want to find the point when you're optimizing the price 00:52:04.250 --> 00:52:07.550 align:middle line:84% p is equal to the marginal cost, which is also 00:52:07.550 --> 00:52:08.920 align:middle line:90% equal to the average cost. 00:52:08.920 --> 00:52:13.220 align:middle line:84% So any price less than that, you're 00:52:13.220 --> 00:52:16.250 align:middle line:90% like, no, I'm not doing it. 00:52:16.250 --> 00:52:19.550 align:middle line:84% And any price greater, they would go to infinity and make 00:52:19.550 --> 00:52:21.420 align:middle line:90% an infinite amount of money. 00:52:21.420 --> 00:52:24.440 align:middle line:84% And at this critical price p, which is exactly 00:52:24.440 --> 00:52:27.590 align:middle line:84% equal to the constant marginal cost, 00:52:27.590 --> 00:52:31.040 align:middle line:84% because the slope is the same on this line everywhere, 00:52:31.040 --> 00:52:34.890 align:middle line:84% they jump from 0 to any old q that you want. 00:52:34.890 --> 00:52:37.250 align:middle line:84% Let me show you the intuition of what's going on here. 00:52:37.250 --> 00:52:39.890 align:middle line:84% Again, keep track of what's in the numerator 00:52:39.890 --> 00:52:41.720 align:middle line:90% and what's in the denominator. 00:52:41.720 --> 00:52:45.530 align:middle line:84% This is supposed to be p1 over p2. 00:52:45.530 --> 00:52:52.400 align:middle line:84% The slope of this isoprofit line is p1 over p2. 00:52:52.400 --> 00:52:56.390 align:middle line:84% If p2 is going up, this isoprofit line slope 00:52:56.390 --> 00:52:58.620 align:middle line:90% will be going down. 00:52:58.620 --> 00:53:01.710 align:middle line:84% So let's compare the three diagrams here. 00:53:01.710 --> 00:53:07.430 align:middle line:84% So here, given a slope associated with p1 over p2, 00:53:07.430 --> 00:53:10.910 align:middle line:84% profits just get higher and higher 00:53:10.910 --> 00:53:14.660 align:middle line:84% as you move further and further along the boundaries. 00:53:14.660 --> 00:53:18.300 align:middle line:90% So 2 is the output. 00:53:18.300 --> 00:53:22.830 align:middle line:84% So as p2 goes up, p1 over p2 is going down. 00:53:22.830 --> 00:53:26.330 align:middle line:84% So we get this isoprofit line, which 00:53:26.330 --> 00:53:30.560 align:middle line:84% is actually coincident with the boundary of the production set. 00:53:30.560 --> 00:53:33.230 align:middle line:84% And finally, for-- so the mistake 00:53:33.230 --> 00:53:35.900 align:middle line:84% here is this should say going down. 00:53:35.900 --> 00:53:38.630 align:middle line:84% I'm constantly catching myself that it wasn't working 00:53:38.630 --> 00:53:43.910 align:middle line:84% that the slope is moving in the direction of the opposite 00:53:43.910 --> 00:53:45.740 align:middle line:90% of what I'm saying. 00:53:45.740 --> 00:53:50.510 align:middle line:84% What we want here is a price where they go-- or infinity 00:53:50.510 --> 00:53:51.440 align:middle line:90% over here. 00:53:51.440 --> 00:53:54.230 align:middle line:84% They're indifferent at this price of output. 00:53:54.230 --> 00:53:56.210 align:middle line:84% And at a lower price of output, they, 00:53:56.210 --> 00:53:57.860 align:middle line:90% don't want to do anything. 00:53:57.860 --> 00:54:01.280 align:middle line:84% So [INAUDIBLE] nothing [? pro ?] profit. 00:54:01.280 --> 00:54:05.090 align:middle line:84% And that's where these lines are steeper, where these isoprofit 00:54:05.090 --> 00:54:06.050 align:middle line:90% lines are steeper. 00:54:06.050 --> 00:54:10.880 align:middle line:84% And they're steeper because it has a slope of p1 or p2. 00:54:10.880 --> 00:54:13.190 align:middle line:84% And the price of output p2 has gotten 00:54:13.190 --> 00:54:18.500 align:middle line:84% so low that these lines are getting so steep that the best 00:54:18.500 --> 00:54:23.980 align:middle line:84% position is zero profit, so I finally set it right that time. 00:54:23.980 --> 00:54:26.020 align:middle line:84% And I did not catch this typo before. 00:54:26.020 --> 00:54:27.990 align:middle line:90% We'll fix that. 00:54:27.990 --> 00:54:31.320 align:middle line:84% The point of the diagram is constant returns to scale 00:54:31.320 --> 00:54:33.990 align:middle line:84% has some very special properties. 00:54:33.990 --> 00:54:36.720 align:middle line:90% Prices have to be just right. 00:54:36.720 --> 00:54:41.190 align:middle line:84% If price ratio is just right, then any production is fine, 00:54:41.190 --> 00:54:43.920 align:middle line:90% and they're making zero profits. 00:54:43.920 --> 00:54:45.220 align:middle line:90% So they don't care. 00:54:45.220 --> 00:54:49.810 align:middle line:84% They will do something if demand makes them do it. 00:54:49.810 --> 00:54:52.590 align:middle line:84% But on these two extremes, this isn't feasible 00:54:52.590 --> 00:54:54.210 align:middle line:90% because it's infinite profit. 00:54:54.210 --> 00:54:56.050 align:middle line:84% And here, they would make losses, 00:54:56.050 --> 00:54:58.980 align:middle line:84% so they minimize their losses and go to 0. 00:54:58.980 --> 00:55:01.890 align:middle line:84% So you get these very extreme solutions. 00:55:01.890 --> 00:55:04.680 align:middle line:84% If you have an equilibrium with positive profits 00:55:04.680 --> 00:55:08.040 align:middle line:84% from a production set which displays constant returns 00:55:08.040 --> 00:55:11.730 align:middle line:84% to scale, you know that the price ratio 00:55:11.730 --> 00:55:14.400 align:middle line:84% has to be coincident with the slope of the boundary 00:55:14.400 --> 00:55:17.340 align:middle line:90% of the production function. 00:55:17.340 --> 00:55:21.280 align:middle line:84% When we get to equilibrium, we can determine prices 00:55:21.280 --> 00:55:24.700 align:middle line:84% from the production side, at least on the premise 00:55:24.700 --> 00:55:26.245 align:middle line:90% that something's being produced. 00:55:26.245 --> 00:55:29.290 align:middle line:90% 00:55:29.290 --> 00:55:32.400 align:middle line:90% That's what I said. 00:55:32.400 --> 00:55:36.450 align:middle line:84% This is a property of constant returns to scale. 00:55:36.450 --> 00:55:41.820 align:middle line:84% Namely, we might just as well solve the cost minimization 00:55:41.820 --> 00:55:47.400 align:middle line:84% problem for an output of 1 and get the ratio 00:55:47.400 --> 00:55:49.450 align:middle line:90% of the inputs that way. 00:55:49.450 --> 00:55:51.510 align:middle line:84% And then, if you want to scale production 00:55:51.510 --> 00:55:56.010 align:middle line:84% up or down, y being greater than or less than 1, that's fine. 00:55:56.010 --> 00:55:58.140 align:middle line:84% You'll change the level of the inputs. 00:55:58.140 --> 00:56:02.130 align:middle line:84% They're going to scale up and down as well by the same scalar 00:56:02.130 --> 00:56:06.840 align:middle line:84% you're using to scale y, but the ratio isn't going to move. 00:56:06.840 --> 00:56:10.080 align:middle line:84% But that's a property of constant returns to scale. 00:56:10.080 --> 00:56:13.380 align:middle line:84% Now, again, I had decided this in advance 00:56:13.380 --> 00:56:17.640 align:middle line:84% and, indeed, have my eyes on the clock here. 00:56:17.640 --> 00:56:19.150 align:middle line:90% There is a proof here. 00:56:19.150 --> 00:56:23.730 align:middle line:84% So if you like, take some time and try to write out 00:56:23.730 --> 00:56:25.260 align:middle line:90% a proof by contradiction. 00:56:25.260 --> 00:56:28.560 align:middle line:84% I think you'll find that to be a useful exercise. 00:56:28.560 --> 00:56:31.050 align:middle line:84% In other words, prove something like 00:56:31.050 --> 00:56:35.260 align:middle line:84% suppose you changed y and the input ratio were to change, 00:56:35.260 --> 00:56:38.760 align:middle line:84% then that would lead to a contradiction, something 00:56:38.760 --> 00:56:41.190 align:middle line:84% inconsistent with constant returns to scale. 00:56:41.190 --> 00:56:44.550 align:middle line:84% All right, here's Robinson Crusoe. 00:56:44.550 --> 00:56:47.550 align:middle line:84% It's like a one-island, one-man economy. 00:56:47.550 --> 00:56:51.840 align:middle line:84% If you don't the story, this means nothing to you. 00:56:51.840 --> 00:56:54.960 align:middle line:84% I think he actually had a helper called Friday. 00:56:54.960 --> 00:56:58.980 align:middle line:84% But anyway, just think of Robinson Crusoe being 00:56:58.980 --> 00:57:04.020 align:middle line:84% a one-man economy, and we plot the production possibility set 00:57:04.020 --> 00:57:08.850 align:middle line:84% with Robinson's input, getting food as output, picking fruit 00:57:08.850 --> 00:57:10.710 align:middle line:90% off the trees, whatever. 00:57:10.710 --> 00:57:15.240 align:middle line:84% And what's the maximal solution for Robinson? 00:57:15.240 --> 00:57:17.460 align:middle line:84% It's to find the utility function which 00:57:17.460 --> 00:57:20.010 align:middle line:90% has the highest tangency. 00:57:20.010 --> 00:57:25.170 align:middle line:84% So these lines are the curves of indifference curves, 00:57:25.170 --> 00:57:30.420 align:middle line:84% and the maximal utility level is [INAUDIBLE].. 00:57:30.420 --> 00:57:33.040 align:middle line:84% So this is kind of a big moment in the course, 00:57:33.040 --> 00:57:36.060 align:middle line:84% even though the concepts probably 00:57:36.060 --> 00:57:38.190 align:middle line:90% look pretty transparent. 00:57:38.190 --> 00:57:43.260 align:middle line:84% We just combined utility maximization with production. 00:57:43.260 --> 00:57:48.000 align:middle line:84% And I'm telling you, the optimum point for the whole economy 00:57:48.000 --> 00:57:50.040 align:middle line:90% is this tangency level. 00:57:50.040 --> 00:57:54.900 align:middle line:84% And, actually, if we drew a line that 00:57:54.900 --> 00:57:57.030 align:middle line:84% were tangent to both the production 00:57:57.030 --> 00:57:59.370 align:middle line:84% boundary and the indifference curve, 00:57:59.370 --> 00:58:03.690 align:middle line:84% that could represent both a profit line 00:58:03.690 --> 00:58:08.190 align:middle line:84% for Robinson Crusoe, the producer, and a budget line 00:58:08.190 --> 00:58:11.140 align:middle line:84% for Robinson Crusoe as a consumer. 00:58:11.140 --> 00:58:12.750 align:middle line:84% So for the first time, we're actually 00:58:12.750 --> 00:58:14.700 align:middle line:84% looking at a competitive equilibrium 00:58:14.700 --> 00:58:15.900 align:middle line:90% for the whole economy. 00:58:15.900 --> 00:58:19.030 align:middle line:84% It's kind of stupid, though, because Robinson's all alone. 00:58:19.030 --> 00:58:21.180 align:middle line:90% There's no one to trade with. 00:58:21.180 --> 00:58:24.650 align:middle line:90% But suppose he could. 00:58:24.650 --> 00:58:27.710 align:middle line:84% So suppose Robinson Crusoe could produce 00:58:27.710 --> 00:58:31.300 align:middle line:84% two goods, some agricultural output, 00:58:31.300 --> 00:58:36.280 align:middle line:84% but also some tools or some manufactured good 00:58:36.280 --> 00:58:38.260 align:middle line:90% on the x-axis. 00:58:38.260 --> 00:58:42.480 align:middle line:84% So the set of production possibilities 00:58:42.480 --> 00:58:49.060 align:middle line:84% is this quarter circle area with its boundary. 00:58:49.060 --> 00:58:54.330 align:middle line:84% And if Robinson Crusoe would be still intent on maximizing 00:58:54.330 --> 00:58:58.260 align:middle line:84% utility, we'd find where the slope of the-- where the-- 00:58:58.260 --> 00:59:00.270 align:middle line:84% we'd find the highest level of the indifference 00:59:00.270 --> 00:59:03.925 align:middle line:90% curve and the tangency. 00:59:03.925 --> 00:59:05.800 align:middle line:84% There's stuff going on in the background here 00:59:05.800 --> 00:59:08.490 align:middle line:84% I'm not showing you to make it simpler. 00:59:08.490 --> 00:59:12.810 align:middle line:90% I'm not talking about his input. 00:59:12.810 --> 00:59:15.780 align:middle line:84% [INAUDIBLE] whether I'm going to work in agriculture or work 00:59:15.780 --> 00:59:19.880 align:middle line:84% in manufacturing, and I'm not showing you that work level. 00:59:19.880 --> 00:59:24.650 align:middle line:84% But it is producing this set of production possibilities 00:59:24.650 --> 00:59:27.230 align:middle line:84% of the two goods, and this would be 00:59:27.230 --> 00:59:30.200 align:middle line:90% the utility maximizing point A. 00:59:30.200 --> 00:59:34.520 align:middle line:84% Now, suppose all of a sudden, he has the opportunity 00:59:34.520 --> 00:59:36.530 align:middle line:90% to trade with another island. 00:59:36.530 --> 00:59:46.280 align:middle line:84% And suppose we represent the isoprofit line in red. 00:59:46.280 --> 00:59:52.930 align:middle line:84% So we want to be as far to the northeast as possible, 00:59:52.930 --> 00:59:58.290 align:middle line:84% maximizing profits, but staying on the production possibility 00:59:58.290 --> 00:59:58.840 align:middle line:90% set. 00:59:58.840 --> 01:00:03.700 align:middle line:84% So that maximized level of production 01:00:03.700 --> 01:00:08.440 align:middle line:84% is here, B, and consumption is here, C. 01:00:08.440 --> 01:00:11.290 align:middle line:84% Now, this is a much more interesting economy 01:00:11.290 --> 01:00:13.960 align:middle line:84% because there's some external agent here. 01:00:13.960 --> 01:00:16.150 align:middle line:84% [INAUDIBLE] Robinson Crusoe and the rest 01:00:16.150 --> 01:00:17.590 align:middle line:90% of the world, that other island. 01:00:17.590 --> 01:00:23.020 align:middle line:84% And they're now trading with each other. 01:00:23.020 --> 01:00:27.710 align:middle line:84% And Robinson Crusoe has decided to specialize more, 01:00:27.710 --> 01:00:33.440 align:middle line:84% though not completely, in manufacturing goods 01:00:33.440 --> 01:00:38.570 align:middle line:84% and to give them up in trade to the other island 01:00:38.570 --> 01:00:42.550 align:middle line:84% in order to acquire more agricultural output. 01:00:42.550 --> 01:00:47.610 align:middle line:84% So we're seeing exports and imports as we move from B to C. 01:00:47.610 --> 01:00:52.980 align:middle line:84% And, obviously, Robinson Crusoe is doing better utility wise 01:00:52.980 --> 01:00:55.860 align:middle line:90% from trade than that autarky. 01:00:55.860 --> 01:00:58.710 align:middle line:90% A might stand for autarky. 01:00:58.710 --> 01:01:02.220 align:middle line:84% So this is our first glimpse at the gains from trade. 01:01:02.220 --> 01:01:03.990 align:middle line:84% With the one-man economy, at least, 01:01:03.990 --> 01:01:07.830 align:middle line:84% Robinson Crusoe can be made better off by trading 01:01:07.830 --> 01:01:09.880 align:middle line:90% with another island. 01:01:09.880 --> 01:01:12.430 align:middle line:90% This is a related picture. 01:01:12.430 --> 01:01:16.240 align:middle line:84% Suppose we have two countries, the UK and the United States, 01:01:16.240 --> 01:01:18.820 align:middle line:84% and their production possibility set 01:01:18.820 --> 01:01:21.730 align:middle line:90% look different from each other. 01:01:21.730 --> 01:01:25.930 align:middle line:84% We're plotting wheat and cloth in both countries, 01:01:25.930 --> 01:01:32.470 align:middle line:84% and we might have an autarky that the UK's maximal utility 01:01:32.470 --> 01:01:38.950 align:middle line:84% position is here at DUK, and that the US's no-trade position 01:01:38.950 --> 01:01:40.900 align:middle line:90% would be at DUS. 01:01:40.900 --> 01:01:43.600 align:middle line:84% Now, you do see that this is deliberately 01:01:43.600 --> 01:01:47.950 align:middle line:84% chosen so the slopes of these lines are different, 01:01:47.950 --> 01:01:50.570 align:middle line:84% and that suggests there are gains to trade. 01:01:50.570 --> 01:01:54.310 align:middle line:84% In fact, what's depicted here is almost common 01:01:54.310 --> 01:01:58.960 align:middle line:84% price ratio, where you could think of it as the prices 01:01:58.960 --> 01:02:00.130 align:middle line:90% in the rest of the world. 01:02:00.130 --> 01:02:02.800 align:middle line:84% But this may be a bilateral country example, 01:02:02.800 --> 01:02:06.130 align:middle line:84% but somehow the prices are fixed for each country 01:02:06.130 --> 01:02:07.700 align:middle line:90% by competitive forces. 01:02:07.700 --> 01:02:11.900 align:middle line:84% So the slope of the dashed line is the same for both countries. 01:02:11.900 --> 01:02:12.950 align:middle line:90% And what happens? 01:02:12.950 --> 01:02:19.300 align:middle line:84% Well, the UK decides to export wheat and import cloth. 01:02:19.300 --> 01:02:22.210 align:middle line:84% I thought it was going to go the other way, actually. 01:02:22.210 --> 01:02:26.005 align:middle line:84% And the United States chooses to-- 01:02:26.005 --> 01:02:27.700 align:middle line:90% did I just say that wrong-- 01:02:27.700 --> 01:02:31.180 align:middle line:84% chooses to-- oh, yeah, I did say it wrong. 01:02:31.180 --> 01:02:35.860 align:middle line:84% The US chooses to produce more wheat and export 01:02:35.860 --> 01:02:40.510 align:middle line:84% part of that increased production back to the UK. 01:02:40.510 --> 01:02:43.990 align:middle line:84% The UK was here in autarky, when it 01:02:43.990 --> 01:02:46.630 align:middle line:84% faced with the possibility of trade, 01:02:46.630 --> 01:02:49.660 align:middle line:84% it produces much more cloth, and then 01:02:49.660 --> 01:02:53.020 align:middle line:90% it exports that cloth to the US. 01:02:53.020 --> 01:02:55.840 align:middle line:84% Cloth and the UK kind go hand-in-hand 01:02:55.840 --> 01:02:58.540 align:middle line:84% because people think of the Industrial Revolution 01:02:58.540 --> 01:03:01.420 align:middle line:84% and manufacturing of textiles, which 01:03:01.420 --> 01:03:04.870 align:middle line:84% was kind over big deal in Britain, and the US and Kansas 01:03:04.870 --> 01:03:07.970 align:middle line:90% and so on having lots of wheat. 01:03:07.970 --> 01:03:11.080 align:middle line:84% So it's meant to be a somewhat realistic example. 01:03:11.080 --> 01:03:15.260 align:middle line:84% Anyway, this is the climax of the slides today 01:03:15.260 --> 01:03:17.620 align:middle line:84% because you're seeing an application of production 01:03:17.620 --> 01:03:20.890 align:middle line:84% theory, in particular, a profit maximizing theory, 01:03:20.890 --> 01:03:24.880 align:middle line:84% but now integrating with utility maximization theory 01:03:24.880 --> 01:03:29.890 align:middle line:84% and looking at an equilibrium of an entire economy, at least 01:03:29.890 --> 01:03:31.720 align:middle line:90% consisting of two countries. 01:03:31.720 --> 01:03:36.580 align:middle line:84% When we talk about tariffs and taxes and so on, 01:03:36.580 --> 01:03:43.420 align:middle line:84% these dashed lines will not be of identical slope. 01:03:43.420 --> 01:03:46.420 align:middle line:84% They're going to have different slopes in different countries, 01:03:46.420 --> 01:03:50.590 align:middle line:84% depending on which good is subject to a tariff. 01:03:50.590 --> 01:03:53.140 align:middle line:84% Imports from China coming into the US, 01:03:53.140 --> 01:03:54.970 align:middle line:84% for example, are subject to a tariff. 01:03:54.970 --> 01:03:58.990 align:middle line:84% And we'll get there, but we're getting very close now. 01:03:58.990 --> 01:04:03.520 align:middle line:84% So there is more to this part of the lecture on production, 01:04:03.520 --> 01:04:09.220 align:middle line:84% but I preplanned to stop here to say, next time, when we start, 01:04:09.220 --> 01:04:15.380 align:middle line:84% we'll finish up this production lecture on Leontief and Google, 01:04:15.380 --> 01:04:23.110 align:middle line:84% Facebook page rank, I mean, and the great Japanese earthquake, 01:04:23.110 --> 01:04:25.090 align:middle line:90% and then we'll-- 01:04:25.090 --> 01:04:28.340 align:middle line:84% which are fun applications for production. 01:04:28.340 --> 01:04:33.700 align:middle line:84% And then we're going to go to lecture 5 and talk 01:04:33.700 --> 01:04:35.620 align:middle line:84% about the consumer theory and then 01:04:35.620 --> 01:04:38.140 align:middle line:84% combine them all over again with some techniques 01:04:38.140 --> 01:04:40.510 align:middle line:90% on linear programming. 01:04:40.510 --> 01:04:43.200 align:middle line:90% OK, see you soon. 01:04:43.200 --> 01:04:56.053 align:middle line:90%