WEBVTT 00:00:00.000 --> 00:00:02.946 align:middle line:90% [SQUEAKING] 00:00:02.946 --> 00:00:04.419 align:middle line:90% [RUSTLING] 00:00:04.419 --> 00:00:05.892 align:middle line:90% [CLICKING] 00:00:05.892 --> 00:00:11.310 align:middle line:90% 00:00:11.310 --> 00:00:13.450 align:middle line:84% ROBERT TOWNSEND: OK, so let me say 00:00:13.450 --> 00:00:16.360 align:middle line:84% a couple of words about where we are in the reading list. 00:00:16.360 --> 00:00:18.040 align:middle line:90% We're going to start today. 00:00:18.040 --> 00:00:22.870 align:middle line:84% We're done with the segment on contracts and mechanism design, 00:00:22.870 --> 00:00:24.070 align:middle line:90% theory and applications. 00:00:24.070 --> 00:00:27.680 align:middle line:84% And we start today, Walrasian General Equilibrium, 00:00:27.680 --> 00:00:29.980 align:middle line:84% which you've seen a bit before already. 00:00:29.980 --> 00:00:33.010 align:middle line:84% But we'll review and give the proper definitions 00:00:33.010 --> 00:00:35.230 align:middle line:84% for the first 20% of the lecture. 00:00:35.230 --> 00:00:37.150 align:middle line:84% And then the rest of the lecture is 00:00:37.150 --> 00:00:40.570 align:middle line:84% devoted to an important application 00:00:40.570 --> 00:00:42.550 align:middle line:90% of General Equilibrium theory. 00:00:42.550 --> 00:00:45.730 align:middle line:84% And in particular, it has to do with trade. 00:00:45.730 --> 00:00:49.150 align:middle line:84% So this lecture today is the theory, 00:00:49.150 --> 00:00:52.540 align:middle line:84% with some really startling and interesting propositions. 00:00:52.540 --> 00:00:54.850 align:middle line:84% And then with trade on our minds, 00:00:54.850 --> 00:00:57.460 align:middle line:84% we'll go back to the Thai data, and take a look 00:00:57.460 --> 00:01:00.490 align:middle line:84% at trade flows and financial flows that 00:01:00.490 --> 00:01:02.620 align:middle line:84% have changed over time as the country became 00:01:02.620 --> 00:01:06.370 align:middle line:84% more liberalized, or more open internally. 00:01:06.370 --> 00:01:11.080 align:middle line:84% And then we're going to go to a parallel lecture on the US 00:01:11.080 --> 00:01:14.260 align:middle line:84% that's going to look at trade flows and potentially 00:01:14.260 --> 00:01:17.320 align:middle line:84% financial flows across states in the US, 00:01:17.320 --> 00:01:20.860 align:middle line:84% where we'll be analyzing the impact of tariffs, 00:01:20.860 --> 00:01:22.690 align:middle line:90% and/or China shocks-- 00:01:22.690 --> 00:01:24.190 align:middle line:90% China imports. 00:01:24.190 --> 00:01:29.470 align:middle line:84% So these, dare I say it, it may or may not be on your mind. 00:01:29.470 --> 00:01:32.710 align:middle line:84% If it's on your mind already, the motivation 00:01:32.710 --> 00:01:37.870 align:middle line:84% here is a US policy decision which is very much in the news, 00:01:37.870 --> 00:01:41.440 align:middle line:84% and in some respects, a big part of the upcoming election. 00:01:41.440 --> 00:01:44.800 align:middle line:84% So I don't know if the timing of these lectures are good or bad. 00:01:44.800 --> 00:01:50.620 align:middle line:84% But anyway, we'll be talking about US trade policy 00:01:50.620 --> 00:01:52.730 align:middle line:90% within two lectures. 00:01:52.730 --> 00:01:54.520 align:middle line:90% So that's what's coming up. 00:01:54.520 --> 00:01:59.500 align:middle line:84% And today's lecture has got two start readings, the Kreps 00:01:59.500 --> 00:02:04.040 align:middle line:84% book on section 6.1, and this Mas-Colell, Whinston, 00:02:04.040 --> 00:02:04.540 align:middle line:90% and Green. 00:02:04.540 --> 00:02:06.460 align:middle line:90% I rarely assign this book. 00:02:06.460 --> 00:02:08.320 align:middle line:84% But on the other hand, the lecture 00:02:08.320 --> 00:02:11.770 align:middle line:84% is based very closely on that chapter. 00:02:11.770 --> 00:02:17.110 align:middle line:84% So I went ahead and just listed it, MWG chapter 15 D, 00:02:17.110 --> 00:02:19.660 align:middle line:84% although it's quite a famous model. 00:02:19.660 --> 00:02:22.670 align:middle line:84% We've been using MWG in the theorems and so on. 00:02:22.670 --> 00:02:26.440 align:middle line:84% But I've never really featured it so heavily. 00:02:26.440 --> 00:02:28.840 align:middle line:90% In terms of the overall-- 00:02:28.840 --> 00:02:31.450 align:middle line:84% just to see it in one spot, again, we're 00:02:31.450 --> 00:02:34.600 align:middle line:84% going to do Walrasian equilibrium, and then 00:02:34.600 --> 00:02:38.080 align:middle line:84% two applications of trade in Thailand and the US. 00:02:38.080 --> 00:02:40.060 align:middle line:84% That's actually going to complete-- 00:02:40.060 --> 00:02:43.360 align:middle line:84% lecture 15 completes the current segment of lectures 00:02:43.360 --> 00:02:45.560 align:middle line:90% before the second exam. 00:02:45.560 --> 00:02:49.510 align:middle line:84% So problem set 4, which was released last-- the last thing 00:02:49.510 --> 00:02:52.210 align:middle line:84% I want to do before getting to the lecture 00:02:52.210 --> 00:02:56.440 align:middle line:84% is to review a bit of what we did last time. 00:02:56.440 --> 00:02:59.890 align:middle line:84% I deliberately didn't pick out very many questions here. 00:02:59.890 --> 00:03:02.590 align:middle line:84% Allow risk aversion on the part of the borrower. 00:03:02.590 --> 00:03:05.200 align:middle line:84% There are several advantages to writing down 00:03:05.200 --> 00:03:07.450 align:middle line:90% programs in lotteries. 00:03:07.450 --> 00:03:10.280 align:middle line:90% List two of them and explain. 00:03:10.280 --> 00:03:14.020 align:middle line:90% So let's see if I can get some-- 00:03:14.020 --> 00:03:17.930 align:middle line:84% Charles, do you want to take a crack at that? 00:03:17.930 --> 00:03:20.570 align:middle line:84% AUDIENCE: I think I can guess one, definitely at least one 00:03:20.570 --> 00:03:22.620 align:middle line:84% of them, which, if I remember correctly, 00:03:22.620 --> 00:03:26.600 align:middle line:84% is that it allows you to make an incentive. 00:03:26.600 --> 00:03:31.040 align:middle line:84% You may select between people with high income 00:03:31.040 --> 00:03:33.650 align:middle line:84% and low income, if you don't know the income already. 00:03:33.650 --> 00:03:38.060 align:middle line:84% So like, if you're risk averse in the correct manner, 00:03:38.060 --> 00:03:44.600 align:middle line:84% then if you have a lottery, then someone with low income 00:03:44.600 --> 00:03:50.900 align:middle line:84% might prefer a lottery with something like-- you might 00:03:50.900 --> 00:03:58.850 align:middle line:84% prefer the different risk return trade off at different incomes. 00:03:58.850 --> 00:04:01.520 align:middle line:84% There's a dislike between various things, 00:04:01.520 --> 00:04:04.010 align:middle line:84% and help with some of the constraints. 00:04:04.010 --> 00:04:08.690 align:middle line:84% The other one, I don't remember the exact details. 00:04:08.690 --> 00:04:12.020 align:middle line:90% I haven't quite got them yet. 00:04:12.020 --> 00:04:14.750 align:middle line:84% ROBERT TOWNSEND: The first one is good. 00:04:14.750 --> 00:04:23.480 align:middle line:84% AUDIENCE: I'm going to guess something like, 00:04:23.480 --> 00:04:26.592 align:middle line:84% maybe making people not as willing to borrow 00:04:26.592 --> 00:04:27.550 align:middle line:90% or something like that. 00:04:27.550 --> 00:04:30.930 align:middle line:90% But I'm not exactly sure. 00:04:30.930 --> 00:04:33.410 align:middle line:90% ROBERT TOWNSEND: Let's see. 00:04:33.410 --> 00:04:36.620 align:middle line:84% Kerry, can you get one more of them? 00:04:36.620 --> 00:04:41.410 align:middle line:90% 00:04:41.410 --> 00:04:43.990 align:middle line:84% AUDIENCE: Is it, like, incentive constraints 00:04:43.990 --> 00:04:46.750 align:middle line:90% could be non-convex. 00:04:46.750 --> 00:04:49.960 align:middle line:90% So it might be hard to solve. 00:04:49.960 --> 00:04:51.550 align:middle line:90% ROBERT TOWNSEND: Yeah, good. 00:04:51.550 --> 00:04:54.210 align:middle line:84% So there's various sources of non-convexity. 00:04:54.210 --> 00:04:56.020 align:middle line:84% So those incentive constraints, when 00:04:56.020 --> 00:05:00.990 align:middle line:84% written without the lotteries, like first order conditions 00:05:00.990 --> 00:05:04.110 align:middle line:84% of the borrower's choice of effort. 00:05:04.110 --> 00:05:07.390 align:middle line:84% Those things can lead to non-convexity. 00:05:07.390 --> 00:05:11.790 align:middle line:84% So by doing it in lotteries, we get that problem solved. 00:05:11.790 --> 00:05:15.840 align:middle line:84% And also, if there is some indivisibilities, 00:05:15.840 --> 00:05:16.990 align:middle line:90% that would be solved. 00:05:16.990 --> 00:05:20.650 align:middle line:84% There are two indivisibilities in this problem. 00:05:20.650 --> 00:05:23.580 align:middle line:84% One has to do with occupation choice. 00:05:23.580 --> 00:05:26.040 align:middle line:84% You're either a wage earner or an entrepreneur. 00:05:26.040 --> 00:05:29.170 align:middle line:90% That's a binary choice. 00:05:29.170 --> 00:05:31.890 align:middle line:84% So the lotteries kind of smooth that over 00:05:31.890 --> 00:05:36.460 align:middle line:84% because you can effectively choose the probability. 00:05:36.460 --> 00:05:40.470 align:middle line:84% And secondly, there could be individual abilities 00:05:40.470 --> 00:05:42.660 align:middle line:84% in the discreteness of the capital grid. 00:05:42.660 --> 00:05:47.820 align:middle line:84% Like, you're going to capitalize a high amount or a low amount. 00:05:47.820 --> 00:05:52.390 align:middle line:84% And again, lotteries help with that indivisibility. 00:05:52.390 --> 00:06:00.120 align:middle line:84% OK, so the next one is really a summary of what we did. 00:06:00.120 --> 00:06:02.760 align:middle line:90% But let me read it out. 00:06:02.760 --> 00:06:06.210 align:middle line:84% Sketch out how one can take the model with the lotteries 00:06:06.210 --> 00:06:10.110 align:middle line:84% to the data on occupation choice, 00:06:10.110 --> 00:06:13.390 align:middle line:90% and estimate parameters. 00:06:13.390 --> 00:06:16.150 align:middle line:84% So can anyone attempt a summary of what we did 00:06:16.150 --> 00:06:18.310 align:middle line:90% last time on that dimension? 00:06:18.310 --> 00:06:28.375 align:middle line:90% 00:06:28.375 --> 00:06:28.875 align:middle line:90% Pablo? 00:06:28.875 --> 00:06:31.880 align:middle line:90% 00:06:31.880 --> 00:06:33.630 align:middle line:84% AUDIENCE: Sure, I can take a stab at this. 00:06:33.630 --> 00:06:38.150 align:middle line:84% So I think the way that we did it was we-- 00:06:38.150 --> 00:06:41.900 align:middle line:84% given these different parameters that we had, of ability, 00:06:41.900 --> 00:06:46.400 align:middle line:84% education, and wealth, we solved for this optimal contract, 00:06:46.400 --> 00:06:47.630 align:middle line:90% theoretically. 00:06:47.630 --> 00:06:51.260 align:middle line:84% And then we made a likelihood function to-- 00:06:51.260 --> 00:06:54.200 align:middle line:84% given these solutions, to see which one of these 00:06:54.200 --> 00:06:58.130 align:middle line:84% is like going to be like the best, essentially. 00:06:58.130 --> 00:07:01.658 align:middle line:84% ROBERT TOWNSEND: Do you remember how we did, quote, "the best?" 00:07:01.658 --> 00:07:04.250 align:middle line:90% The best in terms of what? 00:07:04.250 --> 00:07:06.800 align:middle line:84% AUDIENCE: Just kind of, like, the likelihood function, 00:07:06.800 --> 00:07:07.850 align:middle line:90% trying to maximize that. 00:07:07.850 --> 00:07:09.740 align:middle line:90% ROBERT TOWNSEND: Yeah, OK. 00:07:09.740 --> 00:07:14.960 align:middle line:84% Yeah, so the likelihood function tells us 00:07:14.960 --> 00:07:20.940 align:middle line:84% the probability of what we are seeing in the data 00:07:20.940 --> 00:07:23.190 align:middle line:90% if the model were true. 00:07:23.190 --> 00:07:26.180 align:middle line:84% So we're choosing parameters to make that probability 00:07:26.180 --> 00:07:27.440 align:middle line:90% as high as possible. 00:07:27.440 --> 00:07:31.390 align:middle line:90% 00:07:31.390 --> 00:07:33.840 align:middle line:90% OK, that's good. 00:07:33.840 --> 00:07:38.220 align:middle line:84% Any other questions about that lecture? 00:07:38.220 --> 00:07:40.020 align:middle line:84% All the other questions at the beginning 00:07:40.020 --> 00:07:43.800 align:middle line:84% here were basically, write it down the way we did it 00:07:43.800 --> 00:07:45.480 align:middle line:90% in class, et cetera, et cetera. 00:07:45.480 --> 00:07:49.920 align:middle line:84% So it's not exactly conducive to asking you 00:07:49.920 --> 00:07:51.180 align:middle line:90% these questions in class. 00:07:51.180 --> 00:07:56.010 align:middle line:84% But it is important to be reviewing these review 00:07:56.010 --> 00:07:58.720 align:middle line:90% questions. 00:07:58.720 --> 00:08:04.000 align:middle line:84% All right, so then we come to the lecture today. 00:08:04.000 --> 00:08:08.860 align:middle line:84% This is Walrasian equilibria with an application to trade. 00:08:08.860 --> 00:08:11.890 align:middle line:84% And we're going to consider trade, for the most part, 00:08:11.890 --> 00:08:13.990 align:middle line:90% in a small open economy. 00:08:13.990 --> 00:08:16.900 align:middle line:84% Although we'll say a little bit about putting the world pieces 00:08:16.900 --> 00:08:19.070 align:middle line:90% together at the very end. 00:08:19.070 --> 00:08:22.150 align:middle line:84% So here's the notation, which is largely 00:08:22.150 --> 00:08:25.900 align:middle line:84% a review because you have seen almost all of it before. 00:08:25.900 --> 00:08:29.170 align:middle line:84% When we write down an economy, we mean the commodity space, 00:08:29.170 --> 00:08:32.470 align:middle line:84% in this case, capital L, a finite number of commodities. 00:08:32.470 --> 00:08:34.690 align:middle line:90% I is the number of consumers. 00:08:34.690 --> 00:08:36.460 align:middle line:90% J is the number of firms-- 00:08:36.460 --> 00:08:37.840 align:middle line:90% finite for each. 00:08:37.840 --> 00:08:41.679 align:middle line:84% Household or consumer, I, has a consumption set. 00:08:41.679 --> 00:08:46.060 align:middle line:84% And preferences over bundles in the consumption set, as for, 00:08:46.060 --> 00:08:48.850 align:middle line:84% example, represented by a utility function. 00:08:48.850 --> 00:08:52.480 align:middle line:84% Each firm, J, has a technology production possibility 00:08:52.480 --> 00:08:55.280 align:middle line:84% set in the same L dimensional space. 00:08:55.280 --> 00:08:59.530 align:middle line:84% And the economy has exogenously given endowments, 00:08:59.530 --> 00:09:02.770 align:middle line:84% according to this L dimensional endowment vector, 00:09:02.770 --> 00:09:06.060 align:middle line:84% with the bars being the aggregates. 00:09:06.060 --> 00:09:08.490 align:middle line:90% So that's the economy. 00:09:08.490 --> 00:09:11.580 align:middle line:84% Now, in, say, a private ownership economy, we 00:09:11.580 --> 00:09:15.040 align:middle line:84% have to specify, in addition, who owns what. 00:09:15.040 --> 00:09:20.070 align:middle line:84% So we imagine that consumer I has an endowment vector, L 00:09:20.070 --> 00:09:23.790 align:middle line:84% dimensional, subscript I, for consumer, I. 00:09:23.790 --> 00:09:27.090 align:middle line:84% And it is the sum of these endowments over all 00:09:27.090 --> 00:09:30.840 align:middle line:84% the household that adds up to the omega bar L 00:09:30.840 --> 00:09:33.720 align:middle line:84% dimensional endowment vector on the previous slide. 00:09:33.720 --> 00:09:37.830 align:middle line:84% Aggregates naturally come from the individual's ownership. 00:09:37.830 --> 00:09:41.100 align:middle line:84% Each consumer, in addition to having endowments, 00:09:41.100 --> 00:09:44.650 align:middle line:84% has some share of the profits of firms. 00:09:44.650 --> 00:09:49.860 align:middle line:84% So theta, IJ, being the share that household, I, has, 00:09:49.860 --> 00:09:51.390 align:middle line:90% of J's profits. 00:09:51.390 --> 00:09:54.670 align:middle line:84% So these are meant to be a division of the profit. 00:09:54.670 --> 00:09:58.620 align:middle line:84% So the sum of the shares over households, I, 00:09:58.620 --> 00:10:01.290 align:middle line:84% for a given firm, J, add up to 1. 00:10:01.290 --> 00:10:02.970 align:middle line:90% Everything gets distributed. 00:10:02.970 --> 00:10:07.800 align:middle line:84% And household I is getting theta IJ of pi J, for firm, J. Now, 00:10:07.800 --> 00:10:10.630 align:middle line:84% pi J is obviously an endogenous object. 00:10:10.630 --> 00:10:14.160 align:middle line:84% It could be 0 if there were constant returns to scale. 00:10:14.160 --> 00:10:17.370 align:middle line:84% It could be positive if there are diminishing returns 00:10:17.370 --> 00:10:18.120 align:middle line:90% to scale. 00:10:18.120 --> 00:10:23.010 align:middle line:84% And also, when we did risk and return in village economies, 00:10:23.010 --> 00:10:26.940 align:middle line:84% we kind of naturally had an interesting and relevant 00:10:26.940 --> 00:10:30.900 align:middle line:84% special case that household I gets all the profits 00:10:30.900 --> 00:10:33.390 align:middle line:84% from the activities that it undertakes. 00:10:33.390 --> 00:10:36.330 align:middle line:84% In that case, there was a household, I, and a firm, 00:10:36.330 --> 00:10:38.790 align:middle line:90% I. It was the same entity. 00:10:38.790 --> 00:10:41.190 align:middle line:84% And household I was getting theta IJ 00:10:41.190 --> 00:10:46.020 align:middle line:84% equal to 1 of its own profits, as a special case. 00:10:46.020 --> 00:10:48.840 align:middle line:84% But general equilibrium theory usually 00:10:48.840 --> 00:10:50.700 align:middle line:90% allows something more general. 00:10:50.700 --> 00:10:53.520 align:middle line:84% And you could think of this literally as a market 00:10:53.520 --> 00:10:56.850 align:middle line:84% economy, where the shares are equity shares being traded. 00:10:56.850 --> 00:10:59.640 align:middle line:84% And all of the dividends are claimed 00:10:59.640 --> 00:11:01.380 align:middle line:90% by the ownership of shares. 00:11:01.380 --> 00:11:04.380 align:middle line:84% OK, so a private ownership economy 00:11:04.380 --> 00:11:06.900 align:middle line:84% has a summary of the consumption, 00:11:06.900 --> 00:11:09.210 align:middle line:84% set preferences, endowments of consumer 00:11:09.210 --> 00:11:12.960 align:middle line:84% I, the production sets of all the house firms, J, 00:11:12.960 --> 00:11:16.060 align:middle line:90% and the shares, theta IJ. 00:11:16.060 --> 00:11:19.080 align:middle line:84% Now we get to the big definition here. 00:11:19.080 --> 00:11:22.050 align:middle line:84% In this L dimensional commodity space, 00:11:22.050 --> 00:11:26.790 align:middle line:84% a price vector, p, is the price of all the commodities, inputs, 00:11:26.790 --> 00:11:28.320 align:middle line:90% and outputs, and everything. 00:11:28.320 --> 00:11:31.920 align:middle line:84% And a Walrasian equilibrium, or competitive equilibrium 00:11:31.920 --> 00:11:34.950 align:middle line:84% for short, for this private ownership economy, 00:11:34.950 --> 00:11:39.750 align:middle line:84% is a specification of allocations with stars on them, 00:11:39.750 --> 00:11:42.900 align:middle line:84% x and y star, for consumers and firms-- 00:11:42.900 --> 00:11:49.020 align:middle line:84% and a price vector star here because it's a special price. 00:11:49.020 --> 00:11:52.200 align:middle line:84% Allocations and prices have the special properties. 00:11:52.200 --> 00:11:55.890 align:middle line:84% Each firm, J, has y star J as the maximum 00:11:55.890 --> 00:11:59.550 align:middle line:84% of the firm's problem, which is to maximize profits. 00:11:59.550 --> 00:12:03.660 align:middle line:84% Hence, at y star, the valuation of outputs less inputs 00:12:03.660 --> 00:12:06.990 align:middle line:84% is weakly greater than it would be for any other choice 00:12:06.990 --> 00:12:08.370 align:middle line:90% of production vector. 00:12:08.370 --> 00:12:12.390 align:middle line:84% Each consumer, i, is maximizing relative to the preference 00:12:12.390 --> 00:12:16.410 align:middle line:84% ordering on bundles in the consumption set, but also 00:12:16.410 --> 00:12:17.910 align:middle line:84% subject to the budget constraint, 00:12:17.910 --> 00:12:21.720 align:middle line:84% that expenditures at price p star, for any xi that 00:12:21.720 --> 00:12:23.760 align:middle line:90% could potentially be chosen-- 00:12:23.760 --> 00:12:27.010 align:middle line:84% cannot exceed the right hand side, 00:12:27.010 --> 00:12:28.920 align:middle line:84% which is the private ownership part. 00:12:28.920 --> 00:12:32.490 align:middle line:84% So omega i, being the endowments of consumer, 00:12:32.490 --> 00:12:36.870 align:middle line:84% i, evaluated at price p star, that dot is inner product. 00:12:36.870 --> 00:12:42.060 align:middle line:84% And again, household i's claim on the profits of firm, J. 00:12:42.060 --> 00:12:44.610 align:middle line:84% And the final property is that the allocation 00:12:44.610 --> 00:12:45.495 align:middle line:90% has to be feasible. 00:12:45.495 --> 00:12:49.800 align:middle line:84% And you can just read this as demand cannot exceed supply, 00:12:49.800 --> 00:12:52.500 align:middle line:84% where supply includes both the aggregated endowment 00:12:52.500 --> 00:12:54.540 align:middle line:90% and production. 00:12:54.540 --> 00:12:57.590 align:middle line:84% So we worked a lot with feasibility before. 00:12:57.590 --> 00:13:00.710 align:middle line:84% We did that when we defined Pareto optimality. 00:13:00.710 --> 00:13:02.930 align:middle line:84% The difference here in general equilibrium 00:13:02.930 --> 00:13:05.150 align:middle line:84% or Walrasian equilibrium is we've 00:13:05.150 --> 00:13:08.690 align:middle line:84% added this price vector explicitly, and rewritten 00:13:08.690 --> 00:13:11.270 align:middle line:84% consumer maximization problem, on which we've 00:13:11.270 --> 00:13:12.410 align:middle line:90% had several lectures. 00:13:12.410 --> 00:13:14.990 align:middle line:84% And firms-- sorry, I reversed the order-- 00:13:14.990 --> 00:13:19.050 align:middle line:84% firms max and consumers max problems. 00:13:19.050 --> 00:13:22.906 align:middle line:90% So everything's in one spot. 00:13:22.906 --> 00:13:27.320 align:middle line:84% OK, so a classic example is the Edgeworth Box. 00:13:27.320 --> 00:13:30.740 align:middle line:84% We previously defined Pareto optimal allocations 00:13:30.740 --> 00:13:34.250 align:middle line:84% in this box, as characterized by tangencies 00:13:34.250 --> 00:13:35.720 align:middle line:90% of indifference curves. 00:13:35.720 --> 00:13:38.420 align:middle line:84% Here we add one more ingredient, that if they 00:13:38.420 --> 00:13:42.230 align:middle line:84% start with this endowment, and we draw a price line 00:13:42.230 --> 00:13:46.790 align:middle line:84% through the endowment, then each consumer is maximizing 00:13:46.790 --> 00:13:49.680 align:middle line:90% utility, subject to its budget. 00:13:49.680 --> 00:13:56.550 align:middle line:84% So consumer 1 here has the normal-looking budget line. 00:13:56.550 --> 00:14:01.130 align:middle line:84% And coming from the endowment and the tangency at x star 00:14:01.130 --> 00:14:02.480 align:middle line:90% would be the maximum. 00:14:02.480 --> 00:14:06.500 align:middle line:84% You may remember consumer 2 up here on the northeast 00:14:06.500 --> 00:14:09.920 align:middle line:84% is moving to the southwest in terms of increased utility, 00:14:09.920 --> 00:14:12.590 align:middle line:84% and is also maximizing utility, which 00:14:12.590 --> 00:14:15.780 align:middle line:84% is achieved by a tangency on that consumer's budget line. 00:14:15.780 --> 00:14:18.560 align:middle line:84% So this line here is doing double duty. 00:14:18.560 --> 00:14:20.960 align:middle line:84% It's a budget line for each household. 00:14:20.960 --> 00:14:25.010 align:middle line:84% But the perspective is at or below consumer 1, 00:14:25.010 --> 00:14:28.520 align:middle line:84% at or above the line for consumer 2. 00:14:28.520 --> 00:14:31.850 align:middle line:84% And here you're seeing the star allocations, which 00:14:31.850 --> 00:14:34.010 align:middle line:90% are a competitive equilibrium. 00:14:34.010 --> 00:14:37.700 align:middle line:84% Everything in the box adds up to the aggregates by construction. 00:14:37.700 --> 00:14:39.170 align:middle line:90% There is no production. 00:14:39.170 --> 00:14:41.360 align:middle line:90% They're all maximizing utility. 00:14:41.360 --> 00:14:45.680 align:middle line:84% And the price vector is characterized by the slope 00:14:45.680 --> 00:14:47.300 align:middle line:90% of this budget line. 00:14:47.300 --> 00:14:50.900 align:middle line:84% Now, we could go on with a lot more interesting, 00:14:50.900 --> 00:14:54.710 align:middle line:84% and some nonstandard pictures, of the Edgeworth Box. 00:14:54.710 --> 00:14:58.160 align:middle line:84% But we're going to save a lot of that for another lecture. 00:14:58.160 --> 00:15:00.740 align:middle line:84% Here, I defined Walrasian equilibrium. 00:15:00.740 --> 00:15:04.670 align:middle line:84% I wanted to find something similar but not identical. 00:15:04.670 --> 00:15:08.030 align:middle line:84% It's a price equilibrium with transfers. 00:15:08.030 --> 00:15:11.060 align:middle line:84% So here, the idea is that the right hand 00:15:11.060 --> 00:15:14.030 align:middle line:90% side is a wealth assignment. 00:15:14.030 --> 00:15:16.730 align:middle line:84% Household, i, has wealth, w sub i. 00:15:16.730 --> 00:15:18.903 align:middle line:90% So the rest is similar. 00:15:18.903 --> 00:15:20.820 align:middle line:84% We're going to have an allocation with stars-- 00:15:20.820 --> 00:15:23.840 align:middle line:84% a price vector, p, which should have had stars on it 00:15:23.840 --> 00:15:25.550 align:middle line:90% because it's a special one-- 00:15:25.550 --> 00:15:28.490 align:middle line:84% with the property, and an assignment 00:15:28.490 --> 00:15:32.340 align:middle line:84% of these wealth levels, such that firms maximize profits. 00:15:32.340 --> 00:15:33.800 align:middle line:90% That's the same as before. 00:15:33.800 --> 00:15:36.150 align:middle line:90% Households maximize utility. 00:15:36.150 --> 00:15:40.160 align:middle line:84% But the right hand side of the budget is this wealth object. 00:15:40.160 --> 00:15:43.790 align:middle line:84% And again, when we did partial equilibrium consumer problem, 00:15:43.790 --> 00:15:45.500 align:middle line:84% we talked about prices and wealth. 00:15:45.500 --> 00:15:48.740 align:middle line:84% We had income expansion paths as wealth varied. 00:15:48.740 --> 00:15:51.080 align:middle line:84% So this formulation of the problem 00:15:51.080 --> 00:15:52.700 align:middle line:90% is consistent with that. 00:15:52.700 --> 00:15:54.680 align:middle line:84% Although in partial equilibrium, we 00:15:54.680 --> 00:15:57.170 align:middle line:84% didn't ask where that wealth was coming from. 00:15:57.170 --> 00:15:58.880 align:middle line:90% An allocation is feasible. 00:15:58.880 --> 00:16:03.360 align:middle line:84% Again, demand cannot be greater than the two sources of supply. 00:16:03.360 --> 00:16:06.260 align:middle line:84% Now the wealth assignments have to be feasible. 00:16:06.260 --> 00:16:08.270 align:middle line:84% And here's where we can say something 00:16:08.270 --> 00:16:09.920 align:middle line:84% about where wealth is coming from, 00:16:09.920 --> 00:16:12.710 align:middle line:84% namely, the sum of the wealths, whatever they are, 00:16:12.710 --> 00:16:17.690 align:middle line:84% have to equal the valuation of the aggregated endowment vector 00:16:17.690 --> 00:16:21.425 align:middle line:84% plus the profits of all the firms. 00:16:21.425 --> 00:16:23.300 align:middle line:84% So somehow, this stuff on the right hand side 00:16:23.300 --> 00:16:26.630 align:middle line:84% gets distributed across the households, each household, i, 00:16:26.630 --> 00:16:28.400 align:middle line:90% getting wealth, wi. 00:16:28.400 --> 00:16:30.800 align:middle line:84% And when we add it up, therefore, it 00:16:30.800 --> 00:16:33.360 align:middle line:90% sums up to the right hand side. 00:16:33.360 --> 00:16:37.970 align:middle line:84% Now note that at the bottom here, a particular wealth 00:16:37.970 --> 00:16:41.780 align:middle line:84% assignment that works is what we had on the previous slide, 00:16:41.780 --> 00:16:46.370 align:middle line:84% namely, give household, i, the valuation of its endowment, 00:16:46.370 --> 00:16:50.280 align:middle line:84% plus its share of the profits of the firms that it owns. 00:16:50.280 --> 00:16:54.680 align:middle line:84% So in that sense, a price equilibrium with transfers 00:16:54.680 --> 00:16:59.240 align:middle line:84% is more general, but includes the valuation equilibrium. 00:16:59.240 --> 00:17:03.740 align:middle line:84% That said, the word transfers may catch your eye. 00:17:03.740 --> 00:17:07.609 align:middle line:84% And the idea here is that a Walrasian equilibrium 00:17:07.609 --> 00:17:10.730 align:middle line:84% with ownership specifies one level of wealth. 00:17:10.730 --> 00:17:13.609 align:middle line:84% But we could, in principle, redistribute that wealth 00:17:13.609 --> 00:17:15.800 align:middle line:84% by the government with lump sum taxes 00:17:15.800 --> 00:17:20.720 align:middle line:84% and transfers, and then look for a new priced equilibrium. 00:17:20.720 --> 00:17:23.089 align:middle line:84% And we will-- I think it's on the next slide, yeah. 00:17:23.089 --> 00:17:25.760 align:middle line:84% We will get to why we're doing this. 00:17:25.760 --> 00:17:30.560 align:middle line:84% Namely, any Walrasian equilibrium is Pareto-optimal. 00:17:30.560 --> 00:17:34.880 align:middle line:84% But it's also true that any Pareto-optimal allocation 00:17:34.880 --> 00:17:37.160 align:middle line:84% can be supported as a price equilibrium 00:17:37.160 --> 00:17:40.040 align:middle line:84% with transfers, as in this definition. 00:17:40.040 --> 00:17:43.010 align:middle line:84% However, to achieve the fact that any optimum can 00:17:43.010 --> 00:17:45.980 align:middle line:84% be supported, it's kind of intuitive. 00:17:45.980 --> 00:17:49.570 align:middle line:84% We may have to redistribute wealth. 00:17:49.570 --> 00:17:53.110 align:middle line:84% And again, I have a series of lectures on this. 00:17:53.110 --> 00:17:55.780 align:middle line:84% But if we're back in the Edgeworth Box, 00:17:55.780 --> 00:17:57.850 align:middle line:84% the competitive equilibrium has picked 00:17:57.850 --> 00:18:01.480 align:middle line:84% out one particular Pareto-optimal allocation. 00:18:01.480 --> 00:18:05.530 align:middle line:84% But as there are many optimal allocations-- 00:18:05.530 --> 00:18:07.300 align:middle line:84% and to get to those, we're going to have 00:18:07.300 --> 00:18:10.600 align:middle line:84% to move this budget line up or down, hence, 00:18:10.600 --> 00:18:11.815 align:middle line:90% redistributing the wealth. 00:18:11.815 --> 00:18:14.550 align:middle line:90% 00:18:14.550 --> 00:18:18.510 align:middle line:84% OK, so now that we've got the concept of the Walrasian 00:18:18.510 --> 00:18:21.000 align:middle line:84% equilibrium in hand, I want to focus 00:18:21.000 --> 00:18:24.600 align:middle line:84% on the primary application today, which is production 00:18:24.600 --> 00:18:27.000 align:middle line:90% in a small open economy. 00:18:27.000 --> 00:18:29.610 align:middle line:84% We have to shift gears just a little bit. 00:18:29.610 --> 00:18:32.250 align:middle line:84% It's definitely going to be general equilibrium. 00:18:32.250 --> 00:18:34.290 align:middle line:90% We're going to have J firms. 00:18:34.290 --> 00:18:36.660 align:middle line:90% And firms are producing goods. 00:18:36.660 --> 00:18:40.140 align:middle line:84% So they're J goods because we have J firms. 00:18:40.140 --> 00:18:42.450 align:middle line:90% Firm J produces good, qJ. 00:18:42.450 --> 00:18:44.910 align:middle line:84% And there are L factors of production, 00:18:44.910 --> 00:18:49.080 align:middle line:84% labeled, in this case, Z. So Z1J is 00:18:49.080 --> 00:18:53.500 align:middle line:84% the input of the first factor by firm, J, and so on. 00:18:53.500 --> 00:18:57.450 align:middle line:84% And each firm, J, has a strictly concave production function, 00:18:57.450 --> 00:19:02.520 align:middle line:84% indicated here by f sub J. Output of consumer good, qJ, 00:19:02.520 --> 00:19:05.700 align:middle line:84% is a function of the input vector, ZJ. 00:19:05.700 --> 00:19:08.520 align:middle line:90% So now we specify endowments. 00:19:08.520 --> 00:19:13.320 align:middle line:84% The endowments are going to be given of the factor inputs. 00:19:13.320 --> 00:19:15.060 align:middle line:90% They're owned by households. 00:19:15.060 --> 00:19:17.670 align:middle line:84% And for the sake of argument, let's say they're 00:19:17.670 --> 00:19:19.110 align:middle line:90% supplied in-elastically. 00:19:19.110 --> 00:19:21.340 align:middle line:84% And they're not going to be consumed. 00:19:21.340 --> 00:19:25.290 align:middle line:84% What we're going to try to do is find the factor prices 00:19:25.290 --> 00:19:27.630 align:middle line:84% that's consistent with an equilibrium 00:19:27.630 --> 00:19:29.980 align:middle line:90% within this small open economy. 00:19:29.980 --> 00:19:34.150 align:middle line:84% However, we're going to take the prices of the goods as fixed. 00:19:34.150 --> 00:19:36.660 align:middle line:84% So in that sense, it's still partial equilibrium. 00:19:36.660 --> 00:19:39.990 align:middle line:84% Think of it as a village, as a small open economy 00:19:39.990 --> 00:19:42.420 align:middle line:84% or a country, that is small relative 00:19:42.420 --> 00:19:47.190 align:middle line:84% to the rest of the world so that these price vectors are fixed. 00:19:47.190 --> 00:19:49.860 align:middle line:84% They're not going to be the focus of our attention. 00:19:49.860 --> 00:19:52.800 align:middle line:84% We are going to, however, try to find 00:19:52.800 --> 00:19:56.400 align:middle line:84% the allocation of the inputs across all the firms, how 00:19:56.400 --> 00:19:59.610 align:middle line:84% those endowments of factors get used, 00:19:59.610 --> 00:20:03.390 align:middle line:84% as well as the Walrasian equilibrium price vector. 00:20:03.390 --> 00:20:09.840 align:middle line:84% So firm j producing good qj will maximize the difference 00:20:09.840 --> 00:20:12.580 align:middle line:90% between revenues and costs. 00:20:12.580 --> 00:20:17.010 align:middle line:84% So if w star, say, is a candidate equilibrium price 00:20:17.010 --> 00:20:21.720 align:middle line:84% vector, the firm would choose its input vectors, Zj, 00:20:21.720 --> 00:20:27.180 align:middle line:84% to maximize revenue minus costs at that guessed equilibrium 00:20:27.180 --> 00:20:29.610 align:middle line:90% price vector, w star. 00:20:29.610 --> 00:20:34.650 align:middle line:84% p is, again, a given-- pj, the price of quantity, j, 00:20:34.650 --> 00:20:38.250 align:middle line:90% is externally specified. 00:20:38.250 --> 00:20:44.620 align:middle line:84% So Z star j is the solution to firm j's maximization problem. 00:20:44.620 --> 00:20:46.770 align:middle line:84% And if we're finding an equilibrium, 00:20:46.770 --> 00:20:49.530 align:middle line:84% each firm will be maximizing profits 00:20:49.530 --> 00:20:52.290 align:middle line:84% and the sum of the input demands, 00:20:52.290 --> 00:20:56.610 align:middle line:84% in this case, the demand for the L-th input from firm, 00:20:56.610 --> 00:20:59.670 align:middle line:84% j, summing over all the firms j, and in particular 00:20:59.670 --> 00:21:02.940 align:middle line:84% the stars, which are already profit-maximizing, 00:21:02.940 --> 00:21:07.980 align:middle line:84% add up to the economy's endowment of the factor, Z bar, 00:21:07.980 --> 00:21:09.570 align:middle line:90% for the L-th factor. 00:21:09.570 --> 00:21:12.630 align:middle line:84% And this is true for all, each and every, factor, 00:21:12.630 --> 00:21:16.800 align:middle line:84% L. This one is true for each and every firm, j. 00:21:16.800 --> 00:21:20.610 align:middle line:84% So this is what we're looking for, the Walrasian equilibrium 00:21:20.610 --> 00:21:24.870 align:middle line:84% in the factor markets, taking external prices as given. 00:21:24.870 --> 00:21:28.560 align:middle line:84% All right, you already know something about the solutions 00:21:28.560 --> 00:21:31.140 align:middle line:84% to the profit maximization problem. 00:21:31.140 --> 00:21:34.620 align:middle line:84% Namely, the value of the marginal product 00:21:34.620 --> 00:21:37.920 align:middle line:84% should be equal to-- of the L factor, 00:21:37.920 --> 00:21:39.900 align:middle line:90% should be equal to its cost. 00:21:39.900 --> 00:21:43.800 align:middle line:84% Just differentiate this thing on the previous slide. 00:21:43.800 --> 00:21:47.700 align:middle line:84% With respect to zj, you'll get a derivative in the F function, 00:21:47.700 --> 00:21:52.830 align:middle line:84% and will get the factor input price on the cost side. 00:21:52.830 --> 00:21:53.910 align:middle line:90% That's what this says. 00:21:53.910 --> 00:21:57.450 align:middle line:84% Factor input price-- here's the derivative of F, 00:21:57.450 --> 00:21:59.100 align:middle line:90% with respect to the L factor. 00:21:59.100 --> 00:22:03.540 align:middle line:84% And that p is, of course, part of the exogenously-given price 00:22:03.540 --> 00:22:04.530 align:middle line:90% vector. 00:22:04.530 --> 00:22:07.990 align:middle line:84% How many equations are there like this? 00:22:07.990 --> 00:22:11.070 align:middle line:90% Well, we have to specify here. 00:22:11.070 --> 00:22:12.870 align:middle line:90% It's the L factor. 00:22:12.870 --> 00:22:15.150 align:middle line:90% And we have L-- 00:22:15.150 --> 00:22:17.520 align:middle line:90% capital L factors all together. 00:22:17.520 --> 00:22:20.130 align:middle line:90% And this is for the j firm. 00:22:20.130 --> 00:22:23.680 align:middle line:90% But we have capital J firms. 00:22:23.680 --> 00:22:28.620 align:middle line:84% So we have L times J equations here. 00:22:28.620 --> 00:22:30.480 align:middle line:84% And we have another set of equations 00:22:30.480 --> 00:22:33.930 align:middle line:84% which are the equilibrium, that the demand for the factors 00:22:33.930 --> 00:22:35.370 align:middle line:90% adds up to the total. 00:22:35.370 --> 00:22:38.710 align:middle line:84% But again, this is true for each and every L-th factor. 00:22:38.710 --> 00:22:40.810 align:middle line:90% So there are L of these guys. 00:22:40.810 --> 00:22:44.490 align:middle line:84% So we have L times J equations here. 00:22:44.490 --> 00:22:47.650 align:middle line:90% And we have L equations here. 00:22:47.650 --> 00:22:50.850 align:middle line:84% So the number of equations is equal to the number 00:22:50.850 --> 00:22:52.240 align:middle line:90% of variables. 00:22:52.240 --> 00:22:54.330 align:middle line:84% So in principle, we should be able to solve 00:22:54.330 --> 00:22:57.420 align:middle line:84% this system of equations to determine the equilibrium 00:22:57.420 --> 00:22:58.890 align:middle line:90% factor prices. 00:22:58.890 --> 00:23:01.100 align:middle line:84% It's tedious and boring to keep counting 00:23:01.100 --> 00:23:02.100 align:middle line:90% the number of variables. 00:23:02.100 --> 00:23:06.720 align:middle line:84% But it is actually useful to see if we have fully specified, 00:23:06.720 --> 00:23:09.340 align:middle line:84% and we potentially have a solution. 00:23:09.340 --> 00:23:13.080 align:middle line:84% So now, we've already solved in lecture 4, 00:23:13.080 --> 00:23:17.020 align:middle line:84% which was about production, a cost minimization problem. 00:23:17.020 --> 00:23:21.030 align:middle line:84% So we defined there the cost for firm J 00:23:21.030 --> 00:23:27.120 align:middle line:84% of producing the quantity, qj, when the input vector was w. 00:23:27.120 --> 00:23:30.420 align:middle line:84% I must say, even I went back this morning 00:23:30.420 --> 00:23:33.540 align:middle line:84% in reviewing this lecture to take a look at the earlier 00:23:33.540 --> 00:23:34.590 align:middle line:90% lecture. 00:23:34.590 --> 00:23:38.280 align:middle line:84% Today there's kind of an uneasy tension because I'm partially 00:23:38.280 --> 00:23:41.580 align:middle line:84% telling you things that you've had before, on the other hand, 00:23:41.580 --> 00:23:46.020 align:middle line:84% aware that you may not remember exactly what we had before. 00:23:46.020 --> 00:23:48.630 align:middle line:84% If you go back and look, it'll all come back to you. 00:23:48.630 --> 00:23:50.820 align:middle line:84% And some of these slides are actually 00:23:50.820 --> 00:23:53.820 align:middle line:84% repeats of what was in lecture 4. 00:23:53.820 --> 00:23:57.540 align:middle line:84% Anyway, when we derived the least cost 00:23:57.540 --> 00:24:00.660 align:middle line:84% way of producing an output, qj, we 00:24:00.660 --> 00:24:04.330 align:middle line:84% got that marginal cost was equal to price. 00:24:04.330 --> 00:24:04.830 align:middle line:90% It's equal. 00:24:04.830 --> 00:24:09.660 align:middle line:84% Marginal cost is from your very first economics micro course, 00:24:09.660 --> 00:24:10.560 align:middle line:90% probably. 00:24:10.560 --> 00:24:14.310 align:middle line:84% Now, this second equation is the equation 00:24:14.310 --> 00:24:17.970 align:middle line:84% that the demands for the factors adds up to the total. 00:24:17.970 --> 00:24:19.800 align:middle line:84% But it's written in an interesting 00:24:19.800 --> 00:24:21.990 align:middle line:90% and a bit unconventional way. 00:24:21.990 --> 00:24:26.970 align:middle line:84% The aggregate amount of factor l that's available is z bar l. 00:24:26.970 --> 00:24:30.570 align:middle line:84% And we want the demands of these firms for producing. 00:24:30.570 --> 00:24:33.780 align:middle line:84% But instead, we use the Shephard's lemma, 00:24:33.780 --> 00:24:37.050 align:middle line:84% the derivative of the cost of firm j, 00:24:37.050 --> 00:24:40.560 align:middle line:84% with respect to the price of the l-th factor, 00:24:40.560 --> 00:24:45.820 align:middle line:84% we derived as the amount of the l-th factor to be used. 00:24:45.820 --> 00:24:48.510 align:middle line:84% And again, you may or may not remember that that's 00:24:48.510 --> 00:24:50.610 align:middle line:90% the Shephard's lemma. 00:24:50.610 --> 00:24:53.770 align:middle line:84% Here it is, written down here at the bottom. 00:24:53.770 --> 00:24:57.660 align:middle line:84% So we just substitute what was written as zlj, 00:24:57.660 --> 00:25:02.700 align:middle line:84% summation over j equal to z bar, we substitute in the result 00:25:02.700 --> 00:25:05.550 align:middle line:84% from Shephard's lemma to get this equation 00:25:05.550 --> 00:25:08.670 align:middle line:90% OK, special case, 2 by 2. 00:25:08.670 --> 00:25:11.920 align:middle line:84% And there'll be another two coming later. 00:25:11.920 --> 00:25:17.760 align:middle line:84% But for now, two types of firms, J equals 1, 2-- 00:25:17.760 --> 00:25:19.230 align:middle line:90% two inputs. 00:25:19.230 --> 00:25:23.410 align:middle line:84% Now, it's useful to think of the inputs as labor and capital. 00:25:23.410 --> 00:25:26.040 align:middle line:84% And that, in fact, will anticipate the trade 00:25:26.040 --> 00:25:27.185 align:middle line:90% applications. 00:25:27.185 --> 00:25:28.810 align:middle line:84% Are we reproducing goods with machines? 00:25:28.810 --> 00:25:31.440 align:middle line:84% Are we producing goods with labor? 00:25:31.440 --> 00:25:34.740 align:middle line:84% What happens to the labor when we have 00:25:34.740 --> 00:25:36.720 align:middle line:90% a decline in manufacturing? 00:25:36.720 --> 00:25:38.610 align:middle line:84% It comes from having more inputs. 00:25:38.610 --> 00:25:41.640 align:middle line:84% So these are the kind of ingredients, labor and capital 00:25:41.640 --> 00:25:46.320 align:middle line:84% being kind of obvious labels for us, in the case of this two 00:25:46.320 --> 00:25:47.730 align:middle line:90% factor model. 00:25:47.730 --> 00:25:49.590 align:middle line:84% Production functions, just to write out 00:25:49.590 --> 00:25:54.060 align:middle line:84% the notation, production function for good 1, as it 00:25:54.060 --> 00:25:58.260 align:middle line:84% inputs, the second subscript denotes the firm, 00:25:58.260 --> 00:25:59.650 align:middle line:90% the first one, the input. 00:25:59.650 --> 00:26:02.040 align:middle line:84% So this is the first input in firm 1, 00:26:02.040 --> 00:26:04.635 align:middle line:90% the second input in firm 1. 00:26:04.635 --> 00:26:07.860 align:middle line:84% f2 is output of firm 2 as a function 00:26:07.860 --> 00:26:10.050 align:middle line:84% of the first input in for firm 2, 00:26:10.050 --> 00:26:12.420 align:middle line:90% and the second input for firm 2. 00:26:12.420 --> 00:26:14.460 align:middle line:84% And we're going to assume, in this special 2 00:26:14.460 --> 00:26:17.400 align:middle line:84% by 2 model, that these production functions, each 00:26:17.400 --> 00:26:20.260 align:middle line:84% of them, are constant returns to scale. 00:26:20.260 --> 00:26:22.980 align:middle line:90% So it's homogeneity of degree 1. 00:26:22.980 --> 00:26:26.370 align:middle line:84% So if we're given a vector of input prices 00:26:26.370 --> 00:26:29.670 align:middle line:84% for capital and labor, we can define 00:26:29.670 --> 00:26:34.320 align:middle line:84% the cost of producing one unit of good, j. 00:26:34.320 --> 00:26:38.010 align:middle line:90% This is called the unit cost-- 00:26:38.010 --> 00:26:42.630 align:middle line:84% or you may want to review that the cost of producing arbitrary 00:26:42.630 --> 00:26:45.420 align:middle line:84% levels just scales up or down depending 00:26:45.420 --> 00:26:48.120 align:middle line:84% on whether that other target level is 00:26:48.120 --> 00:26:50.400 align:middle line:90% higher or lower than 1. 00:26:50.400 --> 00:26:53.910 align:middle line:84% So once we have the cost of producing one unit, 00:26:53.910 --> 00:26:57.330 align:middle line:84% we can get the cost of producing any number of units. 00:26:57.330 --> 00:27:00.840 align:middle line:84% And what inputs solve that minimum cost problem? 00:27:00.840 --> 00:27:02.940 align:middle line:90% They're denoted with a's. 00:27:02.940 --> 00:27:07.890 align:middle line:84% So for firm j, a1j and a2j are the way 00:27:07.890 --> 00:27:12.600 align:middle line:84% to achieve the least cost when the input vector is w. 00:27:12.600 --> 00:27:17.190 align:middle line:84% Input price vector is w So here I 00:27:17.190 --> 00:27:20.350 align:middle line:84% belabor a little bit what you've seen before. 00:27:20.350 --> 00:27:22.540 align:middle line:90% It's constant returns to scale. 00:27:22.540 --> 00:27:25.180 align:middle line:90% So the cost function is linear. 00:27:25.180 --> 00:27:27.630 align:middle line:84% Constant returns to scale is this linear production 00:27:27.630 --> 00:27:29.220 align:middle line:90% possibilities frontier. 00:27:29.220 --> 00:27:30.730 align:middle line:90% Cost function is linear. 00:27:30.730 --> 00:27:32.790 align:middle line:84% Marginal cost equals average cost. 00:27:32.790 --> 00:27:35.070 align:middle line:84% If price of a good, say good j, is 00:27:35.070 --> 00:27:38.560 align:middle line:84% less than marginal and average cost, nothing gets produced. 00:27:38.560 --> 00:27:41.940 align:middle line:84% And when price is equal to marginal and average cost, 00:27:41.940 --> 00:27:43.620 align:middle line:84% then they're happy to produce anything. 00:27:43.620 --> 00:27:44.850 align:middle line:90% Although profits are 0. 00:27:44.850 --> 00:27:48.930 align:middle line:84% And price cannot be higher because they would go bananas. 00:27:48.930 --> 00:27:52.170 align:middle line:84% And then the assertion, which I've already said twice, 00:27:52.170 --> 00:27:53.850 align:middle line:90% profits must be zero. 00:27:53.850 --> 00:27:56.860 align:middle line:84% And that's just, again, a review slide from before. 00:27:56.860 --> 00:27:59.610 align:middle line:90% So price equals cost. 00:27:59.610 --> 00:28:01.050 align:middle line:90% This is the unit cost. 00:28:01.050 --> 00:28:03.000 align:middle line:90% Price is the price of a unit. 00:28:03.000 --> 00:28:06.030 align:middle line:90% So we have cost equal to prices. 00:28:06.030 --> 00:28:08.880 align:middle line:90% OK, so another review. 00:28:08.880 --> 00:28:10.500 align:middle line:90% We did this before, too. 00:28:10.500 --> 00:28:12.450 align:middle line:84% What's special about constant returns 00:28:12.450 --> 00:28:17.620 align:middle line:84% to scale is we can solve the problem for the one 00:28:17.620 --> 00:28:20.590 align:middle line:84% unit of output, minimize the cost 00:28:20.590 --> 00:28:23.110 align:middle line:90% of achieving one unit of output. 00:28:23.110 --> 00:28:24.790 align:middle line:90% And that's going to give us-- 00:28:24.790 --> 00:28:26.710 align:middle line:84% I guess I just said this, the cost 00:28:26.710 --> 00:28:30.490 align:middle line:84% for any other level of output, just by scaling up and down 00:28:30.490 --> 00:28:31.420 align:middle line:90% the solution. 00:28:31.420 --> 00:28:35.740 align:middle line:84% And the ratios of input use will not be changing. 00:28:35.740 --> 00:28:38.440 align:middle line:84% I think in the previous lecture, I actually 00:28:38.440 --> 00:28:41.110 align:middle line:84% said I wasn't terribly pleased with how 00:28:41.110 --> 00:28:42.700 align:middle line:90% cryptically-worded it was. 00:28:42.700 --> 00:28:45.850 align:middle line:84% And I gave you as an informal homework assignment 00:28:45.850 --> 00:28:50.770 align:middle line:84% to do a proof by contradiction to prove that the input use 00:28:50.770 --> 00:28:53.590 align:middle line:84% ratio is going to be entirely dictated 00:28:53.590 --> 00:28:57.790 align:middle line:84% by the input prices, regardless of the scale. 00:28:57.790 --> 00:29:01.150 align:middle line:84% That is to say, this input ratio will remain constant. 00:29:01.150 --> 00:29:04.960 align:middle line:84% And again, please, I apologize because I'm throwing at you 00:29:04.960 --> 00:29:06.580 align:middle line:90% things from a previous lecture. 00:29:06.580 --> 00:29:07.570 align:middle line:90% It was lecture 4. 00:29:07.570 --> 00:29:09.370 align:middle line:90% So that was some time ago. 00:29:09.370 --> 00:29:10.870 align:middle line:84% But I don't want to belabor it today 00:29:10.870 --> 00:29:12.740 align:middle line:84% or we won't get to something new. 00:29:12.740 --> 00:29:15.940 align:middle line:84% So for now, we'll just accept this as a fact. 00:29:15.940 --> 00:29:19.600 align:middle line:84% Then we come to a key definition about factor intensity. 00:29:19.600 --> 00:29:25.180 align:middle line:84% Say good 1 is more intensive in factor 1 than good 2. 00:29:25.180 --> 00:29:27.940 align:middle line:84% Then the production of good 2, if naturally, 00:29:27.940 --> 00:29:33.340 align:middle line:84% firm producing good 1 uses relatively more of input 1-- 00:29:33.340 --> 00:29:37.600 align:middle line:84% input 1 relative to input 2, then does firm 2. 00:29:37.600 --> 00:29:41.200 align:middle line:84% So the definition, to say good 1 is relatively more 00:29:41.200 --> 00:29:44.950 align:middle line:84% intensive than factor 1, means that the ratio of factor 1 00:29:44.950 --> 00:29:49.150 align:middle line:84% to factor 2 in firm 1 is higher than it is in firm 2. 00:29:49.150 --> 00:29:51.640 align:middle line:84% So it's a pretty natural definition. 00:29:51.640 --> 00:29:53.620 align:middle line:90% Of course, it can move around. 00:29:53.620 --> 00:29:57.080 align:middle line:84% But the inequality remains the same. 00:29:57.080 --> 00:29:57.880 align:middle line:90% It's intense. 00:29:57.880 --> 00:30:01.510 align:middle line:84% No matter what the input price ratio is, 00:30:01.510 --> 00:30:04.960 align:middle line:84% the optimizing choice of inputs will always 00:30:04.960 --> 00:30:09.610 align:middle line:84% satisfy this inequality when comparing firm 1 to firm 2. 00:30:09.610 --> 00:30:13.840 align:middle line:84% So now something a bit new that was not in lecture 4, 00:30:13.840 --> 00:30:15.340 align:middle line:90% although it's not hard. 00:30:15.340 --> 00:30:20.770 align:middle line:84% We're going to plot the cost of producing one unit of a good. 00:30:20.770 --> 00:30:23.770 align:middle line:84% We're going to plot that cost curve as a function 00:30:23.770 --> 00:30:27.670 align:middle line:90% of the input prices, w1 and w2. 00:30:27.670 --> 00:30:32.380 align:middle line:84% So in particular, if, say, for a given w1 and w2 bar, 00:30:32.380 --> 00:30:35.650 align:middle line:84% your assertion is we're on this cost curve, 00:30:35.650 --> 00:30:41.560 align:middle line:84% then if we increase the price of the first input to w prime 1 00:30:41.560 --> 00:30:44.050 align:middle line:84% and we're supposed to be on an isocost, 00:30:44.050 --> 00:30:46.480 align:middle line:84% meaning same cost curve, the increase 00:30:46.480 --> 00:30:50.000 align:middle line:84% in the first factor price would mean the cost would be higher. 00:30:50.000 --> 00:30:53.530 align:middle line:84% So we have to lower the price of the second factor 00:30:53.530 --> 00:30:55.310 align:middle line:90% in order to compensate. 00:30:55.310 --> 00:30:58.960 align:middle line:84% So that on net, after we make this move, 00:30:58.960 --> 00:31:02.780 align:middle line:84% total costs are the same because it is the isocost curve. 00:31:02.780 --> 00:31:07.090 align:middle line:84% So this down-sloping nature of the isocost curve, 00:31:07.090 --> 00:31:11.570 align:middle line:84% plotted against various input prices, makes a lot of sense. 00:31:11.570 --> 00:31:14.260 align:middle line:84% The other thing is this concave shape. 00:31:14.260 --> 00:31:19.060 align:middle line:84% This comes from a property of the minimized cost. 00:31:19.060 --> 00:31:24.190 align:middle line:84% In particular, this set of input prices at or above the cost 00:31:24.190 --> 00:31:25.780 align:middle line:90% line is a convex set. 00:31:25.780 --> 00:31:27.730 align:middle line:84% But we did this a lot with respect 00:31:27.730 --> 00:31:30.550 align:middle line:84% to the consumer optimization problem. 00:31:30.550 --> 00:31:34.390 align:middle line:84% We defined quasi-concave utility functions and the associated 00:31:34.390 --> 00:31:35.920 align:middle line:90% concave indifference curves. 00:31:35.920 --> 00:31:37.960 align:middle line:84% And we had the weak upper contour 00:31:37.960 --> 00:31:40.090 align:middle line:90% set as being strictly convex. 00:31:40.090 --> 00:31:41.650 align:middle line:90% So these are analogs here. 00:31:41.650 --> 00:31:45.310 align:middle line:84% Now, the assertion is that this cost function is, in fact, 00:31:45.310 --> 00:31:46.330 align:middle line:90% concave. 00:31:46.330 --> 00:31:48.850 align:middle line:90% And again, as a review-- 00:31:48.850 --> 00:31:50.850 align:middle line:90% but who remembers? 00:31:50.850 --> 00:31:54.540 align:middle line:84% We already had derived in lecture 4 00:31:54.540 --> 00:31:58.030 align:middle line:84% that the cost function is concave in the input prices. 00:31:58.030 --> 00:32:01.140 align:middle line:84% So I just pasted in the slide to remind us. 00:32:01.140 --> 00:32:04.170 align:middle line:84% So now we have down-sloping concave. 00:32:04.170 --> 00:32:05.880 align:middle line:84% And the other thing that's going on here 00:32:05.880 --> 00:32:09.330 align:middle line:84% is, at these particular input prices, 00:32:09.330 --> 00:32:11.310 align:middle line:84% if we go back to Shephard's lemma, 00:32:11.310 --> 00:32:14.580 align:middle line:84% the derivative of the minimized cost, 00:32:14.580 --> 00:32:17.760 align:middle line:84% with respect to, say, the price of the first factor, 00:32:17.760 --> 00:32:20.790 align:middle line:84% is the level input of the first factor, 00:32:20.790 --> 00:32:24.040 align:middle line:84% and similarly, for the level input of the second factor. 00:32:24.040 --> 00:32:27.480 align:middle line:84% So the ratio, the slope of this line, which 00:32:27.480 --> 00:32:29.850 align:middle line:84% is the ratio of the derivatives of the cost 00:32:29.850 --> 00:32:33.630 align:middle line:84% curve at an optimum, must, by Shephard's lemma, 00:32:33.630 --> 00:32:37.630 align:middle line:84% be equal to the ratio of the optimizing inputs. 00:32:37.630 --> 00:32:42.870 align:middle line:84% And again, logically, if you increased the input price 00:32:42.870 --> 00:32:46.470 align:middle line:84% of factor 1, keeping cost constant, 00:32:46.470 --> 00:32:49.170 align:middle line:84% you would move along this isocost line. 00:32:49.170 --> 00:32:53.100 align:middle line:84% And you would increase the amount of good 2 00:32:53.100 --> 00:32:57.870 align:middle line:84% and factor 2, and decrease the amount of factor 1. 00:32:57.870 --> 00:32:59.700 align:middle line:84% You substitute away from the factor, which 00:32:59.700 --> 00:33:01.300 align:middle line:90% has now increased in price. 00:33:01.300 --> 00:33:04.350 align:middle line:84% So in the earlier slide, lecture 4, I didn't replicate it. 00:33:04.350 --> 00:33:09.420 align:middle line:84% We had a Walt Disney example about machinery versus humans. 00:33:09.420 --> 00:33:12.600 align:middle line:84% Now I'll just warn you, it is very hard 00:33:12.600 --> 00:33:14.790 align:middle line:84% to remember what's divided by what. 00:33:14.790 --> 00:33:18.300 align:middle line:84% But anyway, the fact that this perpendicular becomes steeper 00:33:18.300 --> 00:33:21.720 align:middle line:84% is consistent with the input of factor 1 00:33:21.720 --> 00:33:25.560 align:middle line:84% going down because it's the inverse ratio. 00:33:25.560 --> 00:33:28.330 align:middle line:90% All right, did that. 00:33:28.330 --> 00:33:32.310 align:middle line:84% So now, how do we solve for equilibrium factor prices? 00:33:32.310 --> 00:33:35.100 align:middle line:84% This is just a picture of a particular firm 00:33:35.100 --> 00:33:37.380 align:middle line:90% and what it would do, firm j-- 00:33:37.380 --> 00:33:39.810 align:middle line:84% what it would do as we vary the input prices? 00:33:39.810 --> 00:33:42.210 align:middle line:84% And we want to find particular input prices, 00:33:42.210 --> 00:33:45.960 align:middle line:84% such that both firms are minimizing costs, 00:33:45.960 --> 00:33:48.390 align:middle line:84% and all the factors get utilized. 00:33:48.390 --> 00:33:51.810 align:middle line:84% So remember, because we have constant returns to scale 00:33:51.810 --> 00:33:55.050 align:middle line:84% and because we've taken the right hand side prices as 00:33:55.050 --> 00:33:57.990 align:middle line:84% arbitrary and given, coming from nowhere, 00:33:57.990 --> 00:34:01.800 align:middle line:84% say, but certainly known to the firms, as part 00:34:01.800 --> 00:34:04.770 align:middle line:84% of the solution, the unit cost has 00:34:04.770 --> 00:34:08.670 align:middle line:84% to equal to the price for both firms 1 and 2. 00:34:08.670 --> 00:34:13.330 align:middle line:84% So we have, from this, two equations and two unknowns, 00:34:13.330 --> 00:34:18.310 align:middle line:84% namely the factor prices, w1 and w2, that we want to solve for. 00:34:18.310 --> 00:34:23.280 align:middle line:84% So visually, we can actually put both firm 1's and firm 2's 00:34:23.280 --> 00:34:26.219 align:middle line:84% isocost lines on the same diagram. 00:34:26.219 --> 00:34:28.150 align:middle line:84% In other words, I've gone from here, 00:34:28.150 --> 00:34:34.050 align:middle line:84% which was firm j, and firm j's particular isocost line-- 00:34:34.050 --> 00:34:36.870 align:middle line:84% talking about how it varies with w1 and w2, 00:34:36.870 --> 00:34:40.260 align:middle line:84% to this diagram, in which we have not only firm 1-- 00:34:40.260 --> 00:34:43.320 align:middle line:90% excuse me, but also firm two. 00:34:43.320 --> 00:34:46.469 align:middle line:84% And if we're going to find an equilibrium, 00:34:46.469 --> 00:34:50.310 align:middle line:84% we're going to find price for factor 1 and a price for factor 00:34:50.310 --> 00:34:51.540 align:middle line:90% 2, each. 00:34:51.540 --> 00:34:54.150 align:middle line:84% So each firm, in an equilibrium, is going 00:34:54.150 --> 00:34:56.670 align:middle line:90% to face the same factor prices. 00:34:56.670 --> 00:35:00.810 align:middle line:84% And each firm must lie on its own unit cost 00:35:00.810 --> 00:35:03.940 align:middle line:84% curve equal to the price from this equation. 00:35:03.940 --> 00:35:10.110 align:middle line:84% So you can see the unit cost for firm 1 equal to p 2, 00:35:10.110 --> 00:35:12.930 align:middle line:84% the unit cost of firm 2 equal to p2. 00:35:12.930 --> 00:35:16.230 align:middle line:84% These are particular isocost curves for firm 1 00:35:16.230 --> 00:35:19.920 align:middle line:84% and firm 2 that are associated with two of the equilibrium 00:35:19.920 --> 00:35:20.710 align:middle line:90% equations. 00:35:20.710 --> 00:35:23.580 align:middle line:84% And if we're going to find common factor prices faced 00:35:23.580 --> 00:35:26.340 align:middle line:84% by both of the firms, we only want 00:35:26.340 --> 00:35:28.890 align:middle line:84% one factor price for the first factor, 00:35:28.890 --> 00:35:31.150 align:middle line:84% and one factor price for the second one. 00:35:31.150 --> 00:35:33.240 align:middle line:84% So where can that happen in this diagram? 00:35:33.240 --> 00:35:38.110 align:middle line:84% Only one spot, and that's where these isocost curves cross. 00:35:38.110 --> 00:35:40.930 align:middle line:84% So this is kind of displaying the equilibrium. 00:35:40.930 --> 00:35:45.270 align:middle line:84% Let me say one more word about why they're shaped in this way. 00:35:45.270 --> 00:35:49.080 align:middle line:84% Remember, firm 1 is intensive in factor 1. 00:35:49.080 --> 00:35:51.900 align:middle line:84% So if we did move along its curve, 00:35:51.900 --> 00:35:55.170 align:middle line:90% lowering w2 and increasing w1-- 00:35:55.170 --> 00:35:57.840 align:middle line:84% because it's intensive in factor 1, 00:35:57.840 --> 00:35:59.730 align:middle line:84% it tends to use a lot of factor 1. 00:35:59.730 --> 00:36:02.160 align:middle line:84% So increasing the price of factor 1 00:36:02.160 --> 00:36:04.740 align:middle line:84% is kind over big burden for firm 1. 00:36:04.740 --> 00:36:07.260 align:middle line:84% And to compensate, we have to lower 00:36:07.260 --> 00:36:10.140 align:middle line:84% the price of the second factor quite a lot, 00:36:10.140 --> 00:36:15.000 align:middle line:84% relative to the compensation we would have to offer firm 2. 00:36:15.000 --> 00:36:19.950 align:middle line:84% For the same increase in w1, the corresponding decrease in w2 00:36:19.950 --> 00:36:24.630 align:middle line:84% is less for firm 2 than firm 1 because, again, firm 1 00:36:24.630 --> 00:36:26.790 align:middle line:90% is intensive in factor 1. 00:36:26.790 --> 00:36:29.100 align:middle line:84% So this factor intensity, which I 00:36:29.100 --> 00:36:31.470 align:middle line:84% said was a very natural definition, 00:36:31.470 --> 00:36:34.860 align:middle line:84% is true everywhere by definition. 00:36:34.860 --> 00:36:38.730 align:middle line:84% Hence, the curve for firm 1 has to be steeper everywhere 00:36:38.730 --> 00:36:40.350 align:middle line:90% than the curve for firm 2. 00:36:40.350 --> 00:36:42.900 align:middle line:84% And that's why they have to cross. 00:36:42.900 --> 00:36:45.150 align:middle line:90% Questions? 00:36:45.150 --> 00:36:48.370 align:middle line:84% OK, so again, just to review this slide, 00:36:48.370 --> 00:36:50.130 align:middle line:84% we're looking for an equilibrium. 00:36:50.130 --> 00:36:53.160 align:middle line:84% The claim is we found it now because we 00:36:53.160 --> 00:36:57.480 align:middle line:84% have found the factor prices for the first and second factor, 00:36:57.480 --> 00:37:01.530 align:middle line:84% such that each firm is minimizing its unit costs 00:37:01.530 --> 00:37:03.870 align:middle line:90% associated with the cost-- 00:37:03.870 --> 00:37:06.590 align:middle line:84% the unit cost being equal to price. 00:37:06.590 --> 00:37:09.460 align:middle line:84% So these must be the equilibrium quantities 00:37:09.460 --> 00:37:13.930 align:middle line:84% of the factors being utilized in each of the two firms. 00:37:13.930 --> 00:37:18.430 align:middle line:84% All right, so we're kind of doing an algorithm here. 00:37:18.430 --> 00:37:23.530 align:middle line:84% The algorithm was define Walrasian equilibrium. 00:37:23.530 --> 00:37:26.650 align:middle line:84% And do that for this 2 by 2 model. 00:37:26.650 --> 00:37:29.500 align:middle line:84% And without telling you that I was going to do this, 00:37:29.500 --> 00:37:31.420 align:middle line:90% we're doing it iteratively. 00:37:31.420 --> 00:37:36.400 align:middle line:84% The very first thing we do is solve for the factor prices. 00:37:36.400 --> 00:37:39.620 align:middle line:84% And now we're going to get the quantities this way. 00:37:39.620 --> 00:37:41.470 align:middle line:84% So the quantities, naturally, are 00:37:41.470 --> 00:37:45.160 align:middle line:84% going to come from a diagram where we have isoquants. 00:37:45.160 --> 00:37:47.350 align:middle line:84% So here we have the input of factor 1, 00:37:47.350 --> 00:37:48.850 align:middle line:90% the input of factor 2. 00:37:48.850 --> 00:37:52.270 align:middle line:84% For firm j, this is an isoquant equal production, 00:37:52.270 --> 00:37:55.340 align:middle line:84% as you move along that isoquant curve. 00:37:55.340 --> 00:37:58.660 align:middle line:84% And again, you may remember, or not, 00:37:58.660 --> 00:38:04.900 align:middle line:84% that to minimize the cost of producing one unit, 00:38:04.900 --> 00:38:08.170 align:middle line:84% we find the tangency, where the isoquant 00:38:08.170 --> 00:38:11.650 align:middle line:90% is tangent to the isocost line. 00:38:11.650 --> 00:38:14.230 align:middle line:84% Now again, it's a bit dizzying because what's 00:38:14.230 --> 00:38:18.880 align:middle line:84% an isocost line in one space is a different-looking cost 00:38:18.880 --> 00:38:19.870 align:middle line:90% line in another. 00:38:19.870 --> 00:38:22.150 align:middle line:84% In this space, the isocost curves 00:38:22.150 --> 00:38:25.300 align:middle line:90% are concave, as I was saying. 00:38:25.300 --> 00:38:27.670 align:middle line:84% That's in the space of factor inputs. 00:38:27.670 --> 00:38:29.950 align:middle line:90% W1 and W2 prices. 00:38:29.950 --> 00:38:34.210 align:middle line:84% w1 and w2 here, the isocost line is 00:38:34.210 --> 00:38:39.880 align:middle line:84% linear, in the space of input quantities, z1 and z2. 00:38:39.880 --> 00:38:44.470 align:middle line:84% So again, the minimized cost of producing one unit of output 00:38:44.470 --> 00:38:49.780 align:middle line:84% here at these factor prices, w, is this tangency. 00:38:49.780 --> 00:38:55.120 align:middle line:84% And it's associated with the inputs, a1 and a2, to firm, j. 00:38:55.120 --> 00:38:58.240 align:middle line:84% And again, the reason is something 00:38:58.240 --> 00:39:00.820 align:middle line:84% we reviewed before, that when you take 00:39:00.820 --> 00:39:07.510 align:middle line:84% the derivative of the isoquant, we're getting dz2, dz1, 00:39:07.510 --> 00:39:09.290 align:middle line:84% the quantity is staying the same. 00:39:09.290 --> 00:39:11.530 align:middle line:90% So we solve for the solution. 00:39:11.530 --> 00:39:14.380 align:middle line:84% And we're getting a relationship between the derivative 00:39:14.380 --> 00:39:17.720 align:middle line:90% of the isocost and the inputs. 00:39:17.720 --> 00:39:22.160 align:middle line:84% OK, so now we want to solve, not only for quantities, 00:39:22.160 --> 00:39:25.160 align:middle line:90% but the equilibrium amounts. 00:39:25.160 --> 00:39:29.540 align:middle line:84% Now remember, again, here we were playing around 00:39:29.540 --> 00:39:30.980 align:middle line:90% with unit costs. 00:39:30.980 --> 00:39:34.280 align:middle line:84% But we don't know the level of the quantities yet. 00:39:34.280 --> 00:39:38.270 align:middle line:84% And likewise, hence, we don't know the levels of the inputs. 00:39:38.270 --> 00:39:40.520 align:middle line:84% Although we do know something about the ratios 00:39:40.520 --> 00:39:45.260 align:middle line:84% of the inputs one once we find the equilibrium factor prices. 00:39:45.260 --> 00:39:47.990 align:middle line:84% So having found the equilibrium factor prices, 00:39:47.990 --> 00:39:51.950 align:middle line:84% we now turn to determining the inputs. 00:39:51.950 --> 00:39:54.470 align:middle line:84% But it almost looks like a review. 00:39:54.470 --> 00:39:57.740 align:middle line:84% The input ratio for firm 1 must be 00:39:57.740 --> 00:40:01.370 align:middle line:84% those a coefficients, the solution to the cost 00:40:01.370 --> 00:40:03.800 align:middle line:84% minimization problem in the quantity 00:40:03.800 --> 00:40:08.270 align:middle line:84% space at the equilibrium factor prices, w star. 00:40:08.270 --> 00:40:09.560 align:middle line:90% This is true for firm 1. 00:40:09.560 --> 00:40:11.490 align:middle line:90% It's also true for firm 2. 00:40:11.490 --> 00:40:14.540 align:middle line:84% So the ratios of inputs being used 00:40:14.540 --> 00:40:17.990 align:middle line:84% must be consistent with the optimized ratio 00:40:17.990 --> 00:40:21.890 align:middle line:84% of inputs being used at the equilibrium factor prices. 00:40:21.890 --> 00:40:23.360 align:middle line:90% There are two equations there. 00:40:23.360 --> 00:40:27.890 align:middle line:84% There's another equation, namely that the inputs being 00:40:27.890 --> 00:40:31.460 align:middle line:84% used by the firm, in this case, input 1, and the second, input 00:40:31.460 --> 00:40:32.060 align:middle line:90% 2-- 00:40:32.060 --> 00:40:34.790 align:middle line:84% input 1 in firm 1, input 1 in firm 2, 00:40:34.790 --> 00:40:37.470 align:middle line:84% must sum up to the aggregate endowments. 00:40:37.470 --> 00:40:41.690 align:middle line:84% So finally, another ingredient of the economy, 00:40:41.690 --> 00:40:44.360 align:middle line:84% the level of the input endowments, 00:40:44.360 --> 00:40:46.500 align:middle line:90% is here on the right hand side. 00:40:46.500 --> 00:40:49.640 align:middle line:84% And when we found an equilibrium in input use, 00:40:49.640 --> 00:40:53.730 align:middle line:84% it has to satisfy this system of equations. 00:40:53.730 --> 00:40:56.300 align:middle line:84% Now I can show you a picture of it. 00:40:56.300 --> 00:40:59.150 align:middle line:84% You may remember that when we did production, 00:40:59.150 --> 00:41:01.650 align:middle line:84% we talked about the production box-- 00:41:01.650 --> 00:41:04.700 align:middle line:84% not the Edgeworth Box of the consumer theory, 00:41:04.700 --> 00:41:08.030 align:middle line:84% but the production box of production theory. 00:41:08.030 --> 00:41:10.610 align:middle line:84% We had, again, like the isoquants, 00:41:10.610 --> 00:41:13.940 align:middle line:84% input 1 and input 2 on the x and y-axis, 00:41:13.940 --> 00:41:17.270 align:middle line:90% firm 1 is oriented around 01. 00:41:17.270 --> 00:41:20.360 align:middle line:84% And we have isoquants for firm 1. 00:41:20.360 --> 00:41:24.080 align:middle line:84% And this ratio, the slope of this line, 00:41:24.080 --> 00:41:27.720 align:middle line:84% is the optimizing ratio of inputs. 00:41:27.720 --> 00:41:30.770 align:middle line:84% And again, that optimizing ratio of inputs 00:41:30.770 --> 00:41:33.410 align:middle line:84% is entirely pinned down by the equilibrium factor 00:41:33.410 --> 00:41:35.400 align:middle line:90% prices, which we already found. 00:41:35.400 --> 00:41:38.600 align:middle line:84% So they have to lie on this line somewhere. 00:41:38.600 --> 00:41:39.710 align:middle line:90% Why here? 00:41:39.710 --> 00:41:44.030 align:middle line:84% Because when we go up to firm 2, the same thing is true, 00:41:44.030 --> 00:41:49.040 align:middle line:84% that the optimizing input ratio of factor 2 to factor 1 must 00:41:49.040 --> 00:41:53.550 align:middle line:84% lie on this line coming out of the 02 origin. 00:41:53.550 --> 00:41:56.810 align:middle line:84% So input ratio for firm 1 is fixed. 00:41:56.810 --> 00:41:59.347 align:middle line:90% Input ratio for firm 2 is fixed. 00:41:59.347 --> 00:41:59.930 align:middle line:90% Fixed at what? 00:41:59.930 --> 00:42:03.410 align:middle line:84% Fixed at these levels, as in the earlier diagrams. 00:42:03.410 --> 00:42:07.430 align:middle line:84% Hence, in an equilibrium, to be consistent with each of them, 00:42:07.430 --> 00:42:10.190 align:middle line:84% it must be where these lines cross. 00:42:10.190 --> 00:42:12.650 align:middle line:84% So this finally determines output 00:42:12.650 --> 00:42:14.840 align:middle line:90% because we have isoquants here. 00:42:14.840 --> 00:42:18.500 align:middle line:84% So firm 1 is producing on this isoquant. 00:42:18.500 --> 00:42:20.870 align:middle line:84% Firm 2 is producing on this isoquant. 00:42:20.870 --> 00:42:24.200 align:middle line:84% So now we've got both, the equilibrium 00:42:24.200 --> 00:42:26.240 align:middle line:90% level of the inputs-- 00:42:26.240 --> 00:42:29.210 align:middle line:84% which is boom boom and boom boom, as well as 00:42:29.210 --> 00:42:32.560 align:middle line:84% the outputs associated with these isoquants. 00:42:32.560 --> 00:42:36.830 align:middle line:90% OK, are there questions so far? 00:42:36.830 --> 00:42:39.400 align:middle line:84% Well, let me just say, when you review, 00:42:39.400 --> 00:42:41.740 align:middle line:84% there is a lot of content in this lecture. 00:42:41.740 --> 00:42:45.130 align:middle line:84% Some of it's review, as I've already been apologizing. 00:42:45.130 --> 00:42:46.750 align:middle line:90% Go back to lecture 4. 00:42:46.750 --> 00:42:49.120 align:middle line:84% But also, it's kind of stacked, in the sense 00:42:49.120 --> 00:42:53.770 align:middle line:84% that each segment builds on the subsequent segments. 00:42:53.770 --> 00:42:55.450 align:middle line:84% And it's very hard to take it in when 00:42:55.450 --> 00:42:58.100 align:middle line:84% I go through these slides for the first time. 00:42:58.100 --> 00:43:01.100 align:middle line:84% But I think if you review it the second round, 00:43:01.100 --> 00:43:03.820 align:middle line:84% you'll see how the ingredients are coming together. 00:43:03.820 --> 00:43:06.760 align:middle line:84% Anyway, where are we at this point? 00:43:06.760 --> 00:43:09.460 align:middle line:84% We had a country, say, an economy, 00:43:09.460 --> 00:43:12.970 align:middle line:84% with two types of firms, j, equal 1, 2. 00:43:12.970 --> 00:43:16.390 align:middle line:84% We have two factors, say labor and capital. 00:43:16.390 --> 00:43:20.740 align:middle line:84% We had this economy facing externally-given arbitrary 00:43:20.740 --> 00:43:25.330 align:middle line:84% prices, p1 and p2, for the firms 1 and 2. 00:43:25.330 --> 00:43:30.370 align:middle line:84% And we have now solved for the market clearing factor prices 00:43:30.370 --> 00:43:35.140 align:middle line:84% in the input markets for the level of the inputs being used 00:43:35.140 --> 00:43:38.510 align:middle line:84% and the equilibrium quantities being produced. 00:43:38.510 --> 00:43:40.560 align:middle line:90% That's how far we got already. 00:43:40.560 --> 00:43:44.240 align:middle line:84% OK, so now I want to get to the trade part. 00:43:44.240 --> 00:43:45.920 align:middle line:84% And I'm going to add the other two. 00:43:45.920 --> 00:43:49.630 align:middle line:84% So this is two firms, two factors, and two countries. 00:43:49.630 --> 00:43:52.390 align:middle line:84% And the two countries are denoted A and B. 00:43:52.390 --> 00:43:56.260 align:middle line:84% So each country has the same technologies. 00:43:56.260 --> 00:43:59.350 align:middle line:84% Each country can produce good 1 and produce 00:43:59.350 --> 00:44:04.450 align:middle line:84% good 2 with these factors, the two factors, labor and capital. 00:44:04.450 --> 00:44:07.090 align:middle line:84% We could have imagined different technologies 00:44:07.090 --> 00:44:09.580 align:middle line:84% in the different countries, and in a minute, 00:44:09.580 --> 00:44:11.080 align:middle line:90% different preferences. 00:44:11.080 --> 00:44:13.240 align:middle line:84% You haven't seen the consumer side of it yet. 00:44:13.240 --> 00:44:18.400 align:middle line:84% But the famous 2 by 2 by 2 model with some amazingly cool 00:44:18.400 --> 00:44:21.430 align:middle line:84% theorems assumes identical technologies 00:44:21.430 --> 00:44:24.400 align:middle line:84% and identical preferences in the two countries. 00:44:24.400 --> 00:44:26.560 align:middle line:84% What's different across the two countries 00:44:26.560 --> 00:44:28.240 align:middle line:90% are the factor endowments. 00:44:28.240 --> 00:44:31.000 align:middle line:84% Namely, one country is capital abundant. 00:44:31.000 --> 00:44:32.890 align:middle line:90% The other is labor abundant. 00:44:32.890 --> 00:44:35.590 align:middle line:84% So you might, for example, have thought-- 00:44:35.590 --> 00:44:37.660 align:middle line:84% although this is no longer as true 00:44:37.660 --> 00:44:40.420 align:middle line:84% as it once was for reasons of the theorem we're 00:44:40.420 --> 00:44:41.560 align:middle line:90% about to see-- 00:44:41.560 --> 00:44:45.040 align:middle line:84% that countries like Mexico or China 00:44:45.040 --> 00:44:47.530 align:middle line:84% are relatively well-endowed with labor. 00:44:47.530 --> 00:44:50.200 align:middle line:84% The US, being quite industrialized, 00:44:50.200 --> 00:44:53.230 align:middle line:84% is relatively well-endowed with capital. 00:44:53.230 --> 00:44:57.010 align:middle line:84% So the US, as distinct from Mexico, 00:44:57.010 --> 00:45:00.010 align:middle line:90% is a capital-abundant country. 00:45:00.010 --> 00:45:02.830 align:middle line:84% And Mexico is a labor-abundant country. 00:45:02.830 --> 00:45:06.520 align:middle line:84% I choose the example carefully because, again, with NAFTA, 00:45:06.520 --> 00:45:08.530 align:middle line:84% there's all kinds of controversies 00:45:08.530 --> 00:45:12.640 align:middle line:84% about what happened to the wages of US domestic workers 00:45:12.640 --> 00:45:16.120 align:middle line:84% when the US increased its trade with Mexico. 00:45:16.120 --> 00:45:18.670 align:middle line:84% We now have a theorem that's going to tell us 00:45:18.670 --> 00:45:20.080 align:middle line:90% something about that. 00:45:20.080 --> 00:45:23.470 align:middle line:84% Oh, anyway, here, two countries, A and B, with different factor 00:45:23.470 --> 00:45:25.780 align:middle line:84% endowments-- one being capital abundant, 00:45:25.780 --> 00:45:27.130 align:middle line:90% and the other labor abundant. 00:45:27.130 --> 00:45:29.170 align:middle line:84% Now, what if there's no trade at all? 00:45:29.170 --> 00:45:30.580 align:middle line:90% We're in autarky. 00:45:30.580 --> 00:45:34.180 align:middle line:84% What's going to determine the price of the consumption goods? 00:45:34.180 --> 00:45:36.580 align:middle line:84% Well, that's where we need the consumer part. 00:45:36.580 --> 00:45:38.740 align:middle line:84% We're going to need the representative consumer 00:45:38.740 --> 00:45:41.440 align:middle line:84% in a given country maximizing utility, 00:45:41.440 --> 00:45:46.580 align:middle line:84% given its ownership of factors, and potentially shares. 00:45:46.580 --> 00:45:49.120 align:middle line:84% Although in this constant returns to scale world, 00:45:49.120 --> 00:45:50.950 align:middle line:90% there are no profits anyway. 00:45:50.950 --> 00:45:53.680 align:middle line:84% So we have an entirely domestic market, 00:45:53.680 --> 00:45:56.980 align:middle line:84% like China completely closed to the rest of the world. 00:45:56.980 --> 00:46:00.010 align:middle line:84% Now we jump all the way to free trade. 00:46:00.010 --> 00:46:02.410 align:middle line:84% And if there's no cost of shipping goods around, 00:46:02.410 --> 00:46:03.640 align:middle line:90% there's an abstraction. 00:46:03.640 --> 00:46:05.320 align:middle line:84% The price of the two goods now have 00:46:05.320 --> 00:46:09.400 align:middle line:84% to be the same in each country because if there's 00:46:09.400 --> 00:46:13.180 align:middle line:84% no cost to shipping goods around, if the price of a good 00:46:13.180 --> 00:46:17.830 align:middle line:84% were lower in one country than another, then 00:46:17.830 --> 00:46:20.290 align:middle line:84% they would basically be exporting 00:46:20.290 --> 00:46:22.270 align:middle line:84% a lot of that good at a low price 00:46:22.270 --> 00:46:24.430 align:middle line:84% to the other country that's willing to buy it 00:46:24.430 --> 00:46:25.340 align:middle line:90% at a high price. 00:46:25.340 --> 00:46:27.040 align:middle line:84% So those prices can't be different 00:46:27.040 --> 00:46:29.530 align:middle line:84% when there's no cost to shipping goods around. 00:46:29.530 --> 00:46:32.860 align:middle line:84% Now exactly where the prices are, 00:46:32.860 --> 00:46:34.910 align:middle line:90% I will show you in a minute. 00:46:34.910 --> 00:46:38.910 align:middle line:84% But first, let me give you the theorem called 00:46:38.910 --> 00:46:40.640 align:middle line:90% Heckscher-Ohlin theorem. 00:46:40.640 --> 00:46:42.640 align:middle line:84% Suppose initially, we're in autarky. 00:46:42.640 --> 00:46:44.740 align:middle line:90% And neither country is trading. 00:46:44.740 --> 00:46:47.470 align:middle line:84% And the price of the capital-intensive good 00:46:47.470 --> 00:46:52.150 align:middle line:84% in the capital-abundant country will be relatively low-- 00:46:52.150 --> 00:46:55.780 align:middle line:84% relative to the price of that capital-intensive good 00:46:55.780 --> 00:46:57.550 align:middle line:90% in the other country. 00:46:57.550 --> 00:47:00.640 align:middle line:84% This is intuitive in terms of the economics. 00:47:00.640 --> 00:47:05.230 align:middle line:84% You have a country that has a lot of capital and relatively 00:47:05.230 --> 00:47:06.370 align:middle line:90% little labor. 00:47:06.370 --> 00:47:08.020 align:middle line:90% That's abundant capital. 00:47:08.020 --> 00:47:10.570 align:middle line:84% That should drive down the price of the capital 00:47:10.570 --> 00:47:14.950 align:middle line:84% input, relative to the wage of the labor input-- 00:47:14.950 --> 00:47:18.400 align:middle line:84% relative also to the other country, 00:47:18.400 --> 00:47:21.700 align:middle line:84% which is abundant in labor, and should have, 00:47:21.700 --> 00:47:27.070 align:middle line:84% because there's a lot of labor around, a relatively low wage. 00:47:27.070 --> 00:47:29.020 align:middle line:84% So those are the two statements, the price 00:47:29.020 --> 00:47:32.080 align:middle line:84% of the capital-intensive good in the capital-abundant country 00:47:32.080 --> 00:47:35.710 align:middle line:84% will be lower, relative to the price 00:47:35.710 --> 00:47:38.500 align:middle line:84% of that capital-intensive good in the other country. 00:47:38.500 --> 00:47:40.450 align:middle line:90% And vise versa for labor. 00:47:40.450 --> 00:47:43.900 align:middle line:84% The price of the labor-intensive good in the labor-abundant 00:47:43.900 --> 00:47:48.490 align:middle line:84% country will be lower, relative to the price of that 00:47:48.490 --> 00:47:50.740 align:middle line:84% labor-intensive good in the other country-- 00:47:50.740 --> 00:47:55.190 align:middle line:84% or in some, when an input is abundant, it has a low price. 00:47:55.190 --> 00:47:57.220 align:middle line:90% Now we go to trade. 00:47:57.220 --> 00:47:59.950 align:middle line:84% We don't allow the inputs to migrate. 00:47:59.950 --> 00:48:02.740 align:middle line:84% But we do allow trade in the goods. 00:48:02.740 --> 00:48:04.210 align:middle line:90% Think about that first line. 00:48:04.210 --> 00:48:06.430 align:middle line:84% The price of the capital-intensive good 00:48:06.430 --> 00:48:10.630 align:middle line:84% is low, relative to the price of the capital-intensive 00:48:10.630 --> 00:48:12.230 align:middle line:90% good in the other country. 00:48:12.230 --> 00:48:15.370 align:middle line:84% So because it has a low price, it 00:48:15.370 --> 00:48:17.980 align:middle line:84% should be a, quote, "competitive export." 00:48:17.980 --> 00:48:22.300 align:middle line:84% So the capital-abundant country will be exporting 00:48:22.300 --> 00:48:24.220 align:middle line:90% the capital-intensive good. 00:48:24.220 --> 00:48:27.340 align:middle line:84% The labor-abundant country will be exporting 00:48:27.340 --> 00:48:29.590 align:middle line:90% the labor-intensive good. 00:48:29.590 --> 00:48:32.290 align:middle line:84% And we're going to find an equilibrium where 00:48:32.290 --> 00:48:36.310 align:middle line:84% this kind of arbitrage is no longer profitable, where 00:48:36.310 --> 00:48:39.460 align:middle line:90% the prices will be the same. 00:48:39.460 --> 00:48:43.870 align:middle line:84% But the adjustment mechanism is coming from the possibility 00:48:43.870 --> 00:48:45.470 align:middle line:90% of exports and imports. 00:48:45.470 --> 00:48:48.220 align:middle line:84% So let me show you an example picture. 00:48:48.220 --> 00:48:52.300 align:middle line:84% Suppose we have country B here with this production 00:48:52.300 --> 00:48:56.350 align:middle line:84% possibilities set in the two goods, G1 and G2. 00:48:56.350 --> 00:48:59.900 align:middle line:84% Now, I'm not telling you exactly why it looks like this. 00:48:59.900 --> 00:49:06.700 align:middle line:84% But it has to do with the fact that country B is relatively 00:49:06.700 --> 00:49:09.040 align:middle line:90% abundant in one of the factors. 00:49:09.040 --> 00:49:11.710 align:middle line:84% You can see, relative to country A, 00:49:11.710 --> 00:49:14.200 align:middle line:84% this country can produce a lot of good 2, 00:49:14.200 --> 00:49:16.420 align:middle line:90% and relatively little of good 1. 00:49:16.420 --> 00:49:19.030 align:middle line:84% And that's because of the different factor intensities. 00:49:19.030 --> 00:49:21.100 align:middle line:84% The flip side of that is country A 00:49:21.100 --> 00:49:24.160 align:middle line:84% over here with its production possibility frontier. 00:49:24.160 --> 00:49:29.380 align:middle line:84% And it can produce relatively more of good 1 than good 2, 00:49:29.380 --> 00:49:31.960 align:middle line:84% in the sense of enumerating all the possibilities. 00:49:31.960 --> 00:49:33.430 align:middle line:90% If you're in autarky-- 00:49:33.430 --> 00:49:36.010 align:middle line:84% now, you remember the Robinson Crusoe economy 00:49:36.010 --> 00:49:38.980 align:middle line:84% where we had Robinson Crusoe producing two goods? 00:49:38.980 --> 00:49:43.630 align:middle line:84% No trade-- we had, in autarky therefore, on the island, 00:49:43.630 --> 00:49:46.120 align:middle line:84% Robinson Crusoe's indifference curve 00:49:46.120 --> 00:49:49.550 align:middle line:84% would be tangent to his production possibility set. 00:49:49.550 --> 00:49:54.490 align:middle line:84% So this would be the autarky allocation in country B. 00:49:54.490 --> 00:49:58.260 align:middle line:84% Likewise, country A, with a different technology, 00:49:58.260 --> 00:50:02.140 align:middle line:84% a different island, so to speak, would have another tangency 00:50:02.140 --> 00:50:05.380 align:middle line:84% of its preference indifference curve, 00:50:05.380 --> 00:50:07.510 align:middle line:84% with a production possibility set. 00:50:07.510 --> 00:50:12.010 align:middle line:84% Now, it's kind of easier to draw with common preferences, 00:50:12.010 --> 00:50:15.340 align:middle line:84% having the same indifference curve tangent in both. 00:50:15.340 --> 00:50:18.760 align:middle line:84% But that's not a necessary part of the argument. 00:50:18.760 --> 00:50:22.150 align:middle line:84% All we care about is tangency for country B 00:50:22.150 --> 00:50:24.070 align:middle line:84% relative to its indifference curves, 00:50:24.070 --> 00:50:27.190 align:middle line:84% and tangency for country A in autarky, 00:50:27.190 --> 00:50:28.825 align:middle line:84% relative to its indifference curves. 00:50:28.825 --> 00:50:31.540 align:middle line:84% It just happens to be the same indifference curve. 00:50:31.540 --> 00:50:34.690 align:middle line:84% Now finally, we go from autarky to free trade. 00:50:34.690 --> 00:50:36.700 align:middle line:90% What must be the solution? 00:50:36.700 --> 00:50:42.460 align:middle line:84% Each country-- B over here will separate its consumption 00:50:42.460 --> 00:50:43.750 align:middle line:90% and production problems. 00:50:43.750 --> 00:50:46.420 align:middle line:84% It will maximize profits by finding 00:50:46.420 --> 00:50:49.570 align:middle line:84% the point on its production possibility frontier 00:50:49.570 --> 00:50:52.960 align:middle line:84% which gives it the highest level of profits, 00:50:52.960 --> 00:50:57.220 align:middle line:84% as captured by this budget line, really-- 00:50:57.220 --> 00:51:01.490 align:middle line:84% trade line, that emanates from the new change production 00:51:01.490 --> 00:51:01.990 align:middle line:90% point. 00:51:01.990 --> 00:51:04.840 align:middle line:84% And I know the old production point was autarky 00:51:04.840 --> 00:51:10.180 align:middle line:84% for country B. It's now shifted to be giving up some of good 1, 00:51:10.180 --> 00:51:15.610 align:middle line:84% and producing more of good 2, and then exporting good 2 00:51:15.610 --> 00:51:17.770 align:middle line:90% and getting more of good 1 back. 00:51:17.770 --> 00:51:21.250 align:middle line:84% So it specializes more in production. 00:51:21.250 --> 00:51:23.560 align:middle line:84% And then achieves balance in consumption 00:51:23.560 --> 00:51:25.840 align:middle line:90% by imports and exports. 00:51:25.840 --> 00:51:29.060 align:middle line:84% And country A down here is doing the opposite. 00:51:29.060 --> 00:51:30.970 align:middle line:84% It specialized more in production 00:51:30.970 --> 00:51:33.970 align:middle line:84% in the other direction, more of good 1, 00:51:33.970 --> 00:51:38.260 align:middle line:84% and then trades in the world economy, giving up good 1 00:51:38.260 --> 00:51:39.940 align:middle line:90% and getting good 2. 00:51:39.940 --> 00:51:44.140 align:middle line:84% And although it's not drawn here, the exports of good 1 00:51:44.140 --> 00:51:48.820 align:middle line:84% for country A are exactly equal to the imports of good 1 00:51:48.820 --> 00:51:53.290 align:middle line:84% for country B. So the prices of the goods associated 00:51:53.290 --> 00:51:55.690 align:middle line:84% with this external price line are 00:51:55.690 --> 00:51:57.760 align:middle line:84% the same for both countries, and have 00:51:57.760 --> 00:52:00.400 align:middle line:84% the property that, finally, we have an equilibrium 00:52:00.400 --> 00:52:01.450 align:middle line:90% in the goods markets. 00:52:01.450 --> 00:52:04.540 align:middle line:84% In sum, we've now made the prices endogenous. 00:52:04.540 --> 00:52:08.260 align:middle line:84% P1 and 2, which were given throughout 3/4 of the lecture, 00:52:08.260 --> 00:52:12.040 align:middle line:84% are now determined internationally by free trade. 00:52:12.040 --> 00:52:14.590 align:middle line:84% All right, so we did Heckscher-Ohlin. 00:52:14.590 --> 00:52:18.130 align:middle line:84% Now we have another theorem, Stopler-Samuelson theorem. 00:52:18.130 --> 00:52:22.090 align:middle line:84% In this 2 by 2 by 2 model with these varying factor 00:52:22.090 --> 00:52:27.260 align:middle line:84% intensities, if the price of good i increases, 00:52:27.260 --> 00:52:29.680 align:middle line:84% then the equilibrium price of the factor 00:52:29.680 --> 00:52:32.360 align:middle line:84% that's used more intensively in good i 00:52:32.360 --> 00:52:36.590 align:middle line:84% also increases, while the price of the other factor decreases. 00:52:36.590 --> 00:52:39.080 align:middle line:90% So let me say it again. 00:52:39.080 --> 00:52:41.570 align:middle line:84% And let me give the motivation here. 00:52:41.570 --> 00:52:45.320 align:middle line:84% We're going from autarky in each country one 00:52:45.320 --> 00:52:49.280 align:middle line:84% at a time, with its internal domestic prices, 00:52:49.280 --> 00:52:52.280 align:middle line:84% to an equilibrium in the world economy 00:52:52.280 --> 00:52:55.680 align:middle line:84% with free trade at different prices. 00:52:55.680 --> 00:52:59.990 align:middle line:84% So for country b here, P1 over P2 00:52:59.990 --> 00:53:04.610 align:middle line:84% is going down, relative to what it will be in the free trade 00:53:04.610 --> 00:53:05.450 align:middle line:90% equilibrium. 00:53:05.450 --> 00:53:07.490 align:middle line:84% And the reverse is true for country A. 00:53:07.490 --> 00:53:10.220 align:middle line:84% So when we go from autarky to trade, 00:53:10.220 --> 00:53:13.550 align:middle line:84% we're moving the price ratio in both countries-- 00:53:13.550 --> 00:53:15.690 align:middle line:84% in different directions, but we're moving it. 00:53:15.690 --> 00:53:19.190 align:middle line:84% So let's say, what is the impact now of moving a price? 00:53:19.190 --> 00:53:23.390 align:middle line:84% If we increase the price of good i relative to the other good, 00:53:23.390 --> 00:53:25.790 align:middle line:84% the equilibrium price of the factor 00:53:25.790 --> 00:53:29.330 align:middle line:84% that was used more intensively in the production of good i 00:53:29.330 --> 00:53:32.210 align:middle line:84% will increase, while the price of the other factor 00:53:32.210 --> 00:53:33.930 align:middle line:90% will decrease. 00:53:33.930 --> 00:53:38.210 align:middle line:84% OK, so let's go back to a proof of this in the earlier diagram, 00:53:38.210 --> 00:53:42.200 align:middle line:84% setting aside the two countries, kind of not asking for a minute 00:53:42.200 --> 00:53:45.860 align:middle line:84% where the price comes from-- except that where previously we 00:53:45.860 --> 00:53:48.530 align:middle line:84% had these isocost lines crossing, 00:53:48.530 --> 00:53:51.590 align:middle line:84% the cost lines associated with P1 and P2-- 00:53:51.590 --> 00:53:54.620 align:middle line:84% we now increase the price of good 1. 00:53:54.620 --> 00:53:59.090 align:middle line:90% So C1 equal to P1 is here. 00:53:59.090 --> 00:54:03.680 align:middle line:84% C1 equal to P prime 1 lies further out. 00:54:03.680 --> 00:54:05.300 align:middle line:84% It kind of stands to reason that it 00:54:05.300 --> 00:54:08.150 align:middle line:84% should lie further out because we've increased the price. 00:54:08.150 --> 00:54:10.200 align:middle line:90% And we want cost equal to price. 00:54:10.200 --> 00:54:12.650 align:middle line:84% So if price is higher, cost can be higher. 00:54:12.650 --> 00:54:17.330 align:middle line:84% And cost can be higher by having higher levels of w1 and w2 00:54:17.330 --> 00:54:18.780 align:middle line:90% as the input prices. 00:54:18.780 --> 00:54:21.620 align:middle line:84% So the isocost line for the price 00:54:21.620 --> 00:54:24.440 align:middle line:84% when the price increases, in this case, good 1's price 00:54:24.440 --> 00:54:26.720 align:middle line:90% increases, shifts outward. 00:54:26.720 --> 00:54:30.380 align:middle line:84% We haven't done anything to the price of the other good. 00:54:30.380 --> 00:54:32.150 align:middle line:90% The C2-- where is it? 00:54:32.150 --> 00:54:35.030 align:middle line:90% The C2 line is staying constant. 00:54:35.030 --> 00:54:39.170 align:middle line:84% The C2 isocost line of price B2 is staying constant. 00:54:39.170 --> 00:54:42.140 align:middle line:84% However, we need a crossing-- that an equilibrium 00:54:42.140 --> 00:54:44.180 align:middle line:84% is described by where the lines cross. 00:54:44.180 --> 00:54:46.700 align:middle line:84% So we would move from this initial crossing 00:54:46.700 --> 00:54:48.950 align:middle line:90% to this second crossing. 00:54:48.950 --> 00:54:51.500 align:middle line:84% And you can see, as a consequence 00:54:51.500 --> 00:54:56.390 align:middle line:84% of the increase in price in P1, that the price of factor 1-- 00:54:56.390 --> 00:54:59.480 align:middle line:90% good 1 is intensive in factor 1. 00:54:59.480 --> 00:55:02.660 align:middle line:84% That's the definition that we were given at the beginning. 00:55:02.660 --> 00:55:05.330 align:middle line:90% The price of good 1 increases. 00:55:05.330 --> 00:55:07.460 align:middle line:84% Therefore, the input, which is used 00:55:07.460 --> 00:55:11.670 align:middle line:84% intensively in good 1, namely factor 1, should increase. 00:55:11.670 --> 00:55:13.320 align:middle line:90% And the other one decreases. 00:55:13.320 --> 00:55:17.390 align:middle line:84% So this diagram illustrates the Stopler-Samuelson theorem. 00:55:17.390 --> 00:55:19.940 align:middle line:84% But again, in the context of free trade, 00:55:19.940 --> 00:55:22.430 align:middle line:84% those price movements come about for a reason. 00:55:22.430 --> 00:55:25.640 align:middle line:84% Namely, we've gone from autarky to free trade. 00:55:25.640 --> 00:55:29.000 align:middle line:84% So you could see earlier which way the price was moving, 00:55:29.000 --> 00:55:32.870 align:middle line:84% depending on the initial country and the autarky. 00:55:32.870 --> 00:55:35.720 align:middle line:84% Now, although we know, again, to repeat 00:55:35.720 --> 00:55:37.520 align:middle line:84% the steps of the algorithm, we now 00:55:37.520 --> 00:55:39.530 align:middle line:84% know what happens to the factor prices 00:55:39.530 --> 00:55:43.040 align:middle line:84% from this diagram, what happens to the levels of the inputs, 00:55:43.040 --> 00:55:44.840 align:middle line:90% and what happens to output. 00:55:44.840 --> 00:55:48.050 align:middle line:84% Well, we have this, again, box for production. 00:55:48.050 --> 00:55:51.890 align:middle line:84% We had the initial allocation ratio of the factors 00:55:51.890 --> 00:55:53.750 align:middle line:90% as quantities crossed. 00:55:53.750 --> 00:55:57.620 align:middle line:84% That is to say, the line for good 1 and the line for good 2 00:55:57.620 --> 00:55:58.730 align:middle line:90% crossed. 00:55:58.730 --> 00:56:00.620 align:middle line:90% And they crossed here. 00:56:00.620 --> 00:56:03.830 align:middle line:84% But now those factor prices have changed. 00:56:03.830 --> 00:56:06.200 align:middle line:84% As a consequence of the price change, 00:56:06.200 --> 00:56:10.400 align:middle line:84% the quantity of the first factor will go down 00:56:10.400 --> 00:56:14.750 align:middle line:84% because its price has gone up from country 1's point of view. 00:56:14.750 --> 00:56:19.010 align:middle line:84% That produces a somewhat steeper sloped line. 00:56:19.010 --> 00:56:21.770 align:middle line:84% And the opposite is going on with country 2. 00:56:21.770 --> 00:56:26.210 align:middle line:84% So the new equilibrium is where these now-changed slope 00:56:26.210 --> 00:56:27.830 align:middle line:90% lines cross. 00:56:27.830 --> 00:56:30.260 align:middle line:84% And you can actually see, in this case, 00:56:30.260 --> 00:56:34.340 align:middle line:84% that firm 1 is producing more because its isoquant will 00:56:34.340 --> 00:56:34.850 align:middle line:90% be higher. 00:56:34.850 --> 00:56:37.760 align:middle line:84% And it's using a higher level of both inputs. 00:56:37.760 --> 00:56:43.490 align:middle line:84% And likewise, the production of good 2 will contract. 00:56:43.490 --> 00:56:47.630 align:middle line:84% So again, this factor price equalization theorem is, 00:56:47.630 --> 00:56:52.550 align:middle line:84% just to restate it, in the 2 by 2 and 2-country model, 00:56:52.550 --> 00:56:56.390 align:middle line:84% as long as we have this factor intensity assumption holding 00:56:56.390 --> 00:56:59.300 align:middle line:84% and the countries are not completely specialized, 00:56:59.300 --> 00:57:03.110 align:middle line:84% the equilibrium price of factors, inputs, 00:57:03.110 --> 00:57:07.790 align:middle line:84% only depends on the technologies and on the output prices-- 00:57:07.790 --> 00:57:10.640 align:middle line:84% determined as it may be, in such a way 00:57:10.640 --> 00:57:15.310 align:middle line:84% that the prices of the outputs equals the input costs. 00:57:15.310 --> 00:57:18.760 align:middle line:84% But now, look, if both countries face the same prices 00:57:18.760 --> 00:57:20.890 align:middle line:84% because they trade in goods, and otherwise there 00:57:20.890 --> 00:57:23.740 align:middle line:84% would be arbitrage, then since the prices are 00:57:23.740 --> 00:57:28.210 align:middle line:84% the same in both countries, we can zero inside each country, 00:57:28.210 --> 00:57:32.020 align:middle line:84% and look at the factor prices, which are clearing the factor 00:57:32.020 --> 00:57:34.250 align:middle line:90% markets in each country. 00:57:34.250 --> 00:57:37.360 align:middle line:84% And since each country has the same technology, 00:57:37.360 --> 00:57:42.100 align:middle line:84% these equations across the two goods in each country hold. 00:57:42.100 --> 00:57:44.290 align:middle line:84% If the prices are the same on the right hand 00:57:44.290 --> 00:57:47.290 align:middle line:84% side as a consequence of free trade, 00:57:47.290 --> 00:57:50.380 align:middle line:84% then the solution to the input prices 00:57:50.380 --> 00:57:53.180 align:middle line:84% has to be the same on the left hand side. 00:57:53.180 --> 00:57:58.360 align:middle line:84% So in short, remarkably, we have not allowed trade in factors-- 00:57:58.360 --> 00:58:00.070 align:middle line:90% no migration. 00:58:00.070 --> 00:58:04.030 align:middle line:84% But we've managed to equalize factor prices without migration 00:58:04.030 --> 00:58:07.280 align:middle line:90% simply by having trade in goods. 00:58:07.280 --> 00:58:11.260 align:middle line:84% So that's the famous factor price equalization theorem. 00:58:11.260 --> 00:58:17.560 align:middle line:84% So vis a vis, the US and Mexico, allowing the NAFTA, free trade 00:58:17.560 --> 00:58:20.110 align:middle line:84% in goods, even with a wall, if I dare 00:58:20.110 --> 00:58:24.070 align:middle line:84% say, prohibiting Mexican migration, 00:58:24.070 --> 00:58:29.680 align:middle line:84% the theorem says that the wages in Mexico should rise in Mexico 00:58:29.680 --> 00:58:33.620 align:middle line:84% and fall to the US, to the point that the wages are the same. 00:58:33.620 --> 00:58:36.610 align:middle line:84% So the factors bear consequences. 00:58:36.610 --> 00:58:40.270 align:middle line:84% It's not necessarily a good thing that wages of labor 00:58:40.270 --> 00:58:42.190 align:middle line:90% are falling in the US-- 00:58:42.190 --> 00:58:43.690 align:middle line:90% not good for those firms. 00:58:43.690 --> 00:58:46.540 align:middle line:84% We have to come back to those households. 00:58:46.540 --> 00:58:48.610 align:middle line:84% We have to come back to the welfare theorems 00:58:48.610 --> 00:58:52.130 align:middle line:84% to think about the winners and losers from trade. 00:58:52.130 --> 00:58:56.440 align:middle line:84% The point here only is that you can see why there is sometimes 00:58:56.440 --> 00:59:00.490 align:middle line:84% resistance to trade because trade does have consequences 00:59:00.490 --> 00:59:01.910 align:middle line:90% for factor prices. 00:59:01.910 --> 00:59:05.440 align:middle line:84% Hence, for the welfare of the providers of those factors, 00:59:05.440 --> 00:59:09.100 align:middle line:84% depending on whether they're capitalists that own wealth, 00:59:09.100 --> 00:59:12.160 align:middle line:84% capital that's used in machinery and so on, 00:59:12.160 --> 00:59:17.100 align:middle line:84% versus laborers that have only their labor endowment. 00:59:17.100 --> 00:59:20.400 align:middle line:84% OK, so next time we're going to go with this kind of theorem 00:59:20.400 --> 00:59:22.560 align:middle line:90% back to the Thai villages. 00:59:22.560 --> 00:59:25.470 align:middle line:84% And then we're going to go, in the subsequent lecture, 00:59:25.470 --> 00:59:28.900 align:middle line:84% to the US, and explore these ideas. 00:59:28.900 --> 00:59:30.900 align:middle line:90% OK, that's all I have for today. 00:59:30.900 --> 00:59:33.470 align:middle line:90% OK, thank you very much. 00:59:33.470 --> 00:59:41.000 align:middle line:90%