1 00:00:00,135 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,059 Commons license. 3 00:00:04,059 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high-quality educational resources for free. 5 00:00:10,720 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,290 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,290 --> 00:00:18,480 at ocw.mit.edu. 8 00:00:27,172 --> 00:00:29,130 ROBERT TOWNSEND: So we've had a mix of lectures 9 00:00:29,130 --> 00:00:35,520 in terms of focusing on specific topics and the details 10 00:00:35,520 --> 00:00:37,650 of models with notation. 11 00:00:37,650 --> 00:00:41,700 Sometimes that's coupled with an overview of the literature 12 00:00:41,700 --> 00:00:45,255 to put the paper in context. 13 00:00:45,255 --> 00:00:47,995 Sometimes-- and today will be an example-- 14 00:00:47,995 --> 00:00:49,620 we're just going to try to do one thing 15 00:00:49,620 --> 00:00:53,040 and do it reasonably well, although there 16 00:00:53,040 --> 00:00:54,390 is a lot of material today. 17 00:00:59,280 --> 00:01:02,250 There are other things that were listed on the reading list-- 18 00:01:02,250 --> 00:01:05,370 Cynthia Kinnan's job market paper, for example. 19 00:01:05,370 --> 00:01:09,360 As we go through this, I'll point out what she was doing 20 00:01:09,360 --> 00:01:12,240 and how that compares to what we're doing today. 21 00:01:12,240 --> 00:01:14,760 I deliberately decided not to go through her paper 22 00:01:14,760 --> 00:01:17,430 on the front end of this because it just chews up more time. 23 00:01:17,430 --> 00:01:21,090 And there's been that macro lecture, 24 00:01:21,090 --> 00:01:24,040 and maybe the labor lecture was-- 25 00:01:24,040 --> 00:01:26,080 filled a lot of material. 26 00:01:26,080 --> 00:01:30,120 And I'm sort of in the mood of doing a smaller 27 00:01:30,120 --> 00:01:33,490 thing well today rather than trying to cover too much. 28 00:01:33,490 --> 00:01:35,610 So there is a background, though, 29 00:01:35,610 --> 00:01:37,470 and that is constraints. 30 00:01:37,470 --> 00:01:41,670 We've been talking about constraints all semester. 31 00:01:41,670 --> 00:01:44,370 Consumption smoothing literature, 32 00:01:44,370 --> 00:01:46,170 including not just full insurance, 33 00:01:46,170 --> 00:01:50,280 but permanent income, buffer stock, those. 34 00:01:50,280 --> 00:01:55,080 We did the standard incomplete markets literature. 35 00:01:55,080 --> 00:01:56,790 There's private information stuff. 36 00:01:56,790 --> 00:02:00,360 We have mentioned it in class from time to time, 37 00:02:00,360 --> 00:02:02,640 have not done too much explicitly, 38 00:02:02,640 --> 00:02:08,340 although the Ben Moll paper and the others that lecture 39 00:02:08,340 --> 00:02:10,560 were macro models based on assumptions 40 00:02:10,560 --> 00:02:13,440 about information structures. 41 00:02:13,440 --> 00:02:16,770 Limited commitment we've actually done quite a lot-- 42 00:02:16,770 --> 00:02:20,910 not just how it affects consumption, but also 43 00:02:20,910 --> 00:02:23,320 in those macro models. 44 00:02:23,320 --> 00:02:25,870 So anyway, there's a consumption literature out there. 45 00:02:25,870 --> 00:02:29,320 There's a bunch of investment literature out there 46 00:02:29,320 --> 00:02:32,890 which includes adjustment cost, sensitivity of investment 47 00:02:32,890 --> 00:02:35,920 to cash flow, structural modeling, and just 48 00:02:35,920 --> 00:02:38,140 outright reduced-form empirical papers. 49 00:02:43,610 --> 00:02:48,620 And I've already mentioned some of this macro literature, 50 00:02:48,620 --> 00:02:51,150 like incomplete markets. 51 00:02:51,150 --> 00:02:55,460 Some of these papers we covered in class, not all of them. 52 00:02:55,460 --> 00:02:57,080 And finally we get to this, which 53 00:02:57,080 --> 00:03:02,000 is small but growing literature trying to test 54 00:03:02,000 --> 00:03:03,350 across different models. 55 00:03:03,350 --> 00:03:07,250 Most of the above pick one thing to do 56 00:03:07,250 --> 00:03:09,810 and try to do it really well. 57 00:03:09,810 --> 00:03:16,190 Some compare across maybe two models and some 58 00:03:16,190 --> 00:03:17,270 even more than two. 59 00:03:17,270 --> 00:03:20,550 So I could probably add two or three more References. 60 00:03:20,550 --> 00:03:23,360 There just isn't that much out there 61 00:03:23,360 --> 00:03:27,590 that systematically are agnostic about what the underlying 62 00:03:27,590 --> 00:03:32,660 constraint is and set out to discover the best 63 00:03:32,660 --> 00:03:37,070 fit against the data, and hence what the obstacles really are. 64 00:03:43,060 --> 00:03:45,730 So the point is we're not going to just look at investment, 65 00:03:45,730 --> 00:03:47,800 not just look at consumption. 66 00:03:47,800 --> 00:03:50,380 Largely we will look at both together. 67 00:03:50,380 --> 00:03:52,780 We're not going to look at just incomplete markets 68 00:03:52,780 --> 00:03:54,790 or endogenous information-constrained 69 00:03:54,790 --> 00:03:55,300 markets. 70 00:03:55,300 --> 00:04:02,380 We're going to test for them, both classes within and across, 71 00:04:02,380 --> 00:04:05,110 and talk about how to do it and what kind of data we need. 72 00:04:08,990 --> 00:04:12,780 The tools-- we're solving these dynamic models. 73 00:04:16,269 --> 00:04:19,360 We can allow any number of financial information regimes. 74 00:04:22,079 --> 00:04:24,240 We're going to use maximum likelihood 75 00:04:24,240 --> 00:04:31,370 to estimate the parameters, which allows us to be more 76 00:04:31,370 --> 00:04:32,910 general in a couple of ways. 77 00:04:32,910 --> 00:04:38,900 First of all, we can back out all the structural parameters, 78 00:04:38,900 --> 00:04:44,810 not just the parameters that are in a particular Euler equation. 79 00:04:44,810 --> 00:04:48,440 And likewise, we can have more than one equation, so to speak. 80 00:04:48,440 --> 00:04:51,350 We can look not only at consumption Euler equations, 81 00:04:51,350 --> 00:04:53,165 but investment rate of return equations. 82 00:04:53,165 --> 00:04:56,210 So again, I think in my mind, at least, 83 00:04:56,210 --> 00:04:58,610 these are familiar themes that we've 84 00:04:58,610 --> 00:05:00,920 been covering in bits and pieces in each 85 00:05:00,920 --> 00:05:03,810 of the various lectures. 86 00:05:03,810 --> 00:05:08,700 What was Cynthia doing in her job market paper? 87 00:05:08,700 --> 00:05:14,040 She was looking at various financial information regimes, 88 00:05:14,040 --> 00:05:16,770 but focusing on the Euler equations. 89 00:05:16,770 --> 00:05:19,800 And as you go from full risk sharing 90 00:05:19,800 --> 00:05:22,230 to limited information about output 91 00:05:22,230 --> 00:05:27,690 to moral hazard, constrained insurance, 92 00:05:27,690 --> 00:05:31,770 the form of the Euler equation varied from one 93 00:05:31,770 --> 00:05:35,340 to the other, including what variable should or should not 94 00:05:35,340 --> 00:05:37,560 show up as lags. 95 00:05:37,560 --> 00:05:41,460 And the basis of her task was to see 96 00:05:41,460 --> 00:05:45,420 whether lagged inverse marginal utility was 97 00:05:45,420 --> 00:05:46,740 a sufficient statistic. 98 00:05:50,110 --> 00:05:52,570 I'll say more about that as we go through. 99 00:05:56,060 --> 00:05:57,820 So there's a long list of regimes. 100 00:05:57,820 --> 00:05:59,590 I think part of the point of this 101 00:05:59,590 --> 00:06:03,910 is that the technology is available to test 102 00:06:03,910 --> 00:06:07,720 almost any regime subject to computational constraints. 103 00:06:07,720 --> 00:06:09,780 So under incomplete markets, we get out 104 00:06:09,780 --> 00:06:12,670 of autarky, which is the worst, savings 105 00:06:12,670 --> 00:06:17,470 only as in buffer stock, maybe borrowing up to a limit. 106 00:06:17,470 --> 00:06:21,670 We've talked about that in the natural borrowing 107 00:06:21,670 --> 00:06:25,000 limit and other limits, and then a single risk-free asset, 108 00:06:25,000 --> 00:06:27,700 which is like permanent income, unlimited borrowing 109 00:06:27,700 --> 00:06:29,530 and lending. 110 00:06:29,530 --> 00:06:33,950 And then we have the endogenously determined 111 00:06:33,950 --> 00:06:40,140 incomplete regimes, namely moral hazard, limited commitment, 112 00:06:40,140 --> 00:06:44,290 hidden output, unobserved investment, 113 00:06:44,290 --> 00:06:47,300 and the least constrained regime, the full information 114 00:06:47,300 --> 00:06:47,800 regime. 115 00:06:47,800 --> 00:06:51,250 So there's six or seven of them here. 116 00:06:51,250 --> 00:06:53,380 And we're going to go through tools 117 00:06:53,380 --> 00:06:57,520 that allow you to test one against the other pairwise 118 00:06:57,520 --> 00:07:05,310 for any pair depending on what data you want to use. 119 00:07:05,310 --> 00:07:11,010 So there is a mechanism design contract theory 120 00:07:11,010 --> 00:07:16,340 part which we've been putting off until today, largely. 121 00:07:16,340 --> 00:07:19,500 There is dynamic programming as in value functions. 122 00:07:19,500 --> 00:07:22,290 We have been seeing versions of that 123 00:07:22,290 --> 00:07:24,850 through various other lectures. 124 00:07:24,850 --> 00:07:26,320 There's linear programming. 125 00:07:26,320 --> 00:07:29,310 I'm going to say why momentarily, 126 00:07:29,310 --> 00:07:31,050 although we were already starting 127 00:07:31,050 --> 00:07:34,740 to do that when we did Rogerson's paper on labor 128 00:07:34,740 --> 00:07:36,360 supply. 129 00:07:36,360 --> 00:07:38,860 And although it was probably hard to figure it out, 130 00:07:38,860 --> 00:07:42,060 that's what Victor Zorn was doing in the TA session 131 00:07:42,060 --> 00:07:42,645 last Thursday. 132 00:07:46,310 --> 00:07:51,500 And maximum likelihood, which sounds familiar, anyway. 133 00:07:51,500 --> 00:07:57,170 So we compute, we estimate, and we test, basically. 134 00:07:57,170 --> 00:08:00,110 We can do it on actual data. 135 00:08:00,110 --> 00:08:04,160 I'm going to focus on the contrast between the urban data 136 00:08:04,160 --> 00:08:06,170 and the rural data. 137 00:08:06,170 --> 00:08:09,620 We can actually use it on simulated data. 138 00:08:09,620 --> 00:08:15,980 So I vote for this technique, which is generate 139 00:08:15,980 --> 00:08:17,660 the data from the model itself. 140 00:08:17,660 --> 00:08:20,690 Then you know for sure what's generating the data 141 00:08:20,690 --> 00:08:22,670 and see whether you get back what 142 00:08:22,670 --> 00:08:26,000 you put in in terms of the financial regime 143 00:08:26,000 --> 00:08:30,240 and the underlying parameters. 144 00:08:30,240 --> 00:08:37,320 And the results are reassuring subject to measurement error. 145 00:08:37,320 --> 00:08:40,590 One comment-- that you have to be patient to get to the-- 146 00:08:43,299 --> 00:08:45,730 get through the next 45 minutes or so-- 147 00:08:45,730 --> 00:08:48,970 the criticism of maximum likelihood 148 00:08:48,970 --> 00:08:50,590 is it's kind of black boxy. 149 00:08:50,590 --> 00:08:56,350 You don't really know, other than trying to fit histograms-- 150 00:08:56,350 --> 00:08:59,705 it's not like you're focusing on ROA 151 00:08:59,705 --> 00:09:03,070 and how it varies with wealth or transitions 152 00:09:03,070 --> 00:09:07,570 in the capital stock, or for that matter, the time series 153 00:09:07,570 --> 00:09:09,317 that the model is generating. 154 00:09:09,317 --> 00:09:10,900 But I'll come back at the end and show 155 00:09:10,900 --> 00:09:12,550 you the pictures of the actual data, 156 00:09:12,550 --> 00:09:14,870 and you'll get-- you've seen it, actually. 157 00:09:14,870 --> 00:09:20,110 But I'll remind you parts of it, and you'll see why-- 158 00:09:20,110 --> 00:09:22,480 what it is that makes it difficult for some 159 00:09:22,480 --> 00:09:27,490 of these financial regimes to fit the data and hence what 160 00:09:27,490 --> 00:09:31,140 the obstacles seem to be out there. 161 00:09:31,140 --> 00:09:35,180 We use the Thai data because we have both the consumption 162 00:09:35,180 --> 00:09:43,560 and the asset and income data, and using both is helpful. 163 00:09:43,560 --> 00:09:47,250 But that doesn't mean these techniques are limited 164 00:09:47,250 --> 00:09:48,960 to rather special databases. 165 00:09:48,960 --> 00:09:52,510 Mostly if you have surveys of firms, 166 00:09:52,510 --> 00:09:56,320 they would not ask about the consumption of the owner. 167 00:09:56,320 --> 00:09:59,440 Fortunately, the reverse is not so-- 168 00:09:59,440 --> 00:10:03,100 is less constraining, which is household-level surveys 169 00:10:03,100 --> 00:10:07,120 done by the World Bank, Living Standards Measurement 170 00:10:07,120 --> 00:10:13,670 and the FLS, Family Life Cycle in Mexico and Indonesia, 171 00:10:13,670 --> 00:10:14,170 and so on. 172 00:10:14,170 --> 00:10:17,650 They do typically ask the household a lot 173 00:10:17,650 --> 00:10:19,450 about their enterprises. 174 00:10:22,640 --> 00:10:29,950 But we've done this in Spain with just data on investment. 175 00:10:29,950 --> 00:10:36,020 So the techniques work even when you don't have the consumption 176 00:10:36,020 --> 00:10:36,710 data itself. 177 00:10:39,500 --> 00:10:44,930 And one of the main findings, reassuringly-- it's 178 00:10:44,930 --> 00:10:48,560 not like we always get the same thing back. 179 00:10:48,560 --> 00:10:52,580 For one thing, we don't get full risk sharing much. 180 00:10:52,580 --> 00:10:53,780 Sometimes we do. 181 00:10:53,780 --> 00:10:55,640 It will not surprise you when we get it 182 00:10:55,640 --> 00:10:59,870 given the other papers we've discovered in class. 183 00:10:59,870 --> 00:11:03,050 The big interesting thing is there 184 00:11:03,050 --> 00:11:06,860 is a difference between the rural data and the urban data. 185 00:11:06,860 --> 00:11:12,020 And the rural data, fairly limited financial regimes, 186 00:11:12,020 --> 00:11:14,630 like savings only or limited borrowing, 187 00:11:14,630 --> 00:11:25,310 they fit the best when you use the investment and income data. 188 00:11:25,310 --> 00:11:27,320 But in the urban areas, even when 189 00:11:27,320 --> 00:11:29,660 you use the investment and income data, 190 00:11:29,660 --> 00:11:32,855 you get something less constraining like moral hazard. 191 00:11:35,910 --> 00:11:39,720 So arguably, the information problem-- 192 00:11:39,720 --> 00:11:42,390 or if you believe in missing markets, 193 00:11:42,390 --> 00:11:45,150 they're more missing in the rural areas 194 00:11:45,150 --> 00:11:48,630 than they are in the urban areas. 195 00:11:48,630 --> 00:11:49,130 All right. 196 00:11:54,540 --> 00:11:59,220 So here's the model, utility over consumption and effort. 197 00:12:02,530 --> 00:12:03,940 Output is stochastic. 198 00:12:03,940 --> 00:12:06,070 Instead of writing output as a function 199 00:12:06,070 --> 00:12:09,310 of effort and capital plus a shock, 200 00:12:09,310 --> 00:12:12,640 this is a more general histogram. 201 00:12:12,640 --> 00:12:15,666 You've seen it at least once before, 202 00:12:15,666 --> 00:12:19,505 the probability of any given output given effort 203 00:12:19,505 --> 00:12:20,005 and capital. 204 00:12:24,440 --> 00:12:26,990 Households are either on their own 205 00:12:26,990 --> 00:12:31,850 or entering into contracts with a financial intermediary. 206 00:12:31,850 --> 00:12:36,415 It's partial equilibrium facing some exogenous outside return. 207 00:12:39,380 --> 00:12:46,700 And you can think about financial intermediaries 208 00:12:46,700 --> 00:12:53,000 as being competitive if you want. 209 00:12:53,000 --> 00:12:55,160 You could also think about it as a stand-in 210 00:12:55,160 --> 00:12:56,390 for the community as a whole. 211 00:12:59,930 --> 00:13:03,320 I'll show you where this stuff matters. 212 00:13:03,320 --> 00:13:05,070 And you're going to solve this contracting 213 00:13:05,070 --> 00:13:07,950 problem for many dates, even potentially 214 00:13:07,950 --> 00:13:12,100 over an infinite horizon. 215 00:13:12,100 --> 00:13:12,600 OK. 216 00:13:12,600 --> 00:13:15,080 And finally, how many people do we have? 217 00:13:15,080 --> 00:13:18,890 You could actually think about this as a risk-neutral 218 00:13:18,890 --> 00:13:21,940 household running a business-- 219 00:13:21,940 --> 00:13:24,700 sorry-- risk-averse household running a business 220 00:13:24,700 --> 00:13:28,180 facing a risk-neutral intermediary as if they're 221 00:13:28,180 --> 00:13:30,390 only two people. 222 00:13:30,390 --> 00:13:33,120 But a lot of what we do is easier 223 00:13:33,120 --> 00:13:36,902 to interpret when there is a continuum-- many, many, 224 00:13:36,902 --> 00:13:41,000 a continuum of household enterprises. 225 00:13:41,000 --> 00:13:44,960 Because then we can talk about the fractions of households 226 00:13:44,960 --> 00:13:47,480 who took effort, had certain capital, 227 00:13:47,480 --> 00:13:50,600 and experienced a certain output. 228 00:13:50,600 --> 00:13:52,400 That actually eliminates uncertainty 229 00:13:52,400 --> 00:13:54,680 from the point of view of the intermediary 230 00:13:54,680 --> 00:13:56,900 because all these things average out. 231 00:13:56,900 --> 00:13:57,822 Yes, Matt. 232 00:13:57,822 --> 00:14:00,830 AUDIENCE: Can they re-contract each period [? for the ?] 233 00:14:00,830 --> 00:14:02,630 [? consumer? ?] 234 00:14:02,630 --> 00:14:06,110 PROFESSOR: Here we're largely ruling it out. 235 00:14:06,110 --> 00:14:08,750 There is some limited commitment in the sense 236 00:14:08,750 --> 00:14:13,100 that they can walk away and go into autarky. 237 00:14:13,100 --> 00:14:17,090 That's about as close as we get in this paper. 238 00:14:17,090 --> 00:14:20,060 I kind of know something about how to extend it, 239 00:14:20,060 --> 00:14:22,762 but it's not going to be in the lecture today. 240 00:14:22,762 --> 00:14:24,220 AUDIENCE: So how do you get-- just, 241 00:14:24,220 --> 00:14:26,720 do you get time variations in those data, 242 00:14:26,720 --> 00:14:31,670 or is it just a cross-section? 243 00:14:31,670 --> 00:14:34,536 PROFESSOR: It's both a cross-section and panel. 244 00:14:34,536 --> 00:14:36,960 You use both aspects. 245 00:14:36,960 --> 00:14:40,170 The best way to think about having multiple intermediaries 246 00:14:40,170 --> 00:14:43,620 is there's ex-ante competition among them 247 00:14:43,620 --> 00:14:45,090 to service households, but there's 248 00:14:45,090 --> 00:14:47,430 something committing households once they sign up 249 00:14:47,430 --> 00:14:50,110 to a long-term agreement. 250 00:14:56,490 --> 00:15:02,080 So what's the initial state for a household saying? 251 00:15:02,080 --> 00:15:06,900 Well, for sure at least the initial capital stock 252 00:15:06,900 --> 00:15:10,020 when we visit them in the baseline survey. 253 00:15:10,020 --> 00:15:14,010 And then the second argument depends 254 00:15:14,010 --> 00:15:15,930 on the financial regime. 255 00:15:15,930 --> 00:15:17,640 If it's like borrowing and lending, 256 00:15:17,640 --> 00:15:21,630 then it's their current assets or their net indebtedness. 257 00:15:26,280 --> 00:15:29,820 Or if it's one of these contract regimes, 258 00:15:29,820 --> 00:15:34,380 it's essentially some utility constraint, some reservation 259 00:15:34,380 --> 00:15:39,210 utility as if it had been promised in the past. 260 00:15:39,210 --> 00:15:40,500 And I'll say more about that. 261 00:15:40,500 --> 00:15:43,770 That's the key to handling the dynamic incentive problem. 262 00:15:47,040 --> 00:15:50,530 Clearly, something like unobserved utility is not seen, 263 00:15:50,530 --> 00:15:53,490 so we're going to have to parameterize it 264 00:15:53,490 --> 00:15:57,990 with the mean and the variance and estimate this unobserved 265 00:15:57,990 --> 00:16:04,890 distribution of debt or promises in the population. 266 00:16:04,890 --> 00:16:08,640 Timing-- OK, so those are initial states. 267 00:16:08,640 --> 00:16:12,660 Then capital is used in production 268 00:16:12,660 --> 00:16:15,300 along with effort, which may or may not 269 00:16:15,300 --> 00:16:17,160 be observed depending on the regime. 270 00:16:17,160 --> 00:16:19,000 Output is realized. 271 00:16:19,000 --> 00:16:22,890 There is this pre-existing financial contract which 272 00:16:22,890 --> 00:16:27,090 determines the debt, say, that they can take on, 273 00:16:27,090 --> 00:16:33,000 or savings for tomorrow, or transfers as an insurance. 274 00:16:33,000 --> 00:16:36,720 And finally, after that financial stuff, 275 00:16:36,720 --> 00:16:42,110 they eat some and invest the rest. 276 00:16:42,110 --> 00:16:45,240 So it's kind of like a standard neoclassical setup. 277 00:16:45,240 --> 00:16:50,730 Capital today, you produce with labor, 278 00:16:50,730 --> 00:16:53,450 then you can either eat or invest. 279 00:16:53,450 --> 00:16:58,250 The fancy stuff is happening with the financial part, which 280 00:16:58,250 --> 00:17:01,820 is the difference between what you have available 281 00:17:01,820 --> 00:17:06,170 and what you either give up or get depending 282 00:17:06,170 --> 00:17:09,349 on the financial contract. 283 00:17:09,349 --> 00:17:10,579 Capital depreciates. 284 00:17:10,579 --> 00:17:11,599 That's pretty standard. 285 00:17:11,599 --> 00:17:13,970 And of course, then you go to the next prime-- 286 00:17:13,970 --> 00:17:17,510 next period with all these prime variables on them, 287 00:17:17,510 --> 00:17:20,630 so the transition from the state today 288 00:17:20,630 --> 00:17:26,500 to the state tomorrow with a financial contract in between. 289 00:17:26,500 --> 00:17:28,339 OK. 290 00:17:28,339 --> 00:17:38,100 In particular, we're going to talk 291 00:17:38,100 --> 00:17:42,100 about what goes on within the period in terms 292 00:17:42,100 --> 00:17:45,880 of the effort, which is induced or assigned, 293 00:17:45,880 --> 00:17:47,920 the output, which happens through Mother 294 00:17:47,920 --> 00:17:51,360 Nature, the decision for capital tomorrow-- 295 00:17:51,360 --> 00:17:55,630 and I'll go more into this promised utility for tomorrow. 296 00:17:55,630 --> 00:17:58,300 This pi thing is like the probability 297 00:17:58,300 --> 00:18:02,260 of this whole quadruple. 298 00:18:02,260 --> 00:18:05,650 And it looks daunting. 299 00:18:05,650 --> 00:18:09,940 Now, from a statistical point of view, it's like a histogram. 300 00:18:09,940 --> 00:18:14,425 If you had a finite number of values that q, 301 00:18:14,425 --> 00:18:17,350 z, k prime and w prime can take on, 302 00:18:17,350 --> 00:18:22,570 then you just have a bunch of points in dimensional space. 303 00:18:22,570 --> 00:18:29,050 And then you can talk about what mass, what height of a bar, 304 00:18:29,050 --> 00:18:33,410 of a histogram bar that those values take on. 305 00:18:33,410 --> 00:18:36,730 So statistically, it's an easy way to summarize the data. 306 00:18:36,730 --> 00:18:39,890 From the point of view of a financial contract, 307 00:18:39,890 --> 00:18:41,710 there's a lot of degeneracy. 308 00:18:41,710 --> 00:18:44,500 So it's not true, for example, typically 309 00:18:44,500 --> 00:18:47,560 that effort is random. 310 00:18:47,560 --> 00:18:51,670 For a given set of parameter values given incentives, 311 00:18:51,670 --> 00:18:54,792 there's one effort that's going to happen 312 00:18:54,792 --> 00:18:56,000 and none of the other happen. 313 00:18:56,000 --> 00:18:57,550 That's not inconsistent with this. 314 00:18:57,550 --> 00:19:00,760 It's just an extreme case where almost everything is zero 315 00:19:00,760 --> 00:19:04,600 except for one point of effort, and that's one. 316 00:19:04,600 --> 00:19:08,170 It's also true that there might be a functional relationship 317 00:19:08,170 --> 00:19:12,430 between consumption and output, maybe a nice, smooth function. 318 00:19:12,430 --> 00:19:14,560 Or if you want to grid it up, you've 319 00:19:14,560 --> 00:19:19,580 got a series of dots that lie on a line. 320 00:19:19,580 --> 00:19:24,490 So then we talk about the probability of Q and C 321 00:19:24,490 --> 00:19:27,310 as if know it was a shotgun. 322 00:19:27,310 --> 00:19:31,420 But in practice, all the mass is going to lie on that line. 323 00:19:31,420 --> 00:19:33,760 So you're used to thinking about, 324 00:19:33,760 --> 00:19:35,380 let c be a function of q. 325 00:19:35,380 --> 00:19:38,110 Let effort be assigned. 326 00:19:38,110 --> 00:19:40,950 We'll take care of how much capital is invested, et cetera. 327 00:19:44,970 --> 00:19:46,920 OK. 328 00:19:46,920 --> 00:19:55,080 It is true also that capital might be indivisible. 329 00:19:55,080 --> 00:19:56,280 We've talked about this. 330 00:19:56,280 --> 00:19:58,380 You get project ideas. 331 00:19:58,380 --> 00:20:00,636 You may or may not want to do it. 332 00:20:00,636 --> 00:20:03,562 The equipment is really chunky, building that warehouse 333 00:20:03,562 --> 00:20:04,270 for the chickens. 334 00:20:06,820 --> 00:20:09,910 So that's actually more realistic. 335 00:20:09,910 --> 00:20:12,730 And then the probability is kind of serious, which 336 00:20:12,730 --> 00:20:16,070 is, what is the probability you're going to do it, 337 00:20:16,070 --> 00:20:18,670 or what fraction of the population are going to do it? 338 00:20:18,670 --> 00:20:21,520 When we talk about fractions of the population, 339 00:20:21,520 --> 00:20:24,220 you should be reminded of Rogerson. 340 00:20:24,220 --> 00:20:27,040 There it was you work overtime or you work 341 00:20:27,040 --> 00:20:28,930 or you don't work at all, and the issue 342 00:20:28,930 --> 00:20:30,880 was what fraction of people are working. 343 00:20:30,880 --> 00:20:34,420 So that was the first experience in this class where 344 00:20:34,420 --> 00:20:36,950 we had a probability number. 345 00:20:36,950 --> 00:20:39,370 And when we have non-convexities, 346 00:20:39,370 --> 00:20:43,360 this is the generalization of it to [INAUDIBLE] probabilities 347 00:20:43,360 --> 00:20:45,490 on-- 348 00:20:45,490 --> 00:20:47,170 potentially on more than one thing. 349 00:20:52,280 --> 00:20:55,280 And there's a reason, and I'll show you when we write it down. 350 00:20:55,280 --> 00:20:59,470 It turns everything into a linear programming problem. 351 00:20:59,470 --> 00:21:01,220 I'm going to talk about particular utility 352 00:21:01,220 --> 00:21:03,380 functions, particular production functions. 353 00:21:03,380 --> 00:21:07,460 But really, we don't need to assume-- 354 00:21:07,460 --> 00:21:09,770 that's both the strength and limitation of this. 355 00:21:09,770 --> 00:21:12,410 I'm not going to show you a lot of closed-form analytic 356 00:21:12,410 --> 00:21:14,270 solutions. 357 00:21:14,270 --> 00:21:17,738 But on the other hand, we can solve anything numerically. 358 00:21:21,230 --> 00:21:24,590 And Cynthia, again, just to alert you, 359 00:21:24,590 --> 00:21:30,260 was backing out from Euler equations 360 00:21:30,260 --> 00:21:34,670 with first-order conditions, which Lagrange multipliers are 361 00:21:34,670 --> 00:21:36,330 binding and why and so on. 362 00:21:36,330 --> 00:21:40,070 So it's not like you can't-- 363 00:21:40,070 --> 00:21:42,410 you can, in principle, do both, actually. 364 00:21:42,410 --> 00:21:44,520 You don't necessarily have to-- 365 00:21:44,520 --> 00:21:45,020 OK. 366 00:21:45,020 --> 00:21:47,780 So let's just think about a standard problem 367 00:21:47,780 --> 00:21:48,800 without the lotteries. 368 00:21:48,800 --> 00:21:52,160 This is the autarky problem. 369 00:21:52,160 --> 00:22:00,050 Household enterprise has a realized output q, 370 00:22:00,050 --> 00:22:03,560 depreciated capital from last period. 371 00:22:03,560 --> 00:22:07,660 Actually, it's to qi it's already realized. 372 00:22:07,660 --> 00:22:10,610 In autarky, there's nothing left other than deciding on what 373 00:22:10,610 --> 00:22:11,870 to eat and what to invest. 374 00:22:11,870 --> 00:22:14,960 So this is the capital stock for tomorrow. 375 00:22:14,960 --> 00:22:15,590 Oh, sorry. 376 00:22:15,590 --> 00:22:18,840 And then there's this effort z. 377 00:22:18,840 --> 00:22:20,520 z is entering in this utility. 378 00:22:20,520 --> 00:22:24,080 It's also entering in this production function. 379 00:22:24,080 --> 00:22:25,450 You want expected utility? 380 00:22:25,450 --> 00:22:25,950 Fine. 381 00:22:25,950 --> 00:22:29,340 Just sum up over all possible outputs, 382 00:22:29,340 --> 00:22:32,550 taking into account Mother Nature's way 383 00:22:32,550 --> 00:22:36,560 of determining stochastically what those outputs are. 384 00:22:39,940 --> 00:22:43,720 And whatever you decide to take over to tomorrow 385 00:22:43,720 --> 00:22:45,320 enters in the value function tomorrow. 386 00:22:45,320 --> 00:22:45,820 OK? 387 00:22:45,820 --> 00:22:49,480 So we're looking for this infinite horizon solution. 388 00:22:49,480 --> 00:22:52,290 We're looking for a value function that's 389 00:22:52,290 --> 00:22:54,850 like a fixed point that solves the functional 390 00:22:54,850 --> 00:22:56,020 equation like this. 391 00:23:03,910 --> 00:23:07,010 Now, again, when qi is realized, you still 392 00:23:07,010 --> 00:23:08,580 have to decide on investment. 393 00:23:08,580 --> 00:23:13,580 That's why i is on both k prime and q. 394 00:23:13,580 --> 00:23:19,610 But z is determined before output is realized. 395 00:23:19,610 --> 00:23:20,930 So that's just one number. 396 00:23:20,930 --> 00:23:22,630 You with me? 397 00:23:22,630 --> 00:23:23,702 OK. 398 00:23:23,702 --> 00:23:28,050 It looks-- this is an equivalent problem. 399 00:23:28,050 --> 00:23:34,160 And this looks familiar, except the i's are missing. 400 00:23:34,160 --> 00:23:38,180 So it's output plus depreciated capital less investment 401 00:23:38,180 --> 00:23:40,910 for tomorrow, deciding on [INAUDIBLE],, 402 00:23:40,910 --> 00:23:46,370 but now we've taken this deterministic-looking problem 403 00:23:46,370 --> 00:23:51,600 and just replaced it with this probability object. 404 00:23:51,600 --> 00:23:54,720 But it allows all these special cases. 405 00:23:54,720 --> 00:23:56,640 Anything that can solve this solves 406 00:23:56,640 --> 00:24:04,880 this subject to grids, subject to approximating 407 00:24:04,880 --> 00:24:09,300 with a finite number of outputs, et cetera. 408 00:24:09,300 --> 00:24:11,510 All right. 409 00:24:11,510 --> 00:24:17,920 Now, it looks here as if the choice object 410 00:24:17,920 --> 00:24:24,630 is this probability number including the probability 411 00:24:24,630 --> 00:24:27,680 of output, but how can that be? 412 00:24:27,680 --> 00:24:31,300 Because Mother Nature plays a role. 413 00:24:31,300 --> 00:24:34,620 So we kind of have to constrain these histograms to respect 414 00:24:34,620 --> 00:24:41,620 the relationship between kz and q, and namely, this object. 415 00:24:41,620 --> 00:24:44,350 We don't want to lose this object. 416 00:24:44,350 --> 00:24:48,280 Well, it's a fancy way of saying the probability of event 417 00:24:48,280 --> 00:24:52,360 A conditioned on event b times the probability event b 418 00:24:52,360 --> 00:24:56,910 is the same thing as a joint event a, comma, b. 419 00:24:56,910 --> 00:25:00,570 So this was the probability of q bar and z bar 420 00:25:00,570 --> 00:25:02,740 because I summed over everything else. 421 00:25:02,740 --> 00:25:06,010 This is a probability of c bar given z bar, 422 00:25:06,010 --> 00:25:08,650 and then this is, summing over everything else, 423 00:25:08,650 --> 00:25:10,990 the probability of z bar. 424 00:25:10,990 --> 00:25:12,160 So it's a cute trick. 425 00:25:15,830 --> 00:25:19,400 So we're not going to cheat on Mother Nature. 426 00:25:19,400 --> 00:25:21,500 We're constrained to follow Mother Nature. 427 00:25:24,800 --> 00:25:31,670 Another thing to-- say what's the dimensionality? 428 00:25:31,670 --> 00:25:36,800 Well, the state variable is k, so the question is, 429 00:25:36,800 --> 00:25:38,290 how many little k's are there? 430 00:25:38,290 --> 00:25:40,830 If we grid up k in the small, medium, and large, 431 00:25:40,830 --> 00:25:43,100 there's three of them, for example. 432 00:25:43,100 --> 00:25:45,470 It could be 10. 433 00:25:45,470 --> 00:25:49,380 That's our choice in terms of the grid. 434 00:25:49,380 --> 00:25:52,490 But whatever it is, we've got to solve this functional equation 435 00:25:52,490 --> 00:25:57,390 v and find the fixed point by iterating. 436 00:25:57,390 --> 00:26:00,090 OK? 437 00:26:00,090 --> 00:26:03,900 And the larger is the dimension of this, 438 00:26:03,900 --> 00:26:06,790 the harder it is to do that. 439 00:26:06,790 --> 00:26:07,290 Jan. 440 00:26:07,290 --> 00:26:13,820 AUDIENCE: [INAUDIBLE] formulations [INAUDIBLE]?? 441 00:26:13,820 --> 00:26:17,040 In the second case, as in the agents 442 00:26:17,040 --> 00:26:20,130 can do some randomization [INAUDIBLE].. 443 00:26:20,130 --> 00:26:21,735 PROFESSOR: Yeah, I should have-- 444 00:26:21,735 --> 00:26:23,860 AUDIENCE: [INAUDIBLE] is the first case [INAUDIBLE] 445 00:26:23,860 --> 00:26:24,527 PROFESSOR: Yeah. 446 00:26:27,180 --> 00:26:29,590 If we don't get the grid right-- for example, 447 00:26:29,590 --> 00:26:34,200 if we have a continuum of values of z, and so on, 448 00:26:34,200 --> 00:26:37,470 then we have to make sure that the solution here 449 00:26:37,470 --> 00:26:39,690 is on a grid point down here. 450 00:26:39,690 --> 00:26:42,300 Or to put it the other way around, 451 00:26:42,300 --> 00:26:54,640 if the grid is serious then the household 452 00:26:54,640 --> 00:26:58,270 may want to optimize in terms of choosing a probability. 453 00:26:58,270 --> 00:27:01,210 And in that case, this is not just going to replicate. 454 00:27:01,210 --> 00:27:02,320 This allows more. 455 00:27:07,160 --> 00:27:10,620 So if it wants to, we can find a solution, 456 00:27:10,620 --> 00:27:14,570 but it may choose to find something else. 457 00:27:14,570 --> 00:27:16,970 There are grid lotteries, but it's too easy 458 00:27:16,970 --> 00:27:19,250 to detect them in the output. 459 00:27:19,250 --> 00:27:22,220 You've got medium effort and high effort, 460 00:27:22,220 --> 00:27:25,770 and then the thing is putting probability on both of those. 461 00:27:25,770 --> 00:27:27,950 They're like adjacent points. 462 00:27:27,950 --> 00:27:29,540 So it's clear where the program's 463 00:27:29,540 --> 00:27:31,820 trying to get something in the middle that's just not 464 00:27:31,820 --> 00:27:33,850 available. 465 00:27:33,850 --> 00:27:39,130 And if the grid is serious, then, as in Rogerson, 466 00:27:39,130 --> 00:27:40,630 that's fine. 467 00:27:40,630 --> 00:27:42,880 If we think the grid ought to be not there, 468 00:27:42,880 --> 00:27:44,710 and it's just meant to be an approximation, 469 00:27:44,710 --> 00:27:46,960 then you probably want to slice and dice it a bit more 470 00:27:46,960 --> 00:27:50,515 and search more intensively between those adjacent points. 471 00:27:53,180 --> 00:27:56,260 AUDIENCE: For the second problem, 472 00:27:56,260 --> 00:28:02,818 is it possible that the solution [INAUDIBLE] random contract? 473 00:28:02,818 --> 00:28:04,110 PROFESSOR: It is possible, yes. 474 00:28:04,110 --> 00:28:07,800 AUDIENCE: But in reality, why the agent can randomly 475 00:28:07,800 --> 00:28:10,680 choose a [INAUDIBLE] So [INAUDIBLE] 476 00:28:10,680 --> 00:28:12,390 the first formulation, a agent can not 477 00:28:12,390 --> 00:28:13,675 choose a random contract. 478 00:28:13,675 --> 00:28:16,003 And in the second one, you can choose. 479 00:28:16,003 --> 00:28:16,920 PROFESSOR: Yeah, yeah. 480 00:28:16,920 --> 00:28:19,920 I agree with your point. 481 00:28:19,920 --> 00:28:23,320 You think it's infeasible to randomize? 482 00:28:23,320 --> 00:28:25,480 AUDIENCE: So I think if your assumption 483 00:28:25,480 --> 00:28:28,890 is that the [INAUDIBLE] agent can do randomization, 484 00:28:28,890 --> 00:28:31,740 then I agree that these [INAUDIBLE] formulations 485 00:28:31,740 --> 00:28:33,220 are equivalent. 486 00:28:33,220 --> 00:28:35,720 But if you don't assume that-- 487 00:28:35,720 --> 00:28:40,420 PROFESSOR: I am assuming they can do the randomization. 488 00:28:40,420 --> 00:28:42,960 And again, as in Rogerson, it's a way 489 00:28:42,960 --> 00:28:46,145 to smooth out the non-convexities. 490 00:28:46,145 --> 00:28:47,520 That's the main-- you know, we've 491 00:28:47,520 --> 00:28:51,600 turned non-convex problems into a linear program. 492 00:28:51,600 --> 00:28:54,960 Oh, why is it a linear programming problem? 493 00:28:54,960 --> 00:28:58,440 Well, these pis are the choice variables. 494 00:28:58,440 --> 00:29:01,480 It's tempting to say, oh, no, it's supposed to be k and z. 495 00:29:01,480 --> 00:29:02,190 No, no, no. 496 00:29:02,190 --> 00:29:04,650 It's the probability of [? qt ?] So this is just 497 00:29:04,650 --> 00:29:09,570 a number, this utility, if you were to do a certain quadruple. 498 00:29:09,570 --> 00:29:12,950 And this is the probability of that quadruple. 499 00:29:12,950 --> 00:29:18,000 So these are the fundamental policy variables. 500 00:29:18,000 --> 00:29:20,220 And what is the dimensionality of them? 501 00:29:20,220 --> 00:29:23,880 Number q cross number z cross number k prime. 502 00:29:23,880 --> 00:29:26,535 So this can get pretty big pretty fast. 503 00:29:29,760 --> 00:29:31,710 And how many constraints there are? 504 00:29:31,710 --> 00:29:35,600 Well, in this case, just the one, although probabilities do 505 00:29:35,600 --> 00:29:36,600 have to add up. 506 00:29:36,600 --> 00:29:39,440 But essentially, one constraint. 507 00:29:39,440 --> 00:29:40,520 Well, not quite. 508 00:29:40,520 --> 00:29:43,970 It's for every q bar and z bar. 509 00:29:43,970 --> 00:29:48,030 So actually, there's number q cross number z constraints 510 00:29:48,030 --> 00:29:48,530 here. 511 00:29:51,220 --> 00:29:52,470 I'm not going to belabor this. 512 00:29:52,470 --> 00:29:54,220 But when you go through the other regimes, 513 00:29:54,220 --> 00:29:57,410 you're going to keep an eye on how many constraints there 514 00:29:57,410 --> 00:29:58,340 are and so on. 515 00:29:58,340 --> 00:30:01,190 And eventually, we'll get to regimes 516 00:30:01,190 --> 00:30:05,180 that are actually nice and challenging to compute. 517 00:30:05,180 --> 00:30:05,930 Here's the-- 518 00:30:12,135 --> 00:30:14,260 AUDIENCE: I still don't understand exactly what you 519 00:30:14,260 --> 00:30:18,220 mean when you say that the pis are the choice variables 520 00:30:18,220 --> 00:30:22,440 and the q and the k's and the z's are just numbers. 521 00:30:22,440 --> 00:30:25,120 I mean, they still ought to be consistent. 522 00:30:25,120 --> 00:30:29,250 So when you choose a pi, then you're doing-- 523 00:30:29,250 --> 00:30:32,070 PROFESSOR: Suppose you had a continuous consumption 524 00:30:32,070 --> 00:30:36,000 schedule, so c is a function of q. 525 00:30:36,000 --> 00:30:39,630 And then the paradigm is either you work hard 526 00:30:39,630 --> 00:30:42,960 or you work very little, so there's just two. 527 00:30:42,960 --> 00:30:45,850 So if you choose not to work hard, 528 00:30:45,850 --> 00:30:47,610 that's going to be reflected in output. 529 00:30:47,610 --> 00:30:50,940 But you're still facing that consumption schedule. 530 00:30:50,940 --> 00:30:54,030 Or you could work hard or potentially randomize 531 00:30:54,030 --> 00:30:57,640 across working hard or not. 532 00:30:57,640 --> 00:30:59,610 So the randomization could be in the effort. 533 00:31:02,900 --> 00:31:04,700 AUDIENCE: So the pis are functions? 534 00:31:04,700 --> 00:31:06,454 I'm choosing a function. 535 00:31:06,454 --> 00:31:07,310 PROFESSOR: Hmm? 536 00:31:07,310 --> 00:31:10,730 AUDIENCE: I'm choosing a function. 537 00:31:10,730 --> 00:31:16,060 PROFESSOR: Actually, here, it's so general 538 00:31:16,060 --> 00:31:17,560 that you don't see any functions. 539 00:31:17,560 --> 00:31:21,280 You're just choosing mass points over a finite number 540 00:31:21,280 --> 00:31:24,850 of q's, z's, and k primes without constraints. 541 00:31:24,850 --> 00:31:27,810 AUDIENCE: I am, quote unquote, [INAUDIBLE] 542 00:31:27,810 --> 00:31:28,642 PROFESSOR: Yeah. 543 00:31:28,642 --> 00:31:30,850 AUDIENCE: Isn't it more like choosing a distribution, 544 00:31:30,850 --> 00:31:32,325 though, instead of saying-- 545 00:31:32,325 --> 00:31:33,700 PROFESSOR: If these were-- yeah-- 546 00:31:33,700 --> 00:31:35,320 if these were a continuum, then this 547 00:31:35,320 --> 00:31:38,310 would be like a density, a multi-dimensional density. 548 00:31:38,310 --> 00:31:40,130 But I don't know how to compute those. 549 00:31:40,130 --> 00:31:42,130 AUDIENCE: It's not-- so correct me if I'm wrong, 550 00:31:42,130 --> 00:31:44,150 but it's not that I can put anything next to it. 551 00:31:44,150 --> 00:31:47,230 I have to plug in to the u next to it 552 00:31:47,230 --> 00:31:48,930 the corresponding stuff in there. 553 00:31:48,930 --> 00:31:52,690 But then I just compute every single thing inside there 554 00:31:52,690 --> 00:31:55,060 and choose the best one after I compute everything. 555 00:31:55,060 --> 00:31:56,060 PROFESSOR: That's right. 556 00:31:56,060 --> 00:31:59,200 So we specify the grids. 557 00:31:59,200 --> 00:32:04,800 So we know the set of feasible choices for all 558 00:32:04,800 --> 00:32:07,530 the quadruples qzk prime. 559 00:32:07,530 --> 00:32:11,880 For any particular one, we know what this real number is. 560 00:32:11,880 --> 00:32:18,080 You use Matlab code to generate these weights. 561 00:32:18,080 --> 00:32:21,260 In this case, it's a weight on it on the objective function, 562 00:32:21,260 --> 00:32:25,580 or you use codes to generate this guy, 563 00:32:25,580 --> 00:32:27,620 and then there's weights on the constraint set. 564 00:32:27,620 --> 00:32:28,542 So 565 00:32:28,542 --> 00:32:30,000 AUDIENCE: So now I understand that. 566 00:32:30,000 --> 00:32:33,030 But then, in some sense, I don't understand 567 00:32:33,030 --> 00:32:38,610 what the difference from the first case is-- 568 00:32:38,610 --> 00:32:40,450 like, for on the first problem. 569 00:32:44,890 --> 00:32:47,270 That the probability of q given k 570 00:32:47,270 --> 00:32:50,170 and z could, in principle, be non-linear. 571 00:32:50,170 --> 00:32:52,840 And so we're just-- 572 00:32:52,840 --> 00:32:54,970 PROFESSOR: It's the extra gain from randomization. 573 00:32:54,970 --> 00:33:14,280 I mean, suppose-- take the point of view 574 00:33:14,280 --> 00:33:16,740 that there are a continuum of-- well, 575 00:33:16,740 --> 00:33:18,690 so far we don't have the social planner. 576 00:33:18,690 --> 00:33:21,570 I mean, this is just an individual optimization. 577 00:33:21,570 --> 00:33:24,780 But I will try to answer your question. 578 00:33:24,780 --> 00:33:28,860 When we have a village-wide resource constraint, 579 00:33:28,860 --> 00:33:31,620 we've got to decide how many resources to use up 580 00:33:31,620 --> 00:33:35,490 in investment and how much to leave over for consumption. 581 00:33:35,490 --> 00:33:37,410 And it may be that you don't want everyone 582 00:33:37,410 --> 00:33:39,090 doing a large, chunky project. 583 00:33:39,090 --> 00:33:40,920 You want to just take some people to do it 584 00:33:40,920 --> 00:33:42,810 and some people not. 585 00:33:42,810 --> 00:33:46,320 And then you know whatever is produced as output, 586 00:33:46,320 --> 00:33:49,350 use that some of that for consumption and some 587 00:33:49,350 --> 00:33:51,060 to save for tomorrow. 588 00:33:51,060 --> 00:33:55,620 So from a social perspective, you rarely 589 00:33:55,620 --> 00:33:57,330 want everyone doing the same thing. 590 00:33:57,330 --> 00:33:59,610 You want to choose the fractions or people doing 591 00:33:59,610 --> 00:34:01,535 one thing or the other. 592 00:34:01,535 --> 00:34:02,160 Does that help? 593 00:34:08,300 --> 00:34:08,800 All right. 594 00:34:08,800 --> 00:34:12,560 So this is with borrowing and lending, 595 00:34:12,560 --> 00:34:16,100 so it's essentially almost the same, 596 00:34:16,100 --> 00:34:19,010 except you can add to your current resources 597 00:34:19,010 --> 00:34:22,219 by borrowing, except you've got to pay back 598 00:34:22,219 --> 00:34:24,960 your loan from yesterday. 599 00:34:24,960 --> 00:34:28,429 So now you've got two ways to intertemporally reallocate 600 00:34:28,429 --> 00:34:33,130 consumption, one through the capital stock, 601 00:34:33,130 --> 00:34:36,710 the equipment, and the other through financial borrowing 602 00:34:36,710 --> 00:34:38,810 and lending. 603 00:34:38,810 --> 00:34:41,000 And it's already written in lotteries. 604 00:34:41,000 --> 00:34:44,389 By the way, it enhances the state vector 605 00:34:44,389 --> 00:34:49,030 from the capital stock to the current debt as well. 606 00:34:58,730 --> 00:35:02,750 Now we can knock off savings only. 607 00:35:02,750 --> 00:35:06,830 That's where a basically, if B means borrowing, 608 00:35:06,830 --> 00:35:14,570 then basically, it can't be positive 609 00:35:14,570 --> 00:35:17,210 if you're not going to allow it. 610 00:35:17,210 --> 00:35:19,520 Or you could allow some small amount 611 00:35:19,520 --> 00:35:24,480 and have a positive but low amount 612 00:35:24,480 --> 00:35:26,950 that you could borrow as b max. 613 00:35:26,950 --> 00:35:30,240 So we can do savings only, et cetera. 614 00:35:30,240 --> 00:35:34,280 If you don't want to limit borrowing at all, that's fine. 615 00:35:34,280 --> 00:35:37,930 Then you just let it take on any value subject 616 00:35:37,930 --> 00:35:39,050 to [? grade ?] issues. 617 00:35:39,050 --> 00:35:39,590 Yep? 618 00:35:39,590 --> 00:35:41,715 AUDIENCE: When you do this kind of stuff in Matlab, 619 00:35:41,715 --> 00:35:44,200 to do these different scenarios, is that just [INAUDIBLE] 620 00:35:44,200 --> 00:35:46,650 one line of code? 621 00:35:46,650 --> 00:35:47,570 PROFESSOR: Which? 622 00:35:47,570 --> 00:35:50,240 AUDIENCE: Like, to have [INAUDIBLE].. 623 00:35:53,003 --> 00:35:53,670 PROFESSOR: Yeah. 624 00:35:53,670 --> 00:35:56,190 Well, actually, you can almost generate it from the grid, 625 00:35:56,190 --> 00:36:01,440 because you might have a large set of possible values 626 00:36:01,440 --> 00:36:03,240 for borrowing. 627 00:36:03,240 --> 00:36:04,960 That would be the unrestricted problem. 628 00:36:04,960 --> 00:36:07,090 If you want savings only, then you just 629 00:36:07,090 --> 00:36:10,360 cut off all the positive borrowings or anything 630 00:36:10,360 --> 00:36:11,320 in between. 631 00:36:11,320 --> 00:36:14,530 So this one you can handle in terms of generating 632 00:36:14,530 --> 00:36:15,460 the underlying grid. 633 00:36:20,690 --> 00:36:24,500 Now we get to these mechanism design models 634 00:36:24,500 --> 00:36:26,030 including full information. 635 00:36:29,080 --> 00:36:35,240 Now we're going to have to go back to the households 636 00:36:35,240 --> 00:36:39,140 as a group dealing with this financial intermediary. 637 00:36:42,840 --> 00:36:44,070 The main thing is this-- 638 00:36:49,545 --> 00:36:51,670 let's look at the household because we were looking 639 00:36:51,670 --> 00:36:54,070 at households a minute ago. 640 00:36:54,070 --> 00:36:59,560 Now it's as if they surrendered all their output to the bank-- 641 00:36:59,560 --> 00:37:01,750 bear with me-- but they get some of it 642 00:37:01,750 --> 00:37:05,150 back in terms of transfers. 643 00:37:05,150 --> 00:37:08,030 Actually, a more natural way to write it 644 00:37:08,030 --> 00:37:11,810 would have been give them q and let them pay back 645 00:37:11,810 --> 00:37:14,840 loans, state-contingent loans. 646 00:37:14,840 --> 00:37:17,370 It's equivalent. 647 00:37:17,370 --> 00:37:21,490 The transfer can be positive or negative. 648 00:37:21,490 --> 00:37:24,080 And it's kind of the basis of consumption, 649 00:37:24,080 --> 00:37:27,410 but it's adjusted, as usual, by investment. 650 00:37:27,410 --> 00:37:31,350 So then the contract has to solve for these transfers, 651 00:37:31,350 --> 00:37:33,270 and the transfers are a function of q. 652 00:37:35,780 --> 00:37:39,350 Now, as I said, the way this is written, 653 00:37:39,350 --> 00:37:43,370 the q ends up with the bank, but the bank 654 00:37:43,370 --> 00:37:44,530 is paying the transfer. 655 00:37:44,530 --> 00:37:50,410 So this is basically a surplus generated from households. 656 00:37:50,410 --> 00:37:54,670 What fraction of households have output q greater than tau, 657 00:37:54,670 --> 00:37:57,010 so the bank is getting money from them? 658 00:37:57,010 --> 00:37:58,377 Well, that's this pi. 659 00:37:58,377 --> 00:37:59,210 That's the fraction. 660 00:38:02,780 --> 00:38:04,580 Well, you know, it's infinite horizon. 661 00:38:04,580 --> 00:38:07,800 So this is the surplus generated today-- 662 00:38:07,800 --> 00:38:11,580 or if negative, the loss-- 663 00:38:11,580 --> 00:38:14,280 summed up over all surpluses and losses 664 00:38:14,280 --> 00:38:17,460 depending on who's at what states 665 00:38:17,460 --> 00:38:19,110 and what the contract assigns. 666 00:38:19,110 --> 00:38:21,820 And then you have tomorrow's profits. 667 00:38:21,820 --> 00:38:24,830 This is a small, open economy. 668 00:38:24,830 --> 00:38:28,920 There's an interest r, 1 plus a little r. 669 00:38:28,920 --> 00:38:32,700 So this is just the present value of tomorrow. 670 00:38:32,700 --> 00:38:36,030 So this is surplus today plus profits today 671 00:38:36,030 --> 00:38:37,860 plus profits tomorrow. 672 00:38:37,860 --> 00:38:42,795 So it's as if the bank is trying to maximize overall profits. 673 00:38:47,180 --> 00:38:48,310 Now, what's the constraint? 674 00:38:48,310 --> 00:38:50,680 Why not just screw the households? 675 00:38:50,680 --> 00:38:54,445 Well, the answer is-- 676 00:38:54,445 --> 00:38:55,820 it's not quite right, but you can 677 00:38:55,820 --> 00:38:59,370 think about this as a reservation utility. 678 00:38:59,370 --> 00:39:00,860 So you can't take too much away. 679 00:39:00,860 --> 00:39:05,300 Otherwise, the households will cry foul and walk away. 680 00:39:05,300 --> 00:39:06,260 OK? 681 00:39:06,260 --> 00:39:09,760 Actually, technically, this promise 682 00:39:09,760 --> 00:39:13,940 was predetermined from the previous period. 683 00:39:13,940 --> 00:39:17,660 Or equivalently, part of the control variable 684 00:39:17,660 --> 00:39:21,970 is the promise from tomorrow on, w prime. 685 00:39:21,970 --> 00:39:26,710 It's easy and yet amazingly powerful. 686 00:39:26,710 --> 00:39:28,870 So just think about incentives. 687 00:39:28,870 --> 00:39:30,970 You have long-term contracts. 688 00:39:30,970 --> 00:39:32,150 Should I work today? 689 00:39:32,150 --> 00:39:34,450 I can get rewarded or penalized depending 690 00:39:34,450 --> 00:39:36,310 on what my output is today. 691 00:39:36,310 --> 00:39:38,200 But it's not just a static contract. 692 00:39:38,200 --> 00:39:39,850 There's tomorrow too. 693 00:39:39,850 --> 00:39:42,340 So maybe my history of observed outputs 694 00:39:42,340 --> 00:39:45,520 will be used tomorrow in terms of the insurance and credit 695 00:39:45,520 --> 00:39:48,160 contract I'm going to get, which in turn could 696 00:39:48,160 --> 00:39:49,980 be used in the third period. 697 00:39:49,980 --> 00:39:51,580 And we're like, oh, my god. 698 00:39:51,580 --> 00:39:54,820 That's a really big object. 699 00:39:54,820 --> 00:39:55,870 But no. 700 00:39:55,870 --> 00:39:57,400 What do the households care about? 701 00:39:57,400 --> 00:40:00,670 They only care about their expected utility. 702 00:40:00,670 --> 00:40:03,880 If I'm looking forward to tomorrow, 703 00:40:03,880 --> 00:40:05,740 I don't have to solve tomorrow's problem 704 00:40:05,740 --> 00:40:09,960 as long as I know what the utility consequences will be. 705 00:40:09,960 --> 00:40:11,790 So this is kind of a reduced form 706 00:40:11,790 --> 00:40:16,560 way of handling the multi-period incentive problem. 707 00:40:16,560 --> 00:40:19,740 What I'm saying is the household's expected utility 708 00:40:19,740 --> 00:40:26,050 is the utility outcome from today plus the expected utility 709 00:40:26,050 --> 00:40:26,550 tomorrow. 710 00:40:29,500 --> 00:40:32,410 It's not only-- it's the probability 711 00:40:32,410 --> 00:40:35,570 of being assigned w prime tomorrow jointly with output 712 00:40:35,570 --> 00:40:36,070 today. 713 00:40:36,070 --> 00:40:39,370 So now, not only is consumption moving up and down 714 00:40:39,370 --> 00:40:41,710 with output today-- 715 00:40:41,710 --> 00:40:44,470 and it will move when there's limited insurance-- 716 00:40:44,470 --> 00:40:49,720 but also, tomorrow's utility will vary up and down. 717 00:40:49,720 --> 00:40:53,620 And because you have concave utility and this force 718 00:40:53,620 --> 00:40:56,500 for inter-temporal smoothing, you won't just do one, 719 00:40:56,500 --> 00:40:58,300 and you won't just do the other. 720 00:40:58,300 --> 00:41:02,090 Actually, you'll load as much into the future as possible. 721 00:41:02,090 --> 00:41:03,600 Front loading is a bad thing. 722 00:41:03,600 --> 00:41:06,650 The longer the horizon, the more powerful 723 00:41:06,650 --> 00:41:08,855 the incentives, because you've got the whole future. 724 00:41:11,500 --> 00:41:15,520 So there's a lot of economics behind this sort 725 00:41:15,520 --> 00:41:18,970 of innocuous-looking two-period problem 726 00:41:18,970 --> 00:41:21,205 where w prime captures tomorrow. 727 00:41:23,890 --> 00:41:24,390 All right. 728 00:41:27,700 --> 00:41:31,900 That's actually the full insurance problem, 729 00:41:31,900 --> 00:41:34,020 and then we can add moral hazard. 730 00:41:34,020 --> 00:41:42,240 So basically, what this says is if z bar is assigned today 731 00:41:42,240 --> 00:41:46,200 in the contract and the guy actually does it, 732 00:41:46,200 --> 00:41:51,000 he has to be wanting to do it relative to-- 733 00:41:51,000 --> 00:41:54,885 and the bank wouldn't know even though z 734 00:41:54,885 --> 00:41:59,290 bar is being recommended, the guy is doing z hat, shirking. 735 00:42:02,660 --> 00:42:06,110 So this is a standard moral hazard constraint. 736 00:42:06,110 --> 00:42:09,350 Of course, the inequality makes them 737 00:42:09,350 --> 00:42:12,840 want to do the recommendation. 738 00:42:12,840 --> 00:42:18,700 But the program has to evaluate any possible deviation 739 00:42:18,700 --> 00:42:21,690 a shirking agent might do and make 740 00:42:21,690 --> 00:42:23,850 sure the utility consequences for the agent 741 00:42:23,850 --> 00:42:29,610 are worse than following the recommended plan. 742 00:42:29,610 --> 00:42:31,290 Now, again, because we've embedded 743 00:42:31,290 --> 00:42:34,440 Mother Nature in this pi thing, we 744 00:42:34,440 --> 00:42:36,390 have to re-normalize the probability. 745 00:42:36,390 --> 00:42:39,400 I actually showed you this once. 746 00:42:39,400 --> 00:42:42,990 But I'm sure with discounting, you can't remember backwards. 747 00:42:42,990 --> 00:42:45,960 More on that next Tuesday. 748 00:42:45,960 --> 00:42:49,500 But we just adjust the likelihood 749 00:42:49,500 --> 00:42:54,950 to reflect the fact that now z hat is taken rather than z bar. 750 00:42:54,950 --> 00:42:58,460 And it's not maybe obvious where this is coming from. 751 00:42:58,460 --> 00:43:00,890 It is nevertheless true. 752 00:43:00,890 --> 00:43:03,560 You just have to write out all the conditional probabilities 753 00:43:03,560 --> 00:43:06,207 and start summing up. 754 00:43:06,207 --> 00:43:07,790 I can give you a reference if you want 755 00:43:07,790 --> 00:43:09,870 to see where it's written out. 756 00:43:09,870 --> 00:43:11,190 Questions? 757 00:43:11,190 --> 00:43:12,000 Yes. 758 00:43:12,000 --> 00:43:13,610 AUDIENCE: I just want to make sure 759 00:43:13,610 --> 00:43:16,140 that I understand this formulation. 760 00:43:16,140 --> 00:43:18,480 So I think the [? language ?] of this formulation 761 00:43:18,480 --> 00:43:22,140 is that if you don't have this formulation, first of all, 762 00:43:22,140 --> 00:43:26,870 you need to solve the incentive-compatible contracts 763 00:43:26,870 --> 00:43:28,120 one by one. 764 00:43:28,120 --> 00:43:31,320 And in this case, the randomizing contract 765 00:43:31,320 --> 00:43:33,330 is like randomize your wealth. 766 00:43:33,330 --> 00:43:38,180 So you may need to solve one problem. 767 00:43:38,180 --> 00:43:39,535 So I [INAUDIBLE] you randomize-- 768 00:43:39,535 --> 00:43:40,910 PROFESSOR: Well, actually, you're 769 00:43:40,910 --> 00:43:43,250 suggesting a simplification that we're not 770 00:43:43,250 --> 00:43:46,100 using, which is any random-- 771 00:43:46,100 --> 00:43:48,710 oh, you mean the w, the promise? 772 00:43:48,710 --> 00:43:50,060 AUDIENCE: Yeah. 773 00:43:50,060 --> 00:43:52,370 I think the advantage of the second formulation 774 00:43:52,370 --> 00:43:54,310 compared to the first line-- 775 00:43:54,310 --> 00:43:57,330 in the first line, you need to solve first 776 00:43:57,330 --> 00:43:59,540 all the incentive-compatible contracts, 777 00:43:59,540 --> 00:44:02,270 and then you'll do an optimal randomization 778 00:44:02,270 --> 00:44:03,750 among these contracts. 779 00:44:03,750 --> 00:44:04,420 Right? 780 00:44:04,420 --> 00:44:05,540 Here you can just-- 781 00:44:05,540 --> 00:44:10,260 x and t, you can randomize over this z and the k, 782 00:44:10,260 --> 00:44:12,480 and that will give you a similar result. 783 00:44:12,480 --> 00:44:15,450 So essentially, you only need to solve one linear programming 784 00:44:15,450 --> 00:44:18,350 problems instead of a lot. 785 00:44:18,350 --> 00:44:19,020 Is that-- 786 00:44:19,020 --> 00:44:19,853 [INTERPOSING VOICES] 787 00:44:19,853 --> 00:44:21,030 PROFESSOR: Well, I guess. 788 00:44:21,030 --> 00:44:24,690 We're searching jointly over all the possibilities. 789 00:44:27,860 --> 00:44:32,860 Now, it's true on the one hand, it's just a linear programming 790 00:44:32,860 --> 00:44:34,040 problem. 791 00:44:34,040 --> 00:44:39,930 So we can just get the best code available and use it. 792 00:44:39,930 --> 00:44:43,200 On the other hand, there's lots of variables, lots of states, 793 00:44:43,200 --> 00:44:44,420 lots of constraints. 794 00:44:44,420 --> 00:44:47,832 So it's not like it comes for free. 795 00:44:47,832 --> 00:44:52,440 If I knew something analytically about the underlying contract, 796 00:44:52,440 --> 00:44:53,730 it would be good to use it. 797 00:44:58,440 --> 00:45:00,930 But if I don't know, then I could just 798 00:45:00,930 --> 00:45:03,240 guess wrong and put some functional form 799 00:45:03,240 --> 00:45:04,230 which is incorrect. 800 00:45:04,230 --> 00:45:07,440 And that's why-- that's what's good about this. 801 00:45:07,440 --> 00:45:14,030 I don't-- now, what was Victor talking about? 802 00:45:14,030 --> 00:45:18,940 Victor was talking about going back to deterministic contracts 803 00:45:18,940 --> 00:45:23,200 and then solving them with a nonlinear optimization problem 804 00:45:23,200 --> 00:45:26,905 and then comparing that solution to these linear codes. 805 00:45:30,200 --> 00:45:34,750 And I'm not sure if he showed you at the end, 806 00:45:34,750 --> 00:45:37,210 but when the grids are really coarse, 807 00:45:37,210 --> 00:45:38,960 the linear program seems to work, 808 00:45:38,960 --> 00:45:41,680 but it can be a really bad approximation. 809 00:45:41,680 --> 00:45:43,750 So there's still trade-offs. 810 00:45:43,750 --> 00:45:46,555 It's not a miracle. 811 00:45:49,280 --> 00:45:50,090 Other questions? 812 00:45:53,770 --> 00:45:54,340 All right. 813 00:45:54,340 --> 00:45:58,030 So now we can do limited commitment. 814 00:45:58,030 --> 00:46:00,820 So this is what I mentioned. 815 00:46:00,820 --> 00:46:03,430 You can go into autarky. 816 00:46:03,430 --> 00:46:06,970 You can decide to do that after your output is realized. 817 00:46:06,970 --> 00:46:10,880 It's like, I'm not paying into the risk-sharing group. 818 00:46:10,880 --> 00:46:11,380 Great. 819 00:46:11,380 --> 00:46:12,970 You financed my project. 820 00:46:12,970 --> 00:46:20,060 I'm the big boss now, and I'd just as soon be on my own. 821 00:46:20,060 --> 00:46:24,040 So what we say is-- 822 00:46:24,040 --> 00:46:27,760 first of all, we compute the value function for autarky. 823 00:46:27,760 --> 00:46:28,510 Well, that's cool. 824 00:46:28,510 --> 00:46:29,350 We already did that. 825 00:46:29,350 --> 00:46:31,520 That was the first financial regime. 826 00:46:31,520 --> 00:46:33,580 So we already know this guy. 827 00:46:33,580 --> 00:46:35,530 And then the contemporary situation 828 00:46:35,530 --> 00:46:40,730 is you've got output plus depreciated capital, 829 00:46:40,730 --> 00:46:43,160 and then you could walk away. 830 00:46:43,160 --> 00:46:44,180 No transfers. 831 00:46:44,180 --> 00:46:45,740 They're gone. 832 00:46:45,740 --> 00:46:48,680 You just keep it all and decide maybe 833 00:46:48,680 --> 00:46:51,830 what you want to carry into tomorrow. 834 00:46:51,830 --> 00:46:55,775 So we can call the solution to this thing omega. 835 00:46:58,430 --> 00:47:00,650 By the way, the z is already foregone. 836 00:47:00,650 --> 00:47:03,000 You've already made that decision. 837 00:47:03,000 --> 00:47:04,540 It was in disutility part. 838 00:47:08,535 --> 00:47:09,660 You've already got capital. 839 00:47:09,660 --> 00:47:11,490 You already got funding. 840 00:47:11,490 --> 00:47:13,260 Now you have output. 841 00:47:13,260 --> 00:47:17,240 Can't go backwards, but you can walk away. 842 00:47:17,240 --> 00:47:21,230 So this is basically the maximizing utility. 843 00:47:21,230 --> 00:47:24,900 And we make sure that if you follow the plan, 844 00:47:24,900 --> 00:47:29,610 you're not going to be tempted to do that. 845 00:47:29,610 --> 00:47:32,220 So compute v, then compute this. 846 00:47:32,220 --> 00:47:35,610 Then we have this, and then you impose this as a constraint. 847 00:47:35,610 --> 00:47:37,720 So this is a limited commitment constraint. 848 00:47:37,720 --> 00:47:41,370 We've talked about collateral constraints. 849 00:47:41,370 --> 00:47:44,000 It's very related to that. 850 00:47:44,000 --> 00:47:46,410 You can walk away, but you might-- 851 00:47:46,410 --> 00:47:48,980 with collateral, you have to sacrifice something. 852 00:47:48,980 --> 00:47:49,488 Not here. 853 00:47:49,488 --> 00:47:51,155 They actually keep all of their capital. 854 00:47:54,100 --> 00:47:58,690 If we had financial savings, then they would lose that. 855 00:47:58,690 --> 00:48:01,378 They would lose the stuff they had in the bank. 856 00:48:01,378 --> 00:48:03,155 AUDIENCE: So what keeps me from doing 857 00:48:03,155 --> 00:48:04,780 that is that the continued valuation is 858 00:48:04,780 --> 00:48:06,130 going to be on that side? 859 00:48:06,130 --> 00:48:07,870 PROFESSOR: Yeah. 860 00:48:07,870 --> 00:48:12,700 This is playing ball, and this is being tempted to pull out. 861 00:48:12,700 --> 00:48:15,940 It doesn't mean that it isn't binding on the solution. 862 00:48:15,940 --> 00:48:17,560 This can do damage. 863 00:48:17,560 --> 00:48:19,990 It could be a binding constraint and have a big Lagrange 864 00:48:19,990 --> 00:48:21,190 multiplier. 865 00:48:21,190 --> 00:48:25,450 So the solution will look different. 866 00:48:25,450 --> 00:48:28,120 But you never see the out-of-equilibrium event 867 00:48:28,120 --> 00:48:30,300 that they walk away. 868 00:48:30,300 --> 00:48:33,030 But the damage can be done. 869 00:48:33,030 --> 00:48:34,680 Not getting a whole lot of capital, 870 00:48:34,680 --> 00:48:36,390 not having a whole lot of insurance-- 871 00:48:36,390 --> 00:48:38,870 those things happen. 872 00:48:38,870 --> 00:48:42,950 I mean, again, it's rich guys who would prefer, say, 873 00:48:42,950 --> 00:48:45,743 not to pay into the system. 874 00:48:45,743 --> 00:48:46,910 They're going to be tempted. 875 00:48:46,910 --> 00:48:52,010 So you start eliminating the people that pay in, 876 00:48:52,010 --> 00:48:54,400 and that starts limiting the insurance, et cetera, 877 00:48:54,400 --> 00:48:54,900 et cetera. 878 00:48:57,730 --> 00:49:04,770 And then finally, we have this hidden output. 879 00:49:04,770 --> 00:49:08,370 So here, the idea is income is produced, all right, 880 00:49:08,370 --> 00:49:12,420 but the outsiders don't see what it is. 881 00:49:12,420 --> 00:49:12,920 OK. 882 00:49:12,920 --> 00:49:16,510 But then you say, well, tell me anyway. 883 00:49:16,510 --> 00:49:19,410 Send me a message. 884 00:49:19,410 --> 00:49:22,890 So the idea here is q is actually-- 885 00:49:22,890 --> 00:49:30,510 q bar is actually realized, and the business says so. 886 00:49:30,510 --> 00:49:31,150 Believe me. 887 00:49:31,150 --> 00:49:34,440 My profits are low. 888 00:49:34,440 --> 00:49:38,500 Or-- and they are low. 889 00:49:38,500 --> 00:49:44,260 Or q bar is realized, but we have this counterfactual where 890 00:49:44,260 --> 00:49:47,212 he says something else about q. 891 00:49:47,212 --> 00:49:48,670 Maybe in that particular period, it 892 00:49:48,670 --> 00:49:52,330 was advantageous to say profits are really, really high, 893 00:49:52,330 --> 00:49:55,690 or it could be vice versa. 894 00:49:55,690 --> 00:49:57,310 When they're high, you say high. 895 00:49:57,310 --> 00:49:59,210 When they're low, you're tempted-- 896 00:49:59,210 --> 00:49:59,710 sorry. 897 00:49:59,710 --> 00:50:00,970 When they're low, you say low. 898 00:50:00,970 --> 00:50:04,230 And when they're high, you're tempted to say low. 899 00:50:04,230 --> 00:50:05,170 All right? 900 00:50:05,170 --> 00:50:09,940 This allows for any q that they will, quote, tell the truth. 901 00:50:09,940 --> 00:50:10,720 Well, great. 902 00:50:10,720 --> 00:50:12,430 It's just another inequality. 903 00:50:12,430 --> 00:50:15,070 No problemo. 904 00:50:15,070 --> 00:50:17,780 We know how to generate constraints. 905 00:50:17,780 --> 00:50:19,780 But again, it will do different damage. 906 00:50:19,780 --> 00:50:27,420 There's going to be consequences for the underlying contract. 907 00:50:27,420 --> 00:50:31,810 And remember, the goal here is to get to the data. 908 00:50:31,810 --> 00:50:34,320 You can assume these parametric utility functions-- 909 00:50:34,320 --> 00:50:39,540 constant relative risk aversion, power sigma, disutility 910 00:50:39,540 --> 00:50:46,240 of effort, a power of Frisch elasticity. 911 00:50:46,240 --> 00:50:47,350 OK? 912 00:50:47,350 --> 00:50:50,020 So 3-- we're going to be limited, actually. 913 00:50:50,020 --> 00:50:54,160 We're going to estimate sigma and theta, xe equal 1. 914 00:50:54,160 --> 00:50:56,080 Here's a production function. 915 00:50:56,080 --> 00:50:59,170 We actually load in an observed histogram. 916 00:50:59,170 --> 00:51:02,500 But you can do constant elasticity of substitution 917 00:51:02,500 --> 00:51:03,340 if you want. 918 00:51:03,340 --> 00:51:03,840 Yep? 919 00:51:03,840 --> 00:51:07,493 AUDIENCE: So you [? test ?] one where you can verify a state? 920 00:51:07,493 --> 00:51:09,160 Because now the output is hidden, right? 921 00:51:09,160 --> 00:51:09,340 PROFESSOR: Yeah. 922 00:51:09,340 --> 00:51:10,270 I didn't do it. 923 00:51:10,270 --> 00:51:13,010 AUDIENCE: Why not? 924 00:51:13,010 --> 00:51:15,220 PROFESSOR: I forgot about costly state application. 925 00:51:15,220 --> 00:51:17,590 But anyway, it's doable. 926 00:51:20,480 --> 00:51:21,440 Hong's working on it. 927 00:51:24,620 --> 00:51:26,960 And these other things, like the discount rate, 928 00:51:26,960 --> 00:51:29,250 the depreciation rate, the outside interest rate, 929 00:51:29,250 --> 00:51:30,090 and so on. 930 00:51:30,090 --> 00:51:32,600 So this is very much in the spirit of calibration. 931 00:51:32,600 --> 00:51:35,600 These numbers are similar to numbers 932 00:51:35,600 --> 00:51:38,720 you've seen in various papers in the classes trying 933 00:51:38,720 --> 00:51:40,800 to be somewhat realistic. 934 00:51:40,800 --> 00:51:42,410 Yep. 935 00:51:42,410 --> 00:51:45,580 AUDIENCE: When we're [INAUDIBLE] looking at the matrix p of q 936 00:51:45,580 --> 00:51:46,535 given c an dk. 937 00:51:46,535 --> 00:51:51,857 Is that just for setting up the moral hazard constraint? 938 00:51:51,857 --> 00:51:52,440 PROFESSOR: No. 939 00:51:52,440 --> 00:51:54,100 We need that in general. 940 00:51:54,100 --> 00:51:55,420 AUDIENCE: [INAUDIBLE] 941 00:51:55,420 --> 00:51:59,540 PROFESSOR: So we're going to say that we see effort even though, 942 00:51:59,540 --> 00:52:04,370 in the models, sometimes it's unobserved. 943 00:52:04,370 --> 00:52:06,860 I mean, the question is what an outside lender would see, 944 00:52:06,860 --> 00:52:10,450 not what the households tell us when we interview them. 945 00:52:10,450 --> 00:52:12,190 So anyway, we take the stand that we 946 00:52:12,190 --> 00:52:15,820 see a good version of it, and capital as well, 947 00:52:15,820 --> 00:52:16,960 and we see output. 948 00:52:16,960 --> 00:52:18,480 And I think that-- 949 00:52:18,480 --> 00:52:20,020 let's see where that is. 950 00:52:23,610 --> 00:52:24,240 Oh, my god. 951 00:52:24,240 --> 00:52:28,320 It's way-- there it was. 952 00:52:28,320 --> 00:52:32,610 So there is an empirical histogram 953 00:52:32,610 --> 00:52:35,910 of capital, labor, and output. 954 00:52:35,910 --> 00:52:39,750 I don't know if you really get a three-dimensional view 955 00:52:39,750 --> 00:52:41,280 of that thing. 956 00:52:41,280 --> 00:52:44,700 I can kind of see it's bending over here, and then [INAUDIBLE] 957 00:52:44,700 --> 00:52:45,950 flip on me. 958 00:52:45,950 --> 00:52:50,700 But-- so we can load that in, or we can actually say, 959 00:52:50,700 --> 00:52:54,840 no, no, no, it's CES, some elasticity of substitution, 960 00:52:54,840 --> 00:52:59,026 and estimate whether it's [INAUDIBLE] for linear 961 00:52:59,026 --> 00:53:02,050 or stuck in between. 962 00:53:02,050 --> 00:53:02,747 Yep? 963 00:53:02,747 --> 00:53:04,340 AUDIENCE: [INAUDIBLE] question. 964 00:53:04,340 --> 00:53:09,470 On graphic views, is there a way to understand [INAUDIBLE] 965 00:53:09,470 --> 00:53:11,580 whether those-- 966 00:53:11,580 --> 00:53:14,340 PROFESSOR: Yeah, but it's kind of hard 967 00:53:14,340 --> 00:53:17,160 because I don't know what this thing is doing. 968 00:53:17,160 --> 00:53:20,250 It doesn't look visually like it kept going down. 969 00:53:20,250 --> 00:53:22,260 It may actually come back. 970 00:53:22,260 --> 00:53:25,470 By the way, it makes my point. 971 00:53:25,470 --> 00:53:29,070 We don't have to assume any kind of concavity 972 00:53:29,070 --> 00:53:31,950 in the underlying primitives. 973 00:53:31,950 --> 00:53:35,880 If it has this sort of scallop shape, wonderful. 974 00:53:35,880 --> 00:53:37,830 Bring on the lotteries. 975 00:53:37,830 --> 00:53:41,460 Then it will span the arc line. 976 00:53:41,460 --> 00:53:43,950 We can even allow risk-loving households here. 977 00:53:43,950 --> 00:53:45,750 I mean, there's really no restriction 978 00:53:45,750 --> 00:53:47,100 in the underlying perimeters. 979 00:53:47,100 --> 00:53:51,970 We have parameterized utilities to make people risk-averse, 980 00:53:51,970 --> 00:53:55,620 but in principle, we don't have to put restrictions 981 00:53:55,620 --> 00:53:56,610 on what we load in. 982 00:54:01,100 --> 00:54:01,620 OK. 983 00:54:01,620 --> 00:54:05,320 So preferences, technology-- 984 00:54:05,320 --> 00:54:08,920 I'll just say a few words about dimensionality. 985 00:54:08,920 --> 00:54:13,240 As we go through autarky savings, full information, 986 00:54:13,240 --> 00:54:15,730 moral hazard, et cetera, you can start 987 00:54:15,730 --> 00:54:17,650 counting the number of linear programs that 988 00:54:17,650 --> 00:54:22,030 need to be solved, the number of variables in each program, 989 00:54:22,030 --> 00:54:24,520 the numbers of constraints. 990 00:54:24,520 --> 00:54:27,640 I didn't even dare show you this unobserved investment one, 991 00:54:27,640 --> 00:54:31,590 but the hidden output is getting up there too. 992 00:54:31,590 --> 00:54:35,260 We actually have a technology to compute these things 993 00:54:35,260 --> 00:54:37,780 with hundreds of thousands of constraints. 994 00:54:37,780 --> 00:54:40,750 I mean, the commercial code is CPLEX, 995 00:54:40,750 --> 00:54:44,350 and there's open freeware that's comparable to it. 996 00:54:44,350 --> 00:54:48,100 It's the latest Princeton interior point. 997 00:54:48,100 --> 00:54:51,430 It's not just a simplex algorithm. 998 00:54:51,430 --> 00:54:54,460 You can use it as a student, actually. 999 00:54:54,460 --> 00:54:56,530 It cost me a couple thousand dollars. 1000 00:54:56,530 --> 00:55:02,920 But anyway, so we can handle fairly large numbers. 1001 00:55:02,920 --> 00:55:08,600 Now, that said, what's the tension here? 1002 00:55:08,600 --> 00:55:10,910 Well, you're going to see not only 1003 00:55:10,910 --> 00:55:12,830 do you have to iterate off the value functions 1004 00:55:12,830 --> 00:55:17,930 and solve the linear program at each iteration and solve-- 1005 00:55:17,930 --> 00:55:19,340 this is just one step. 1006 00:55:22,370 --> 00:55:25,340 But then we don't even know what the parameters are. 1007 00:55:25,340 --> 00:55:30,470 So then we got to do it for all the set of possible parameters 1008 00:55:30,470 --> 00:55:33,770 and generate a likelihood. 1009 00:55:33,770 --> 00:55:36,740 So these guys start to get demanding not because you can't 1010 00:55:36,740 --> 00:55:39,860 solve it once in 20 or 30 seconds, 1011 00:55:39,860 --> 00:55:45,620 but because you've got to do it hundreds and hundreds of times. 1012 00:55:45,620 --> 00:55:48,890 So you can understand why there's 1013 00:55:48,890 --> 00:55:52,760 a big interest in having relatively efficient code. 1014 00:55:52,760 --> 00:55:58,370 It starts to constrain you in terms of the kinds of problems 1015 00:55:58,370 --> 00:56:01,970 you really want to consider. 1016 00:56:01,970 --> 00:56:03,260 All right. 1017 00:56:03,260 --> 00:56:08,400 So how does it work computationally? 1018 00:56:08,400 --> 00:56:13,980 Well, once you solve for the optimizing policy pi star, 1019 00:56:13,980 --> 00:56:16,530 you have a transition function, basically. 1020 00:56:16,530 --> 00:56:18,990 You start with promised utility and capital 1021 00:56:18,990 --> 00:56:21,930 today, integrate out over everything 1022 00:56:21,930 --> 00:56:24,990 else other than capital and utility tomorrow, 1023 00:56:24,990 --> 00:56:27,030 and you get the probability of that. 1024 00:56:27,030 --> 00:56:29,970 So you get this Markov object, right? 1025 00:56:29,970 --> 00:56:32,280 It's simple enough conceptually-- 1026 00:56:32,280 --> 00:56:35,160 the probability of states tomorrow given states today. 1027 00:56:37,790 --> 00:56:40,700 So that's kind of the underlying dynamic engine 1028 00:56:40,700 --> 00:56:44,590 that's chugging along for each one of these financial regimes. 1029 00:56:44,590 --> 00:56:48,470 And then any time you pick up certain w and k 1030 00:56:48,470 --> 00:56:51,680 in the solution, you can generate the contract, 1031 00:56:51,680 --> 00:56:54,950 because that's the stuff that we just integrated out. 1032 00:56:54,950 --> 00:56:55,910 But it's still there. 1033 00:56:55,910 --> 00:56:58,430 You can still use it. 1034 00:56:58,430 --> 00:57:03,440 So we can generate histograms, CQ configurations. 1035 00:57:03,440 --> 00:57:05,180 We can have two cross-sections. 1036 00:57:05,180 --> 00:57:07,190 We can have panel. 1037 00:57:07,190 --> 00:57:09,680 There are some limits in terms of the length of the panel 1038 00:57:09,680 --> 00:57:10,775 that we can actually use. 1039 00:57:14,950 --> 00:57:15,700 You still with me? 1040 00:57:19,750 --> 00:57:22,480 We've got to estimate some parameters, 1041 00:57:22,480 --> 00:57:25,380 so we're going to have these underlying structural 1042 00:57:25,380 --> 00:57:27,840 parameters. 1043 00:57:27,840 --> 00:57:30,450 We don't see, say, the distribution of promised 1044 00:57:30,450 --> 00:57:34,200 utility, so we're going to imagine that's generated maybe 1045 00:57:34,200 --> 00:57:36,810 like a normal distribution with a certain mean 1046 00:57:36,810 --> 00:57:40,140 and a certain variance, and we have to estimate those. 1047 00:57:40,140 --> 00:57:42,090 Better yet would be a mixture of normals 1048 00:57:42,090 --> 00:57:45,010 which can approximate any old thing you want, 1049 00:57:45,010 --> 00:57:47,760 but that raises the dimensions. 1050 00:57:47,760 --> 00:57:52,290 And we're trying to be lean in terms of numbers of parameters. 1051 00:57:52,290 --> 00:57:55,020 So what do I mean by likelihood? 1052 00:57:55,020 --> 00:57:58,260 The probability of getting observables y, 1053 00:57:58,260 --> 00:58:03,120 which could be in a static cross-section, 1054 00:58:03,120 --> 00:58:06,990 values of consumption, output, investment, and capital. 1055 00:58:06,990 --> 00:58:10,140 What's the probability of seeing a particular configuration 1056 00:58:10,140 --> 00:58:14,430 like that given the observed capital today 1057 00:58:14,430 --> 00:58:16,770 as a function of these underlying parameter values? 1058 00:58:20,020 --> 00:58:26,170 So because the model is already using lotteries, 1059 00:58:26,170 --> 00:58:28,960 you already get the probability of these objects. 1060 00:58:28,960 --> 00:58:34,260 It just sort of comes for free as part 1061 00:58:34,260 --> 00:58:36,990 of the optimizing solution. 1062 00:58:36,990 --> 00:58:41,220 Now, the next thing is, do we see things perfectly? 1063 00:58:41,220 --> 00:58:44,100 No, not necessarily. 1064 00:58:44,100 --> 00:58:46,200 We can put in-- 1065 00:58:46,200 --> 00:58:48,600 let me jump a second. 1066 00:58:48,600 --> 00:58:55,950 We can put in measurement error and say if c star at j 1067 00:58:55,950 --> 00:58:58,590 were the true value, we don't see it. 1068 00:58:58,590 --> 00:59:01,090 We see some measured version with error. 1069 00:59:01,090 --> 00:59:05,070 So this is classic econometric contamination, 1070 00:59:05,070 --> 00:59:06,600 classical measurement error. 1071 00:59:06,600 --> 00:59:12,260 We can put that on everything that you like. 1072 00:59:12,260 --> 00:59:18,970 And then the program would say, what 1073 00:59:18,970 --> 00:59:23,590 is the probability the underlying output 1074 00:59:23,590 --> 00:59:26,530 and consumption would be c star and q star? 1075 00:59:26,530 --> 00:59:28,870 That we generate from the code. 1076 00:59:28,870 --> 00:59:32,470 And then we can say, well, what we see is c having q hat, 1077 00:59:32,470 --> 00:59:35,870 so that could come from any c star and q star. 1078 00:59:35,870 --> 00:59:38,530 So you get a new sort of distribution of observables. 1079 00:59:44,660 --> 00:59:46,340 Well, that's what I just said in words. 1080 00:59:54,800 --> 00:59:58,970 Well, what's the point of the likelihood? 1081 00:59:58,970 --> 01:00:02,390 We actually have the data, and we see people 1082 01:00:02,390 --> 01:00:05,180 with a certain capital stock getting a certain output 1083 01:00:05,180 --> 01:00:07,640 and having a certain consumption, 1084 01:00:07,640 --> 01:00:08,930 investing a certain amount. 1085 01:00:08,930 --> 01:00:11,380 These are our observables. 1086 01:00:11,380 --> 01:00:15,190 And so we see the histogram in the data, 1087 01:00:15,190 --> 01:00:18,310 and now we have a histogram in the model. 1088 01:00:18,310 --> 01:00:20,980 So we can say, does it-- 1089 01:00:20,980 --> 01:00:25,480 what parameters would best rationalize the data 1090 01:00:25,480 --> 01:00:29,050 if the data came from that financial regime, 1091 01:00:29,050 --> 01:00:30,130 a particular one? 1092 01:00:37,600 --> 01:00:46,490 So let me-- so I said we have these data. 1093 01:00:46,490 --> 01:00:51,380 This is Kansas and the Rocky Mountains for Thailand. 1094 01:00:51,380 --> 01:00:58,700 A mixed metaphor, but it's also the investment. 1095 01:00:58,700 --> 01:01:02,510 Consumption is, again, pretty flat, 1096 01:01:02,510 --> 01:01:04,820 so it's going to suggest a regime where 1097 01:01:04,820 --> 01:01:06,500 there's a lot of consumption smoothing 1098 01:01:06,500 --> 01:01:08,600 against income fluctuations. 1099 01:01:08,600 --> 01:01:13,010 On the other hand, investment isn't flat. 1100 01:01:13,010 --> 01:01:15,920 And I haven't even shown you the transitions in the capital 1101 01:01:15,920 --> 01:01:16,890 stock. 1102 01:01:16,890 --> 01:01:37,300 So when you-- you pick the rural data, and you use, say-- 1103 01:01:37,300 --> 01:01:40,780 say choose consumption and output only. 1104 01:01:43,750 --> 01:01:46,690 There's a tie. 1105 01:01:46,690 --> 01:01:50,620 If you believed it was moral hazard regime-- for example, 1106 01:01:50,620 --> 01:01:52,990 it's estimating the degree of risk aversion 1107 01:01:52,990 --> 01:01:57,010 at 1.02, the Frisch elasticity at 1.6. 1108 01:01:57,010 --> 01:01:59,860 It's got a certain mean and variance of this underlying 1109 01:01:59,860 --> 01:02:03,310 unseen distribution of promised utilities and an estimate 1110 01:02:03,310 --> 01:02:04,390 of the measurement error. 1111 01:02:07,840 --> 01:02:12,110 And it does this for each financial regime 1112 01:02:12,110 --> 01:02:14,720 against the same data. 1113 01:02:14,720 --> 01:02:18,920 And then we use this sort of information criterion, 1114 01:02:18,920 --> 01:02:23,270 which Vong created, which allows testing 1115 01:02:23,270 --> 01:02:26,160 across non-nested regimes. 1116 01:02:26,160 --> 01:02:29,630 In this case, it's a tie. 1117 01:02:29,630 --> 01:02:34,640 But if you went to the investment data alone, 1118 01:02:34,640 --> 01:02:36,780 it would be savings, savings only. 1119 01:02:36,780 --> 01:02:42,590 So in the rural data, we get a limited regime. 1120 01:02:42,590 --> 01:02:45,870 I probably won't have time to show you the Monte Carlos. 1121 01:02:45,870 --> 01:02:47,750 The Monte Carlos are like this. 1122 01:02:47,750 --> 01:02:50,210 You pick a set of parameter values, maybe the ones 1123 01:02:50,210 --> 01:02:52,700 we estimate in the data, for example, 1124 01:02:52,700 --> 01:02:57,120 and then generate the data from the model, 1125 01:02:57,120 --> 01:02:58,490 and then go through all of this. 1126 01:02:58,490 --> 01:03:03,140 And depending on the degree of measurement area-- 1127 01:03:03,140 --> 01:03:07,070 error that you use to contaminate the data, you can-- 1128 01:03:07,070 --> 01:03:09,530 especially when you use joint consumption and investment 1129 01:03:09,530 --> 01:03:14,070 data, you pretty much get back what you put in, 1130 01:03:14,070 --> 01:03:15,260 which is reassuring. 1131 01:03:15,260 --> 01:03:16,670 But it's a bit black boxy. 1132 01:03:16,670 --> 01:03:19,010 We don't have an analytic proof. 1133 01:03:22,850 --> 01:03:28,790 So if you use consumption data alone, two cross-sections, 1134 01:03:28,790 --> 01:03:32,180 you can use two-year panels. 1135 01:03:32,180 --> 01:03:35,120 Then there's-- in the rural data, 1136 01:03:35,120 --> 01:03:37,850 it's kind of hard to pin down too much. 1137 01:03:37,850 --> 01:03:40,220 In terms of the regime, moral hazard is in there. 1138 01:03:40,220 --> 01:03:42,110 Sometimes full information is in there. 1139 01:03:42,110 --> 01:03:44,090 Limited commitment is in there. 1140 01:03:44,090 --> 01:03:48,080 But again, when you use the investment data alone 1141 01:03:48,080 --> 01:03:51,140 or use the joint data, it's pretty clear 1142 01:03:51,140 --> 01:03:55,520 that savings only, no borrowing, buffer stock model 1143 01:03:55,520 --> 01:03:59,040 is the one that the data like the best. 1144 01:04:03,880 --> 01:04:08,380 But when we-- if we use the network alone-- 1145 01:04:08,380 --> 01:04:10,750 you're saying, well, you've contradicted-- no. 1146 01:04:10,750 --> 01:04:13,310 Like, what Cynthia and I do, and so on, 1147 01:04:13,310 --> 01:04:17,260 we can actually get the full information regime out 1148 01:04:17,260 --> 01:04:18,700 of the consumption data. 1149 01:04:18,700 --> 01:04:22,270 So that's very reassuring because in those other papers, 1150 01:04:22,270 --> 01:04:25,140 we didn't use this method. 1151 01:04:25,140 --> 01:04:30,610 There's a lot of consumption smoothing among the networks. 1152 01:04:30,610 --> 01:04:33,330 And if we go to the urban data, we 1153 01:04:33,330 --> 01:04:40,410 get this result that even when we use the investment data, 1154 01:04:40,410 --> 01:04:43,560 it's the moral hazard regime, not the limited savings 1155 01:04:43,560 --> 01:04:45,620 only regime. 1156 01:04:45,620 --> 01:04:49,910 So they're different across the two specifications, 1157 01:04:49,910 --> 01:04:53,600 although it is true that if you used the investment data alone, 1158 01:04:53,600 --> 01:04:56,900 it would still like savings. 1159 01:04:56,900 --> 01:04:59,120 So there's still this-- 1160 01:04:59,120 --> 01:05:00,860 and we've talked about this-- 1161 01:05:00,860 --> 01:05:05,750 money doesn't flow from unproductive to productive 1162 01:05:05,750 --> 01:05:12,730 people the way these relatively unconstrained regimes would 1163 01:05:12,730 --> 01:05:13,390 imply. 1164 01:05:13,390 --> 01:05:16,690 However, if you looked at the consumption data jointly, 1165 01:05:16,690 --> 01:05:18,430 then it's one likelihood. 1166 01:05:18,430 --> 01:05:22,120 And it decides in the urban area that, oh, well, moral hazard 1167 01:05:22,120 --> 01:05:24,820 actually fits better than savings only, 1168 01:05:24,820 --> 01:05:28,750 but the reverse is true in the rural data. 1169 01:05:28,750 --> 01:05:33,550 So although the investment data doesn't fit perfectly well 1170 01:05:33,550 --> 01:05:43,870 in the urban data, the verdict is 1171 01:05:43,870 --> 01:05:47,600 we do a ton of robustness checks. 1172 01:05:47,600 --> 01:05:48,820 And I'll skip it. 1173 01:05:52,610 --> 01:05:53,350 So let me just-- 1174 01:05:58,060 --> 01:06:00,500 I really want to say something about heterogeneity, 1175 01:06:00,500 --> 01:06:05,480 but I'd rather show you three slides before I quit-- namely, 1176 01:06:05,480 --> 01:06:09,980 in the actual rural data, this sort of diagonal 1177 01:06:09,980 --> 01:06:15,020 here represents the persistence of the capital stock. 1178 01:06:15,020 --> 01:06:17,970 So it doesn't move much. 1179 01:06:17,970 --> 01:06:19,750 It moves very slowly. 1180 01:06:19,750 --> 01:06:24,060 You can hardly see mass off the diagonal. 1181 01:06:24,060 --> 01:06:28,710 In the urban data, it's still sluggish, 1182 01:06:28,710 --> 01:06:32,640 but at least you can move away more. 1183 01:06:32,640 --> 01:06:35,020 If you have this wonderful financial regime, 1184 01:06:35,020 --> 01:06:37,680 you're going to adjust almost instantaneously to the observed 1185 01:06:37,680 --> 01:06:39,400 productivity. 1186 01:06:39,400 --> 01:06:41,910 So this speed of adjustment of the capital stock 1187 01:06:41,910 --> 01:06:44,610 is really the thing that's kind of pushing 1188 01:06:44,610 --> 01:06:47,940 the likelihood toward limited financial regimes. 1189 01:06:47,940 --> 01:06:51,330 By the way, these are the best-fitting regimes 1190 01:06:51,330 --> 01:06:53,610 through the lens of the model, which 1191 01:06:53,610 --> 01:06:57,944 still have an excessive amount of smoothing. 1192 01:07:02,120 --> 01:07:05,110 These are the time series-- 1193 01:07:05,110 --> 01:07:09,550 levels of consumption over time, levels of the capital stock, 1194 01:07:09,550 --> 01:07:10,870 levels of income. 1195 01:07:10,870 --> 01:07:13,270 We did not use all of this data. 1196 01:07:13,270 --> 01:07:15,940 This is like setting aside certain things in the data 1197 01:07:15,940 --> 01:07:21,070 and then looking after the fact as a new criterion. 1198 01:07:21,070 --> 01:07:22,890 And we're doing pretty well. 1199 01:07:22,890 --> 01:07:27,970 The standard deviations, we do less well, 1200 01:07:27,970 --> 01:07:31,540 but for obvious reasons that there are stuff in the tails 1201 01:07:31,540 --> 01:07:35,020 that the model doesn't like. 1202 01:07:35,020 --> 01:07:39,040 But if we smooth off 1% of the tails 1203 01:07:39,040 --> 01:07:41,470 and kind of get rid of it, we actually 1204 01:07:41,470 --> 01:07:45,230 bring the standard deviations generated by the model much, 1205 01:07:45,230 --> 01:07:50,740 much closer to the data. 1206 01:07:50,740 --> 01:07:55,300 And we can actually use the level on borrowing, 1207 01:07:55,300 --> 01:07:57,800 which we didn't use [? an ?] [? estimate, ?] and got really, 1208 01:07:57,800 --> 01:08:00,510 really close to that. 1209 01:08:00,510 --> 01:08:04,200 And finally, here is this data on the return 1210 01:08:04,200 --> 01:08:10,290 on assets with these low-wealth households having high returns. 1211 01:08:10,290 --> 01:08:15,140 And the savings only regime which fits best 1212 01:08:15,140 --> 01:08:17,877 is trying to do that. 1213 01:08:17,877 --> 01:08:19,710 And this is the urban data, which is already 1214 01:08:19,710 --> 01:08:21,330 a little more dispersed, but it still 1215 01:08:21,330 --> 01:08:26,100 has this low-wealth, high-return guise. 1216 01:08:26,100 --> 01:08:30,359 And it can generate that somewhat 1217 01:08:30,359 --> 01:08:31,560 with this moral hazard. 1218 01:08:31,560 --> 01:08:33,330 So these are the familiar objects 1219 01:08:33,330 --> 01:08:35,189 that we've been looking at. 1220 01:08:40,510 --> 01:08:42,420 And finally, let me just end with this. 1221 01:08:45,670 --> 01:08:46,170 Sorry. 1222 01:08:46,170 --> 01:08:48,212 I guess that discussion at the beginning of class 1223 01:08:48,212 --> 01:08:51,000 took a chunk of time. 1224 01:08:51,000 --> 01:08:53,240 So if you believe the structural model, 1225 01:08:53,240 --> 01:08:56,300 you can do policy evaluation. 1226 01:08:56,300 --> 01:08:58,640 And here we're doing something simple 1227 01:08:58,640 --> 01:09:02,220 like just changing the interest rate. 1228 01:09:02,220 --> 01:09:07,310 Now, the point is some people win, and some people lose. 1229 01:09:07,310 --> 01:09:11,870 And also, who are the winners and the losers 1230 01:09:11,870 --> 01:09:15,020 depends on the financial regime. 1231 01:09:15,020 --> 01:09:19,689 So this is the consumption equivalent welfare gain 1232 01:09:19,689 --> 01:09:23,990 of lowering the interest rate. 1233 01:09:23,990 --> 01:09:27,200 Negative and positive-- this is as we 1234 01:09:27,200 --> 01:09:28,880 vary the level of current savings 1235 01:09:28,880 --> 01:09:30,590 and the level of assets. 1236 01:09:30,590 --> 01:09:34,970 But this is what it would look like if it were not the savings 1237 01:09:34,970 --> 01:09:38,840 regime, but the borrowing regime at the estimated parameters 1238 01:09:38,840 --> 01:09:41,240 we get. 1239 01:09:41,240 --> 01:09:44,330 And then this is the difference of those two. 1240 01:09:44,330 --> 01:09:47,490 So it's another difference and difference, 1241 01:09:47,490 --> 01:09:50,210 but it's a difference in the welfare criteria 1242 01:09:50,210 --> 01:09:52,399 across two financial regimes. 1243 01:09:52,399 --> 01:09:56,490 And the point is it matters quite a lot, 1244 01:09:56,490 --> 01:09:59,690 if you did this interest rate subsidy, what 1245 01:09:59,690 --> 01:10:04,700 the financial regime is in terms of who wins and who loses. 1246 01:10:04,700 --> 01:10:08,580 These are non-trivial gains, and so on. 1247 01:10:08,580 --> 01:10:09,080 All right. 1248 01:10:09,080 --> 01:10:11,240 I'll quit there.