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ROBERT TOWNSEND: So
today, we're going

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00:00:28,740 --> 00:00:32,259
to talk about labor
and development.

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00:00:32,259 --> 00:00:35,130
There are several
pieces to this,

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00:00:35,130 --> 00:00:42,840
just to sort of get into it.

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00:00:42,840 --> 00:00:47,160
One, number one, back to
risk sharing, full benchmark,

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00:00:47,160 --> 00:00:51,240
but this time with endogenous
labor supply and wage risk.

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00:00:51,240 --> 00:00:55,470
And I'll mention sort of some
tests using that framework.

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00:00:55,470 --> 00:00:57,330
I dumped a lot of
the excess slides,

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00:00:57,330 --> 00:01:00,090
because I don't want that to be
the sole focus of the lecture

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00:01:00,090 --> 00:01:03,990
today, although it's kind of
the obvious starting point.

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00:01:03,990 --> 00:01:10,440
With those lotteries,
basically, you

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00:01:10,440 --> 00:01:14,460
end up dealing with
a non-convexity.

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00:01:14,460 --> 00:01:23,070
And then that has a lot to do
with estimating elasticities

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00:01:23,070 --> 00:01:25,350
of labor supply from
micro data versus what

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00:01:25,350 --> 00:01:27,570
you see in aggregate data.

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00:01:27,570 --> 00:01:29,430
And we've learned
a lot about that.

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00:01:29,430 --> 00:01:32,430
So there is sort of the
complete market's version.

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00:01:32,430 --> 00:01:36,000
And then, a la lecture last
time, the incomplete market's

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00:01:36,000 --> 00:01:41,660
version, where the aggregate
elasticity is higher, sometimes

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00:01:41,660 --> 00:01:45,770
a lot higher, than it is in
the micro data for reasons

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00:01:45,770 --> 00:01:47,660
of the aggregation.

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00:01:51,000 --> 00:01:53,730
And since we've talked
about zoom in, zoom out,

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00:01:53,730 --> 00:01:57,420
you know, micro, macro, that
seems like a relevant thing.

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00:01:57,420 --> 00:02:02,110
And then we, in fact, end
the lecture with Seema's,

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00:02:02,110 --> 00:02:04,570
Jayachandran, paper
on labor supply.

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00:02:04,570 --> 00:02:07,680
And I think it was
India, with rainfall

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00:02:07,680 --> 00:02:08,870
and agricultural shock.

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00:02:08,870 --> 00:02:18,420
So sharing wage risk,
I just said this.

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00:02:18,420 --> 00:02:22,740
We have non-labor income, which
is like the y moving around

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00:02:22,740 --> 00:02:23,580
we've had before.

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00:02:23,580 --> 00:02:25,740
But we also have
income from wages,

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00:02:25,740 --> 00:02:27,840
but hours are endogenous.

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00:02:27,840 --> 00:02:30,110
Labor is not supplied
inelastically.

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00:02:30,110 --> 00:02:38,470
And we're going to allow for
heterogeneity among individuals

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00:02:38,470 --> 00:02:42,610
in a household and
heterogeneity across households.

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00:02:42,610 --> 00:02:44,440
So let me just lead
off with an example,

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00:02:44,440 --> 00:02:46,790
because I think it makes
the points pretty well.

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00:02:46,790 --> 00:02:49,630
And then we'll generalize.

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00:02:49,630 --> 00:02:53,710
So you have two people,
or two households.

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00:02:53,710 --> 00:02:56,260
One is risk neutral.

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00:02:56,260 --> 00:02:59,420
The other risk averse.

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00:02:59,420 --> 00:03:00,920
Why is this an obvious example?

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00:03:00,920 --> 00:03:04,970
Because the risk neutral guy
should absorb all the risk

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00:03:04,970 --> 00:03:10,040
and would in every setup
that we've seen so far

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00:03:10,040 --> 00:03:11,840
and take it away
from the risk averse

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00:03:11,840 --> 00:03:14,390
guy, who's got a strictly
concave utility function.

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00:03:14,390 --> 00:03:18,380
However, this guy cares about
leisure and consumption, not

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00:03:18,380 --> 00:03:20,730
just consumption.

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00:03:20,730 --> 00:03:23,160
So typical Cobb-Douglas
specification

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00:03:23,160 --> 00:03:27,695
with some curvature, making
him strictly concave guy.

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00:03:30,510 --> 00:03:36,060
If you don't consume leisure
in your work, that's supplied.

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00:03:36,060 --> 00:03:39,588
You get this wage, W2.

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00:03:39,588 --> 00:03:41,880
When we generalize, we're
going to allow different wage

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00:03:41,880 --> 00:03:46,920
processes for different people,
not modeling necessarily

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00:03:46,920 --> 00:03:48,660
why they have different talent.

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00:03:48,660 --> 00:03:52,530
But for now, there's only
one guy supplying labor.

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00:03:52,530 --> 00:03:54,540
Again, y is this
non-labor income.

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00:03:54,540 --> 00:03:57,000
T is the time endowment.

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00:03:57,000 --> 00:03:59,100
Consumption good
is the numeraire.

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00:03:59,100 --> 00:04:01,770
Wages are real.

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00:04:01,770 --> 00:04:04,860
And there is a budget constraint
for this two-person programming

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00:04:04,860 --> 00:04:12,560
problem, which is like a small
open economy in the sense

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00:04:12,560 --> 00:04:16,070
that it's partial
equilibrium, and they

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00:04:16,070 --> 00:04:17,810
face these outside wages.

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00:04:17,810 --> 00:04:20,060
In fact, these wages
may be moving around.

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00:04:20,060 --> 00:04:23,960
It could be a village with two
people, something like that.

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00:04:23,960 --> 00:04:27,890
Wages are determined in the
district or national level.

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00:04:27,890 --> 00:04:31,130
This is the standard
budget constraint though.

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00:04:31,130 --> 00:04:35,520
Call this thing full income,
if you remember Becker price

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00:04:35,520 --> 00:04:36,020
theory.

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00:04:39,150 --> 00:04:42,170
So basically, it's as if
you supplied all the time

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00:04:42,170 --> 00:04:46,880
you have, cap T, hours
and days times the wage.

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00:04:46,880 --> 00:04:49,550
You have this non-labor income.

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00:04:49,550 --> 00:04:51,210
The community as
a whole has this.

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00:04:51,210 --> 00:04:54,660
And then, you have
consumption and Mr. 2 at least

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00:04:54,660 --> 00:04:56,710
buys back leisure.

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00:04:56,710 --> 00:04:58,650
You know, well, it's
obvious, you could just

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00:04:58,650 --> 00:05:00,270
take time minus leisure.

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00:05:00,270 --> 00:05:03,660
That would be labor supply and
put it on the right hand side.

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00:05:03,660 --> 00:05:07,192
But it's kind of useful to
talk about leisure as a good,

88
00:05:07,192 --> 00:05:10,950
because that's what enters
the utility function.

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00:05:10,950 --> 00:05:16,950
And we have some standard sort
of consumption sharing rules,

90
00:05:16,950 --> 00:05:18,700
division rules.

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00:05:18,700 --> 00:05:21,520
Who's going to bear the risk?

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00:05:21,520 --> 00:05:26,780
Well, it's going to turn out
that as we've done so far,

93
00:05:26,780 --> 00:05:30,350
this non-labor income risk
is going to be shared--

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00:05:30,350 --> 00:05:34,400
taken away entirely
by household 1.

95
00:05:34,400 --> 00:05:36,350
But things are much
more complicated

96
00:05:36,350 --> 00:05:38,150
when it comes to the wage.

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00:05:38,150 --> 00:05:41,480
The wage moves around,
and it changes income.

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00:05:41,480 --> 00:05:45,170
But the wage also
represents a rate

99
00:05:45,170 --> 00:05:47,870
of substitution between
consumption and leisure

100
00:05:47,870 --> 00:05:50,750
for the group as a whole,
even though person 2

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00:05:50,750 --> 00:05:52,630
has to supply it.

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00:05:52,630 --> 00:05:54,410
You know, the wage
goes way up, there's

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00:05:54,410 --> 00:05:57,590
kind of a change to
opportunity set for the group.

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00:05:57,590 --> 00:05:59,870
And you can imagine
things should

105
00:05:59,870 --> 00:06:03,485
move in order to be efficient
to take that into account.

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00:06:07,840 --> 00:06:11,010
So we solved the
problem in two steps.

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00:06:11,010 --> 00:06:17,850
And that's also going to make
it familiar in the sense of it

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00:06:17,850 --> 00:06:20,820
integrates with what we've
been talking about so far.

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00:06:20,820 --> 00:06:24,270
First, ex post
inefficiency, suppose--

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00:06:24,270 --> 00:06:27,700
ex post efficiency-- this
is like the second welfare

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00:06:27,700 --> 00:06:28,200
theorem.

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00:06:28,200 --> 00:06:30,370
But I'll tell you what I mean.

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00:06:30,370 --> 00:06:32,710
Any Pareto optimal
allocation can

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00:06:32,710 --> 00:06:37,780
be supported among households
with a suitable lump sum

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00:06:37,780 --> 00:06:39,700
taxes and subsidies.

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00:06:39,700 --> 00:06:41,730
Remember that theorem?

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00:06:41,730 --> 00:06:44,040
OK, so what does it mean for us?

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00:06:44,040 --> 00:06:49,700
You can kind of decentralize
the problem of the two people,

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00:06:49,700 --> 00:06:55,220
as if one or later both can
supply labor on their own.

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00:06:55,220 --> 00:06:57,590
But this lump sum
tax or transfer

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00:06:57,590 --> 00:06:59,540
is like an internal
redistribution

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00:06:59,540 --> 00:07:01,200
within the household.

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00:07:01,200 --> 00:07:03,590
So for any target Pareto
optimal allocation,

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00:07:03,590 --> 00:07:08,300
you can find a transfer and
then have each act in isolation,

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00:07:08,300 --> 00:07:12,240
maximizing utility.

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00:07:12,240 --> 00:07:14,980
And then step 2 is to take
sort of those indirect utility

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00:07:14,980 --> 00:07:18,350
functions, take a
step back in time

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00:07:18,350 --> 00:07:21,200
and maximize the ex
ante expected utility.

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00:07:21,200 --> 00:07:24,700
And that's going to look like a
standard risk sharing problem.

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00:07:24,700 --> 00:07:28,610
And in this case, for
this baby example,

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00:07:28,610 --> 00:07:31,880
only member 2 is
supplying labor.

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00:07:31,880 --> 00:07:36,890
Remember it was Cobb-Douglas,
and the power was basically 2.

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00:07:36,890 --> 00:07:40,230
So basically here it's easier.

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00:07:40,230 --> 00:07:42,530
You take 1/2 of
your full income.

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00:07:42,530 --> 00:07:44,700
And that's your consumption.

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00:07:44,700 --> 00:07:51,700
So those are those easy kind of
consumption expenditure rules.

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00:07:51,700 --> 00:07:52,900
This is actually the same.

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00:07:52,900 --> 00:07:54,940
You put the wage over here.

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00:07:54,940 --> 00:07:59,510
The value of leisure is again
1/2 of your full income.

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00:07:59,510 --> 00:08:02,050
So those are with
that functional

141
00:08:02,050 --> 00:08:06,940
form standard leisure
consumption for household 2,

142
00:08:06,940 --> 00:08:10,750
plug it back into the
underlying utility function,

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00:08:10,750 --> 00:08:16,450
raise it to a power, and you
may remember working this out

144
00:08:16,450 --> 00:08:18,290
at some point in
some other course.

145
00:08:18,290 --> 00:08:20,620
You get a standard
indirect utility function

146
00:08:20,620 --> 00:08:29,090
with all this stuff going on
in terms of these coefficients.

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00:08:29,090 --> 00:08:31,910
Well, it's kind of obvious
that it had to end up this way.

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00:08:31,910 --> 00:08:37,730
Note that the wage is in
here a couple of times,

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00:08:37,730 --> 00:08:40,880
partly because the
full income and partly

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00:08:40,880 --> 00:08:43,840
because this guy's optimizing--

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00:08:43,840 --> 00:08:49,130
as the wage changed,
say, the allocation,

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00:08:49,130 --> 00:08:52,670
the maximizing
allocation would change.

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00:08:52,670 --> 00:08:54,290
You've got an
indifference curve,

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00:08:54,290 --> 00:08:56,623
and you're going to change
the slope of the budget line.

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00:08:59,770 --> 00:09:04,180
So now we take that
step back and consider

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00:09:04,180 --> 00:09:05,470
the overall maximum.

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00:09:05,470 --> 00:09:10,750
But instead of having utility of
consumption directly of the two

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00:09:10,750 --> 00:09:14,140
members, we can use
the indirect functions.

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00:09:14,140 --> 00:09:16,810
I mean, if you believed me,
then the indirect function

160
00:09:16,810 --> 00:09:19,590
is without any restrictions.

161
00:09:19,590 --> 00:09:25,080
It's a natural object that
represents efficient ex post

162
00:09:25,080 --> 00:09:25,980
optimization.

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00:09:29,700 --> 00:09:31,730
Here's that row.

164
00:09:31,730 --> 00:09:35,250
Did I-- I should
have emphasized it.

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00:09:35,250 --> 00:09:39,390
Row is the amount that
say 2 is getting from 1.

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00:09:39,390 --> 00:09:41,700
Those are the
internal transfers.

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00:09:41,700 --> 00:09:46,650
And household 2 optimizes
for a given row.

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00:09:46,650 --> 00:09:55,570
So the ex ante allocation is
set the marginal utilities

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00:09:55,570 --> 00:10:01,280
basically equal to each
other, or equal to a constant.

170
00:10:01,280 --> 00:10:04,540
This constant would be the
ratio of Pareto weights.

171
00:10:04,540 --> 00:10:07,720
I'm not saying that two members
have to be treated equally

172
00:10:07,720 --> 00:10:09,480
within the household.

173
00:10:09,480 --> 00:10:12,160
It could be very asymmetric.

174
00:10:12,160 --> 00:10:15,700
We usually have lambda 1 over
lambda 2, or mu 1 over mu 2,

175
00:10:15,700 --> 00:10:19,990
or some ratio of those ex
ante given Pareto weights.

176
00:10:19,990 --> 00:10:21,970
That's all embedded up
here in the constant.

177
00:10:21,970 --> 00:10:26,740
Of course, convenient part
is the indirect utility

178
00:10:26,740 --> 00:10:31,580
of household 1 is just linear,
because that person had

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00:10:31,580 --> 00:10:33,910
a utility that depended
only on consumption.

180
00:10:33,910 --> 00:10:36,800
And it was linear
in consumption.

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00:10:36,800 --> 00:10:39,040
So we can now,
with this constant,

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00:10:39,040 --> 00:10:43,000
if we carry it around
and multiply, you know,

183
00:10:43,000 --> 00:10:47,390
keep track that parameters
get added or multiplied

184
00:10:47,390 --> 00:10:50,840
to it, that we can get a
closed form solution from this

185
00:10:50,840 --> 00:10:53,090
for the transfer rule.

186
00:10:53,090 --> 00:10:55,370
And we already know what
consumption and leisure

187
00:10:55,370 --> 00:10:59,690
are for person 2 as a
function of the transfer.

188
00:10:59,690 --> 00:11:00,870
So we solve that.

189
00:11:00,870 --> 00:11:03,420
And I don't know,
it's just algebra.

190
00:11:03,420 --> 00:11:06,120
What do you want
to see down here?

191
00:11:06,120 --> 00:11:11,580
Well, person 1's consumption
is moving around one to one

192
00:11:11,580 --> 00:11:14,380
with non-labor income.

193
00:11:14,380 --> 00:11:19,050
So as anticipated, member 1
is absorbing all of that risk

194
00:11:19,050 --> 00:11:22,120
and member 2 is
absorbing none of it.

195
00:11:22,120 --> 00:11:32,990
But the wage is affecting
consumption and more obviously

196
00:11:32,990 --> 00:11:35,620
labor supply of person 2.

197
00:11:35,620 --> 00:11:41,840
And that wage is actually
affecting person 2 as well.

198
00:11:41,840 --> 00:11:43,130
Here it is down here in--

199
00:11:43,130 --> 00:11:46,980
person 1, sorry, in terms
of person 1's consumption.

200
00:11:51,240 --> 00:11:56,190
And it's doing it in a
particular way basically.

201
00:11:56,190 --> 00:11:59,310
So this paper is called
sharing wage risk.

202
00:11:59,310 --> 00:12:03,360
And both these members
will share the wage risk,

203
00:12:03,360 --> 00:12:07,020
even though only person 2
can work more or less hours

204
00:12:07,020 --> 00:12:07,890
at that wage.

205
00:12:13,510 --> 00:12:16,480
So in some sense,
leisure pooling

206
00:12:16,480 --> 00:12:26,680
is a bit different from
consumption pooling in the way

207
00:12:26,680 --> 00:12:28,660
that they're going
to want to adjust

208
00:12:28,660 --> 00:12:33,000
their hours is going to depend
on the outside opportunities.

209
00:12:33,000 --> 00:12:33,980
So that's the summary.

210
00:12:33,980 --> 00:12:38,130
And then we'll
generalize this case.

211
00:12:38,130 --> 00:12:42,950
It's not general, because
person 1 was risk neutral.

212
00:12:42,950 --> 00:12:44,210
Here's the general case.

213
00:12:44,210 --> 00:12:46,410
There are S states of the world.

214
00:12:46,410 --> 00:12:49,910
There's a risk sharing group
of households, as in a village

215
00:12:49,910 --> 00:12:53,260
or township or something.

216
00:12:53,260 --> 00:12:58,655
And every household H
has individual members.

217
00:12:58,655 --> 00:13:00,230
It could be more than one.

218
00:13:00,230 --> 00:13:02,750
It's likely to be two or three.

219
00:13:02,750 --> 00:13:04,670
This is meant to be realistic.

220
00:13:04,670 --> 00:13:09,090
And we actually go to the data
with these specifications.

221
00:13:09,090 --> 00:13:12,500
And we have hours and
labor force participation

222
00:13:12,500 --> 00:13:14,780
for every member
of the household.

223
00:13:14,780 --> 00:13:18,740
So we normally only have
consumption for the household.

224
00:13:18,740 --> 00:13:20,550
So we kind of throw
up our hands a bit,

225
00:13:20,550 --> 00:13:23,498
but here, we have labor at
the level of individuals.

226
00:13:23,498 --> 00:13:25,040
And we definitely
want to keep track.

227
00:13:28,250 --> 00:13:29,990
The consumption
of household H is

228
00:13:29,990 --> 00:13:33,200
a sum of consumption
of individual members.

229
00:13:33,200 --> 00:13:36,530
That's true in theory,
even if we don't measure it

230
00:13:36,530 --> 00:13:38,930
individually in practice.

231
00:13:38,930 --> 00:13:40,700
Aggregate consumption
in the village

232
00:13:40,700 --> 00:13:44,240
is a sum of consumption
across the households.

233
00:13:44,240 --> 00:13:45,470
Nothing surprising there.

234
00:13:45,470 --> 00:13:48,170
Utilities of everyone
stays strictly increasing

235
00:13:48,170 --> 00:13:49,385
and concave.

236
00:13:52,190 --> 00:13:58,880
And household H faces a vector
of wage and non-labor income

237
00:13:58,880 --> 00:14:00,960
variation.

238
00:14:00,960 --> 00:14:06,170
You know the S is a subscript
on the wages of person

239
00:14:06,170 --> 00:14:10,590
1, the wages of person
2, for how many people

240
00:14:10,590 --> 00:14:12,120
there are in this household.

241
00:14:12,120 --> 00:14:13,920
And again, here
although it really

242
00:14:13,920 --> 00:14:17,302
doesn't matter because we're
going to added up anyway,

243
00:14:17,302 --> 00:14:19,260
you could say different
people in the household

244
00:14:19,260 --> 00:14:20,580
have different occupations.

245
00:14:20,580 --> 00:14:22,110
Some are helping on the farm.

246
00:14:22,110 --> 00:14:24,900
Other people run the
business exclusively.

247
00:14:24,900 --> 00:14:28,300
You could talk about
the non-wage income.

248
00:14:28,300 --> 00:14:32,460
If there's perfect markets,
which we're going to assume,

249
00:14:32,460 --> 00:14:36,000
then the profits
of the firm are as

250
00:14:36,000 --> 00:14:39,870
if they had supply their
own labor to the market

251
00:14:39,870 --> 00:14:44,490
and hire other labor to run the
business, which is problematic.

252
00:14:44,490 --> 00:14:48,680
But, you know, then you get
the usual sort of separation.

253
00:14:48,680 --> 00:14:52,350
And so these are really
like profit numbers.

254
00:14:57,990 --> 00:15:01,630
But this is the aggregate
of overall individuals

255
00:15:01,630 --> 00:15:03,280
in the household.

256
00:15:03,280 --> 00:15:06,520
This is what we've been
calling household profits.

257
00:15:06,520 --> 00:15:10,060
And finally, the
resources available

258
00:15:10,060 --> 00:15:13,390
are household profits plus
gifts and other things.

259
00:15:13,390 --> 00:15:16,870
These taus are not
yet determined.

260
00:15:16,870 --> 00:15:22,607
They're given for this household
H in state S. In a minute

261
00:15:22,607 --> 00:15:24,440
they're going to get
determined endogenously

262
00:15:24,440 --> 00:15:26,355
from the rest of the village.

263
00:15:26,355 --> 00:15:27,980
So, again, we're
going to kind of solve

264
00:15:27,980 --> 00:15:33,312
individual problems, household
problems, village problems,

265
00:15:33,312 --> 00:15:34,395
and sort of build upwards.

266
00:15:37,820 --> 00:15:40,940
So what does the household do?

267
00:15:40,940 --> 00:15:45,620
It maximizes these
new weighted sums

268
00:15:45,620 --> 00:15:52,310
of utilities of its individual
members, summing over states.

269
00:15:52,310 --> 00:15:54,320
And actually, we
could sum over dates.

270
00:15:54,320 --> 00:15:57,570
But you already know
that doesn't matter.

271
00:15:57,570 --> 00:15:59,960
There's nothing you know
particularly inter-temporal

272
00:15:59,960 --> 00:16:02,300
about this--

273
00:16:02,300 --> 00:16:04,550
well, these sub-problems
apply even when

274
00:16:04,550 --> 00:16:06,560
the household as a whole
can borrow and lend

275
00:16:06,560 --> 00:16:09,210
with the outside, et cetera.

276
00:16:09,210 --> 00:16:12,740
And just simplify by
getting rid of that for now.

277
00:16:12,740 --> 00:16:17,000
Common probability over states
S. Utilities are allowed

278
00:16:17,000 --> 00:16:20,720
to depend on individual i.

279
00:16:20,720 --> 00:16:25,900
We're going to try to retain as
much heterogeneity as possible,

280
00:16:25,900 --> 00:16:28,390
although it's sometimes
problematic how much we

281
00:16:28,390 --> 00:16:30,250
can allow in the data.

282
00:16:30,250 --> 00:16:34,030
And here's the sort of
collective resource constraint

283
00:16:34,030 --> 00:16:39,200
for the household H.
It's the same as before.

284
00:16:39,200 --> 00:16:41,710
We're just summing over
all the individual members.

285
00:16:41,710 --> 00:16:47,950
This is the sum consumption,
sum of the valuation of leisure,

286
00:16:47,950 --> 00:16:52,150
equal to full income,
including any taxes

287
00:16:52,150 --> 00:16:54,632
or subsidies the
household is getting

288
00:16:54,632 --> 00:16:55,840
from the rest of the village.

289
00:16:58,720 --> 00:17:03,250
These Pareto weights mu
are kind of taken as given.

290
00:17:03,250 --> 00:17:06,190
That's not to say there aren't
forces that determine them--

291
00:17:06,190 --> 00:17:10,089
you know, the marriage
market, one person

292
00:17:10,089 --> 00:17:12,700
has a lot more income than the
other, all this bargaining.

293
00:17:12,700 --> 00:17:17,660
But any bargaining that takes
place here is entirely ex ante.

294
00:17:17,660 --> 00:17:24,710
And it is assumed to be fixed
and unalterable subsequently

295
00:17:24,710 --> 00:17:27,869
as states of the
world are realized.

296
00:17:27,869 --> 00:17:31,370
You know, we haven't talked
about it in this class.

297
00:17:31,370 --> 00:17:33,590
I think you've probably
seen it in other classes,

298
00:17:33,590 --> 00:17:35,960
this whole literature
on the unitary household

299
00:17:35,960 --> 00:17:39,380
and, you know, ex
post bargaining,

300
00:17:39,380 --> 00:17:43,160
when opportunities of the
spouse increase and so on.

301
00:17:43,160 --> 00:17:44,300
They may take more goods.

302
00:17:44,300 --> 00:17:47,690
They may better dictate
what household is consuming.

303
00:17:47,690 --> 00:17:49,070
That's all suppressed here.

304
00:17:49,070 --> 00:17:54,030
It's buried in these
fixed Pareto weights.

305
00:17:54,030 --> 00:17:59,810
The idea is to see how far we
can get even assuming that.

306
00:17:59,810 --> 00:18:04,850
OK, well, then we have
the Marshallian demand.

307
00:18:04,850 --> 00:18:07,280
I only say that with
emphasis because we're

308
00:18:07,280 --> 00:18:10,130
going to go Hicks and
Frisch in a minute.

309
00:18:10,130 --> 00:18:14,030
But this is the regular
demand for leisure

310
00:18:14,030 --> 00:18:18,450
that solves the individual's
optimization problem.

311
00:18:18,450 --> 00:18:21,410
This is an individual
i in household H.

312
00:18:21,410 --> 00:18:24,350
This, apart from the
more cumbersome notation,

313
00:18:24,350 --> 00:18:26,870
is exactly what we
went through before.

314
00:18:26,870 --> 00:18:31,370
This is some transfer that
individual i is getting

315
00:18:31,370 --> 00:18:34,013
in state S from the rest--

316
00:18:34,013 --> 00:18:35,930
giving or getting from
the rest of the members

317
00:18:35,930 --> 00:18:36,680
of the household.

318
00:18:36,680 --> 00:18:39,590
This is the
valuation of the time

319
00:18:39,590 --> 00:18:41,210
endowment for individual i.

320
00:18:41,210 --> 00:18:44,900
And in the left-hand side
is the sum of expenditures

321
00:18:44,900 --> 00:18:48,800
on consumption and leisure.

322
00:18:48,800 --> 00:18:53,720
And that after maximization
gives you the indirect utility

323
00:18:53,720 --> 00:18:57,560
function, which clearly
depends on the wage

324
00:18:57,560 --> 00:18:59,780
and on this transfer.

325
00:19:10,040 --> 00:19:16,310
So for the household to
have an objective function,

326
00:19:16,310 --> 00:19:22,570
we can call it sort of
the Pareto mu weighted sum

327
00:19:22,570 --> 00:19:26,410
of the value functions
of the individuals.

328
00:19:26,410 --> 00:19:29,050
Again, we went through
this in the baby example

329
00:19:29,050 --> 00:19:30,205
at the beginning.

330
00:19:32,880 --> 00:19:35,730
And then the problem
of the village--

331
00:19:35,730 --> 00:19:38,640
note that now we summed over H--

332
00:19:38,640 --> 00:19:45,780
is to maximize this household
by household expected utility,

333
00:19:45,780 --> 00:19:49,920
and now making it expected
because it's conditioned

334
00:19:49,920 --> 00:19:55,590
on state S. And here's the
constraint that these transfers

335
00:19:55,590 --> 00:19:58,230
stay within the
village have to add up

336
00:19:58,230 --> 00:20:01,800
to zero if the village is
the risk sharing group you

337
00:20:01,800 --> 00:20:04,590
want to be considering.

338
00:20:08,510 --> 00:20:12,800
One first order
condition, you know,

339
00:20:12,800 --> 00:20:15,740
basically says that the marginal
utility of the household

340
00:20:15,740 --> 00:20:23,990
respective transfers in state
S is just pre-multiplied

341
00:20:23,990 --> 00:20:26,900
by the probability and
then this lambda HS--

342
00:20:26,900 --> 00:20:30,740
so this lambda HS is
like the shadow price

343
00:20:30,740 --> 00:20:33,520
on the budget constrain
of the household at state

344
00:20:33,520 --> 00:20:38,740
S. It reflects
how tight or loose

345
00:20:38,740 --> 00:20:40,600
the optimization
problem is, just

346
00:20:40,600 --> 00:20:43,360
like in standard price theory.

347
00:20:43,360 --> 00:20:45,040
It's just the
Lagrange multiplier

348
00:20:45,040 --> 00:20:46,780
on the budget constraint.

349
00:20:46,780 --> 00:20:49,820
It's also going to
equal something else.

350
00:20:49,820 --> 00:20:55,452
But so far I haven't talked
about it in the notation.

351
00:20:55,452 --> 00:20:56,910
It's going to equal
something else,

352
00:20:56,910 --> 00:20:59,820
because these sum of
the transfers are zero.

353
00:20:59,820 --> 00:21:03,720
So there's like a collective
resource constraint that isn't

354
00:21:03,720 --> 00:21:05,250
written out here explicitly.

355
00:21:05,250 --> 00:21:08,970
And that thing is going to
pick up a Lagrange multiplier.

356
00:21:08,970 --> 00:21:10,610
And that Lagrange
multiplier is going

357
00:21:10,610 --> 00:21:15,630
to be common over
all the households,

358
00:21:15,630 --> 00:21:19,800
just like it was for
the consumption problem.

359
00:21:19,800 --> 00:21:27,160
OK, so here's an example,
Cobb-Douglas coefficients

360
00:21:27,160 --> 00:21:33,350
are depending on i and
H raised to a power.

361
00:21:33,350 --> 00:21:37,570
This obvious generalization
of the example.

362
00:21:37,570 --> 00:21:40,990
We have a little corner
to worry about here.

363
00:21:40,990 --> 00:21:45,160
You know, the way to see
this is putting w over here,

364
00:21:45,160 --> 00:21:48,250
this is the
expenditures on leisure.

365
00:21:48,250 --> 00:21:50,815
And expenditures on leisure
ought to be equal to--

366
00:21:54,780 --> 00:21:57,642
well, these add up
to 1, basically--

367
00:21:57,642 --> 00:22:00,400
I almost think that's a typo.

368
00:22:00,400 --> 00:22:03,490
Should be 1 minus alpha--

369
00:22:03,490 --> 00:22:05,180
the expenditure
share on leisure.

370
00:22:09,150 --> 00:22:16,140
But what happens if this
amount, given this full income,

371
00:22:16,140 --> 00:22:21,480
this leisure that individual
i is supposed to be consuming,

372
00:22:21,480 --> 00:22:24,450
exceeds his time endowment?

373
00:22:24,450 --> 00:22:25,825
Then you've hit
a binding corner.

374
00:22:28,820 --> 00:22:34,150
So hence the min, which is
this cannot exceed this.

375
00:22:34,150 --> 00:22:35,760
When it's less, it's no problem.

376
00:22:35,760 --> 00:22:40,830
You just when hours are
less, you do those hours.

377
00:22:40,830 --> 00:22:43,860
When hours are more,
you do the max.

378
00:22:43,860 --> 00:22:50,200
Now, let me just say this is
very natural in this problem

379
00:22:50,200 --> 00:22:52,500
when the time
endowment is fixed.

380
00:22:52,500 --> 00:22:57,390
But you could also imagine
other indivisibilities.

381
00:22:57,390 --> 00:23:00,810
You could imagine that
you could only supply

382
00:23:00,810 --> 00:23:02,280
discrete amount of hours.

383
00:23:02,280 --> 00:23:06,930
Or something like in
the US, typically you

384
00:23:06,930 --> 00:23:09,660
have 8-hour days and 40-hour
weeks, and that's it.

385
00:23:09,660 --> 00:23:12,150
And you can't vary your hours.

386
00:23:12,150 --> 00:23:15,210
So you can load other
restrictions in here.

387
00:23:15,210 --> 00:23:20,310
And in the data, there
is enormous variability

388
00:23:20,310 --> 00:23:23,230
in non-participation.

389
00:23:23,230 --> 00:23:25,510
Month by month,
households will sit out

390
00:23:25,510 --> 00:23:28,660
for substantial periods
of times and then re-enter

391
00:23:28,660 --> 00:23:29,540
the labor force.

392
00:23:29,540 --> 00:23:32,215
So something is
making that happen.

393
00:23:36,470 --> 00:23:38,000
And actually, it's
the other corner.

394
00:23:38,000 --> 00:23:42,380
It's more like zeros, not
working up to the max.

395
00:23:42,380 --> 00:23:43,420
Yes?

396
00:23:43,420 --> 00:23:47,420
AUDIENCE: So it only
gets a corner solution

397
00:23:47,420 --> 00:23:52,160
if that household's Pareto bases
is too large for that agent?

398
00:23:52,160 --> 00:23:54,440
ROBERT TOWNSEND: That's
one of the factors.

399
00:23:54,440 --> 00:23:56,450
No, too small.

400
00:23:56,450 --> 00:23:59,000
So-- oh, too large,
sorry, leisure.

401
00:23:59,000 --> 00:24:01,730
And I just made a
mistake, absolutely, yeah.

402
00:24:01,730 --> 00:24:05,180
So when the Pareto weight is
high and leisure is valued,

403
00:24:05,180 --> 00:24:07,130
then they want to
do maximum leisure.

404
00:24:07,130 --> 00:24:09,710
They set leisure equal to the
time endowment and work hours

405
00:24:09,710 --> 00:24:11,030
are zero.

406
00:24:11,030 --> 00:24:16,645
So that's one thing, but
it depends on the wage.

407
00:24:16,645 --> 00:24:18,020
And it's actually
going to depend

408
00:24:18,020 --> 00:24:21,770
on how binding the household's
resource constraint is.

409
00:24:21,770 --> 00:24:24,360
So there's two other factors.

410
00:24:24,360 --> 00:24:29,610
But you're right about
the Pareto weight.

411
00:24:29,610 --> 00:24:33,800
So we can solve out for the
value functions in these work

412
00:24:33,800 --> 00:24:35,390
and no work branches.

413
00:24:35,390 --> 00:24:39,280
I'm not sure that is
very enlightening.

414
00:24:39,280 --> 00:24:41,840
There is going to be a
critical wage, as I was just

415
00:24:41,840 --> 00:24:45,080
anticipating, where that
constraint is binding

416
00:24:45,080 --> 00:24:50,480
or not if the wages are higher
than this critical number,

417
00:24:50,480 --> 00:24:53,090
the individual would be working.

418
00:24:53,090 --> 00:24:55,110
And if it's lower than
that critical number,

419
00:24:55,110 --> 00:24:57,290
they're not going to be working.

420
00:24:57,290 --> 00:24:58,550
So there is a threshold.

421
00:25:03,530 --> 00:25:06,020
And it turns out it's
smooth as the wage-- there's

422
00:25:06,020 --> 00:25:08,630
no jumps in this
problem, at least.

423
00:25:08,630 --> 00:25:10,490
You can see it in
the value functions.

424
00:25:10,490 --> 00:25:14,000
You can see it in the
way the wage moves.

425
00:25:14,000 --> 00:25:16,760
But again, I don't
think it's worth getting

426
00:25:16,760 --> 00:25:20,870
bogged down in the algebra
to make that point.

427
00:25:24,010 --> 00:25:26,820
In other words, hours
will kindly smoothly

428
00:25:26,820 --> 00:25:29,610
stay at zero until the wage
goes above this threshold.

429
00:25:29,610 --> 00:25:35,920
And then hours will start
to pick up in a regular way.

430
00:25:35,920 --> 00:25:39,940
So here's what I just
mentioned to Yeng,

431
00:25:39,940 --> 00:25:44,170
which is that households are
less likely to participate when

432
00:25:44,170 --> 00:25:50,610
the wage is low, when the
household is doing well,

433
00:25:50,610 --> 00:25:54,410
and when the Pareto
weight is large.

434
00:25:54,410 --> 00:25:58,310
Or conversely, you'll see
greater participation,

435
00:25:58,310 --> 00:26:02,400
and condition on
participation, greater hours

436
00:26:02,400 --> 00:26:07,040
when the wage is high, when
the household is doing badly,

437
00:26:07,040 --> 00:26:15,670
meaning that the shadow price on
the budget constraint is high,

438
00:26:15,670 --> 00:26:18,160
and when this person's
Pareto weight is small.

439
00:26:18,160 --> 00:26:19,630
You know, make the slaves work.

440
00:26:26,460 --> 00:26:27,480
This looks awful.

441
00:26:30,780 --> 00:26:33,330
It supposed to be a
good thing that we

442
00:26:33,330 --> 00:26:37,740
are able to analytically solve.

443
00:26:37,740 --> 00:26:40,080
So let me highlight something.

444
00:26:40,080 --> 00:26:44,190
There's two equations
on the slide.

445
00:26:44,190 --> 00:26:48,130
There's a reservation wage
and an hour's equation.

446
00:26:48,130 --> 00:26:48,630
OK?

447
00:26:53,810 --> 00:26:55,150
Actually look down here first.

448
00:26:59,860 --> 00:27:03,450
The reservation wage for
individual i in the household h

449
00:27:03,450 --> 00:27:07,440
has a constant
term and something

450
00:27:07,440 --> 00:27:14,230
that depends on the shadow
price of the budget.

451
00:27:14,230 --> 00:27:16,770
Now, when we did the
consumption risk sharing thing,

452
00:27:16,770 --> 00:27:20,070
we talked about the shadow
price of consumption.

453
00:27:20,070 --> 00:27:21,300
I'm kind of like tongue tied.

454
00:27:21,300 --> 00:27:22,920
I don't quite know
what to call it.

455
00:27:22,920 --> 00:27:25,620
It's more than
consumption, because we

456
00:27:25,620 --> 00:27:26,970
have leisure in this problem.

457
00:27:26,970 --> 00:27:29,050
But we did have a
budget constraint.

458
00:27:29,050 --> 00:27:32,340
So it's the shadow price
on the budget constraint.

459
00:27:32,340 --> 00:27:36,480
And there's no H on this guy,
because this thing represents

460
00:27:36,480 --> 00:27:39,570
that all the individuals and all
the households in the village

461
00:27:39,570 --> 00:27:41,920
are pooling all this risk.

462
00:27:41,920 --> 00:27:45,270
So there's like a common
community level resource

463
00:27:45,270 --> 00:27:46,740
constraint.

464
00:27:46,740 --> 00:27:50,510
That's the content
of the theory.

465
00:27:50,510 --> 00:27:55,580
And here's hours actually
worked when working.

466
00:27:55,580 --> 00:27:57,780
And again, it has
a constant term.

467
00:27:57,780 --> 00:28:03,170
It has this same shadow
price object, DS,

468
00:28:03,170 --> 00:28:06,200
in it, and also on the wage.

469
00:28:06,200 --> 00:28:08,760
Now, the wage just
reflects the fact that,

470
00:28:08,760 --> 00:28:11,720
as we said before,
the higher the wage,

471
00:28:11,720 --> 00:28:15,110
the more you're likely
to make someone work.

472
00:28:28,460 --> 00:28:32,450
And so now we can maybe get
the courage to go back up

473
00:28:32,450 --> 00:28:34,910
and look at what all
these objects are.

474
00:28:34,910 --> 00:28:36,710
Look at all the
beautiful restrictions

475
00:28:36,710 --> 00:28:42,230
that theory is placing on
these regression coefficients.

476
00:28:42,230 --> 00:28:44,750
First of all, why
all the i's and h's?

477
00:28:44,750 --> 00:28:48,380
Well, we allowed a
lot of heterogeneity

478
00:28:48,380 --> 00:28:51,410
in those Cobb-Douglas
coefficients and the curvatures

479
00:28:51,410 --> 00:28:52,760
for each of the households.

480
00:28:58,640 --> 00:29:02,560
Let's spot some familiar--
this is all preference stuff.

481
00:29:09,990 --> 00:29:13,330
This is cool for me.

482
00:29:15,960 --> 00:29:19,830
This is the log of the
probability weighted shadow

483
00:29:19,830 --> 00:29:26,190
price in the whole village.

484
00:29:26,190 --> 00:29:26,970
And what's this?

485
00:29:26,970 --> 00:29:29,020
This is the risk aversion.

486
00:29:29,020 --> 00:29:32,770
So you remember that stuff
about risk tolerance?

487
00:29:32,770 --> 00:29:40,050
So, you know, the greater gamma,
the less risk tolerant they

488
00:29:40,050 --> 00:29:40,740
are--

489
00:29:40,740 --> 00:29:43,085
sorry, going the wrong way.

490
00:29:43,085 --> 00:29:46,730
This gamma is the curvature
in the utility function.

491
00:29:46,730 --> 00:29:49,890
So-- oh, there's a
negative sign there, right.

492
00:29:49,890 --> 00:29:53,400
So basically what it's
saying is the more risk

493
00:29:53,400 --> 00:29:56,370
averse that an
individual is, the less

494
00:29:56,370 --> 00:29:59,310
you should see that its
hours are moving around

495
00:29:59,310 --> 00:30:00,660
with this aggregate state.

496
00:30:03,920 --> 00:30:07,780
So that's the leisure version
of the consumption equations

497
00:30:07,780 --> 00:30:08,470
we had before.

498
00:30:24,980 --> 00:30:28,340
I mean here's Pareto weights
in here, for example,

499
00:30:28,340 --> 00:30:31,610
in the critical wage threshold.

500
00:30:31,610 --> 00:30:33,590
This is the Pareto
weight of the household.

501
00:30:33,590 --> 00:30:36,020
This is the Pareto
weight of the individual.

502
00:30:36,020 --> 00:30:39,260
And the higher this thing
is, the higher that wage

503
00:30:39,260 --> 00:30:41,810
is going to be, because
that's basically

504
00:30:41,810 --> 00:30:44,370
the way you take your leisure.

505
00:30:44,370 --> 00:30:45,122
Yes?

506
00:30:45,122 --> 00:30:47,180
AUDIENCE: So before
we were talking

507
00:30:47,180 --> 00:30:50,690
about it's great to see all
these restrictions because

508
00:30:50,690 --> 00:30:53,340
of being familiar with the
data and doing regression.

509
00:30:53,340 --> 00:30:57,740
But we will never have data
on a Pareto weight, right?

510
00:30:57,740 --> 00:31:00,350
That's not a real thing?

511
00:31:00,350 --> 00:31:01,700
ROBERT TOWNSEND: That's true.

512
00:31:01,700 --> 00:31:02,390
But--

513
00:31:02,390 --> 00:31:04,640
AUDIENCE: If that's correlated
with other stuff that's

514
00:31:04,640 --> 00:31:06,830
going to start moving
around the other betas

515
00:31:06,830 --> 00:31:09,553
that we're estimating on
the stuff we can see, right?

516
00:31:09,553 --> 00:31:11,720
ROBERT TOWNSEND: Yeah, but
basically-- that's right.

517
00:31:11,720 --> 00:31:13,850
And that's what
this is basically.

518
00:31:13,850 --> 00:31:17,510
So you start looking at what
the regression coefficients are.

519
00:31:20,770 --> 00:31:24,970
For example, here's this
one thing, the risk aversion

520
00:31:24,970 --> 00:31:28,900
of individual i and household
h, and there's an F on there.

521
00:31:28,900 --> 00:31:30,960
So where was the F thing?

522
00:31:30,960 --> 00:31:34,450
The F thing was what I was
highlighting right here.

523
00:31:37,830 --> 00:31:39,290
So now this is a fixed effect.

524
00:31:39,290 --> 00:31:40,430
We don't see it.

525
00:31:40,430 --> 00:31:43,370
But you can put a time
bearing fixed effect

526
00:31:43,370 --> 00:31:45,290
for the whole village.

527
00:31:45,290 --> 00:31:48,170
And then estimate this
regression coefficient.

528
00:31:48,170 --> 00:31:51,830
And the theory is telling you
that's 1 over the coefficient

529
00:31:51,830 --> 00:31:53,750
of relative risk aversion.

530
00:31:53,750 --> 00:31:58,610
And so now you can work
backwards, you know--

531
00:32:03,268 --> 00:32:04,810
AUDIENCE: Wouldn't
that be correlated

532
00:32:04,810 --> 00:32:06,448
with the Pareto weight on me?

533
00:32:06,448 --> 00:32:07,750
ROBERT TOWNSEND: Mm, hmm.

534
00:32:07,750 --> 00:32:10,270
No.

535
00:32:10,270 --> 00:32:12,840
No, we didn't see that in the--

536
00:32:12,840 --> 00:32:14,970
AUDIENCE: Oh, no,
sorry, what I meant

537
00:32:14,970 --> 00:32:17,730
is if I don't have
the Pareto weight--

538
00:32:17,730 --> 00:32:19,635
so that'll be like
fixed effect, right?

539
00:32:19,635 --> 00:32:21,760
ROBERT TOWNSEND: The Pareto
weights is like a fix--

540
00:32:21,760 --> 00:32:24,140
this is a time
bearing fixed effect.

541
00:32:24,140 --> 00:32:26,070
And the Pareto weights
are going to show up--

542
00:32:30,705 --> 00:32:31,330
where are they?

543
00:32:31,330 --> 00:32:32,860
Over here.

544
00:32:32,860 --> 00:32:34,142
Here's one.

545
00:32:34,142 --> 00:32:35,350
AUDIENCE: Can you have some--

546
00:32:35,350 --> 00:32:37,642
ROBERT TOWNSEND: This is the
households' Pareto weight.

547
00:32:37,642 --> 00:32:39,790
This is the individual's
Pareto weight.

548
00:32:39,790 --> 00:32:42,850
Granted there's a little bit
of these alphas to worry about.

549
00:32:42,850 --> 00:32:44,540
So we'll have to grab them.

550
00:32:44,540 --> 00:32:47,170
But part of this
fixed effect, this

551
00:32:47,170 --> 00:32:51,990
is not varying with time
or states or anything else.

552
00:32:51,990 --> 00:32:54,240
AUDIENCE: So you can have
like an individual effect is

553
00:32:54,240 --> 00:32:55,370
what you're saying.

554
00:32:55,370 --> 00:32:56,287
ROBERT TOWNSEND: Yeah.

555
00:33:00,490 --> 00:33:02,500
So that's sort
of-- the algebra is

556
00:33:02,500 --> 00:33:05,290
if you had all these regression
coefficients and so on,

557
00:33:05,290 --> 00:33:13,450
then you basically can identify
almost all these parameters.

558
00:33:13,450 --> 00:33:15,780
The thing that you
can actually get

559
00:33:15,780 --> 00:33:21,050
is the absolute value
of the risk tolerance.

560
00:33:21,050 --> 00:33:22,840
And we actually
saw that earlier.

561
00:33:22,840 --> 00:33:25,440
It's the ratio of the risk
tolerance of household i

562
00:33:25,440 --> 00:33:28,560
relative to other members
that you're actually

563
00:33:28,560 --> 00:33:30,270
able to identify.

564
00:33:30,270 --> 00:33:34,300
You can see who's more risk
averse than others, but not--

565
00:33:34,300 --> 00:33:39,360
OK, and actually the paper
itself goes through--

566
00:33:39,360 --> 00:33:40,090
it's long.

567
00:33:40,090 --> 00:33:42,060
I'm afraid it's quite tedious--

568
00:33:42,060 --> 00:33:44,160
goes through, you know,
this parametric example.

569
00:33:44,160 --> 00:33:46,650
It actually goes through
non-parametric stuff.

570
00:33:46,650 --> 00:33:50,310
Pierre-Andre has figured
out with his co-authors

571
00:33:50,310 --> 00:33:53,760
how to identify tons of stuff,
just based on labor supply,

572
00:33:53,760 --> 00:33:57,060
even though there's all
this underlying consumption

573
00:33:57,060 --> 00:33:58,820
stuff going on.

574
00:33:58,820 --> 00:34:02,560
You use data from Britain.

575
00:34:02,560 --> 00:34:05,010
OK.

576
00:34:05,010 --> 00:34:06,491
So yep?

577
00:34:06,491 --> 00:34:08,699
AUDIENCE: So you starting
out with the multiplication

578
00:34:08,699 --> 00:34:13,309
that consumption will
have to equal to waste.

579
00:34:13,309 --> 00:34:14,790
ROBERT TOWNSEND: Yeah.

580
00:34:14,790 --> 00:34:18,643
AUDIENCE: But your
paper in 1994--

581
00:34:18,643 --> 00:34:19,560
ROBERT TOWNSEND: Yeah.

582
00:34:19,560 --> 00:34:21,822
AUDIENCE: So how do you
reconcile with this?

583
00:34:25,190 --> 00:34:28,090
ROBERT TOWNSEND: Well, I mean,
the first answer was, you know,

584
00:34:28,090 --> 00:34:30,889
I assumed the problem away.

585
00:34:30,889 --> 00:34:31,909
Actually, not quite.

586
00:34:31,909 --> 00:34:35,989
If you look, you'll see that
I had put measures of hours

587
00:34:35,989 --> 00:34:38,840
in the consumption equations.

588
00:34:38,840 --> 00:34:41,150
I was aware of this even then.

589
00:34:44,750 --> 00:34:47,820
But it certainly would be
easy to make a mistake.

590
00:34:47,820 --> 00:34:52,260
And people are often critical,
like puzzled, how can you

591
00:34:52,260 --> 00:34:54,940
say income shouldn't matter?

592
00:34:54,940 --> 00:34:56,880
And here, we are
distinguishing the income

593
00:34:56,880 --> 00:34:59,520
that's coming from labor
supply versus the income that's

594
00:34:59,520 --> 00:35:01,230
non-labor.

595
00:35:01,230 --> 00:35:02,460
And it matters.

596
00:35:02,460 --> 00:35:05,450
The theory places
restrictions on how.

597
00:35:05,450 --> 00:35:08,650
So in some sense, consumption
does move with income.

598
00:35:08,650 --> 00:35:12,150
If the wage is moving income,
then it's moving consumption.

599
00:35:12,150 --> 00:35:14,874
And it's moving
consumption of everyone.

600
00:35:14,874 --> 00:35:17,580
AUDIENCE: Not quite generally.

601
00:35:17,580 --> 00:35:20,370
ROBERT TOWNSEND: No, it's just
from the first order conditions

602
00:35:20,370 --> 00:35:23,550
of the optimum problem,
the baby example

603
00:35:23,550 --> 00:35:25,433
that I gave at the
very beginning.

604
00:35:31,110 --> 00:35:35,150
AUDIENCE: So why are we
testing a social planning

605
00:35:35,150 --> 00:35:37,750
problem instead of
competitive equilibrium.

606
00:35:37,750 --> 00:35:41,365
Is that because Thailand
economy is centrally--

607
00:35:41,365 --> 00:35:43,240
ROBERT TOWNSEND: There
is no social planning.

608
00:35:43,240 --> 00:35:46,030
AUDIENCE: Isn't just to see how
close we get to the benchmark?

609
00:35:46,030 --> 00:35:47,800
ROBERT TOWNSEND: Yeah.

610
00:35:47,800 --> 00:35:50,080
First of all, it's not
a particular benchmark,

611
00:35:50,080 --> 00:35:53,680
because those fixed effects--

612
00:35:53,680 --> 00:35:59,264
we want to see where we are
in the whole frontier, right?

613
00:35:59,264 --> 00:36:01,295
AUDIENCE: I guess there's
a menu of benchmarks.

614
00:36:01,295 --> 00:36:02,170
ROBERT TOWNSEND: Yes.

615
00:36:02,170 --> 00:36:05,680
So the idea is somehow like
sort of a closed thing,

616
00:36:05,680 --> 00:36:10,180
that you wouldn't leave,
quote, "surplus on the table."

617
00:36:10,180 --> 00:36:13,450
That there must be
markets or institutions

618
00:36:13,450 --> 00:36:18,010
or other arrangements that lead
them toward efficient outcomes.

619
00:36:18,010 --> 00:36:20,560
But, yeah, it's not like--

620
00:36:20,560 --> 00:36:23,520
it's interesting that you're
against the social planner.

621
00:36:23,520 --> 00:36:31,390
It's-- and I often slip, and
you caught me the other day,

622
00:36:31,390 --> 00:36:33,910
saying, you know, social
planner this and that when I did

623
00:36:33,910 --> 00:36:37,060
the capital asset model.

624
00:36:37,060 --> 00:36:39,250
That's just a metaphor for--

625
00:36:39,250 --> 00:36:41,440
it's as if the
community as a whole

626
00:36:41,440 --> 00:36:46,380
is allocating resources so
as to not lose any surplus.

627
00:36:46,380 --> 00:36:49,720
Now, you know, exactly
what the institution--

628
00:36:49,720 --> 00:36:51,630
so I should tell you
about what we find.

629
00:36:51,630 --> 00:36:53,550
First of all, in these
villages, there's

630
00:36:53,550 --> 00:36:57,090
something called labor
exchange arrangements.

631
00:36:57,090 --> 00:36:59,640
And it's partly
with the rice thing.

632
00:36:59,640 --> 00:37:03,240
You know, the low land
gets flooded early.

633
00:37:03,240 --> 00:37:08,310
They all go down and help
that guy farm his land.

634
00:37:08,310 --> 00:37:13,450
And then as the water rises,
they flood the other plots.

635
00:37:13,450 --> 00:37:16,420
But they don't charge
each other for the labor.

636
00:37:16,420 --> 00:37:20,120
It looks like free
labor exchange.

637
00:37:20,120 --> 00:37:23,410
And it doesn't necessarily
balance out either.

638
00:37:23,410 --> 00:37:26,530
Now, I'm not saying in any given
economy you'd be lucky enough

639
00:37:26,530 --> 00:37:28,820
to observe something like that.

640
00:37:28,820 --> 00:37:31,120
But in these villages
you actually do

641
00:37:31,120 --> 00:37:36,370
have a real institutional
counterpart that might explain

642
00:37:36,370 --> 00:37:38,710
how they managed to do so well.

643
00:37:38,710 --> 00:37:39,290
Yes?

644
00:37:39,290 --> 00:37:41,665
AUDIENCE: So I'm not sure it
would matter for this model.

645
00:37:41,665 --> 00:37:43,410
But the story you
told is that there's

646
00:37:43,410 --> 00:37:45,160
several risk
sharing arrangements

647
00:37:45,160 --> 00:37:46,250
for a different domain.

648
00:37:46,250 --> 00:37:48,490
That one was labor
opportunities.

649
00:37:48,490 --> 00:37:51,500
Maybe there's a separate
one for consumption.

650
00:37:51,500 --> 00:37:54,100
Would it matter-- from
what perspective would it

651
00:37:54,100 --> 00:37:56,250
matter if there were
one unified arrangement?

652
00:37:56,250 --> 00:37:58,130
ROBERT TOWNSEND: No,
it wouldn't matter.

653
00:37:58,130 --> 00:38:01,480
But, you know, it's nice
to be able to say something

654
00:38:01,480 --> 00:38:03,610
about what the mechanism is.

655
00:38:03,610 --> 00:38:06,310
You know, for some people
just looking at outcomes

656
00:38:06,310 --> 00:38:08,480
isn't totally convincing.

657
00:38:08,480 --> 00:38:09,940
I must say when
I started this, I

658
00:38:09,940 --> 00:38:13,480
thought it was a virtue to
say we don't care how it's

659
00:38:13,480 --> 00:38:18,100
done, I'm not going to look
separately as savings accounts,

660
00:38:18,100 --> 00:38:22,150
selling bullocks, borrowing,
running down your grain stores,

661
00:38:22,150 --> 00:38:24,700
let's just look at the total.

662
00:38:24,700 --> 00:38:30,140
But still people want to know
how it could possibly happen.

663
00:38:30,140 --> 00:38:31,970
So what's going
on in these data?

664
00:38:31,970 --> 00:38:35,590
Well, I spare you
all the details.

665
00:38:35,590 --> 00:38:42,510
But we put non-labor income
on the right-hand side

666
00:38:42,510 --> 00:38:46,890
of the participation equations
and the labor supply equations.

667
00:38:46,890 --> 00:38:53,290
And the coefficient on it is
small and quite negligible.

668
00:38:53,290 --> 00:38:55,080
I mean, it's not zero.

669
00:38:55,080 --> 00:38:56,910
So we reject.

670
00:38:56,910 --> 00:38:59,730
But, you know, if
you come at this--

671
00:38:59,730 --> 00:39:02,700
and we'll see this
in Seema's paper--

672
00:39:02,700 --> 00:39:06,870
if you come at this from the
traditional point of view

673
00:39:06,870 --> 00:39:10,920
in development, it would be
like labor supply is your safety

674
00:39:10,920 --> 00:39:12,450
net.

675
00:39:12,450 --> 00:39:16,270
If you're getting your income
is really low from crops,

676
00:39:16,270 --> 00:39:19,580
then migrate out,
work for wages.

677
00:39:19,580 --> 00:39:21,510
And this says,
well, not so fast.

678
00:39:21,510 --> 00:39:25,860
It might be done at
a community level.

679
00:39:25,860 --> 00:39:29,010
And the test of it would
be whether how well you're

680
00:39:29,010 --> 00:39:32,580
doing on your own land has any
influence on your participation

681
00:39:32,580 --> 00:39:33,540
and your labor supply.

682
00:39:33,540 --> 00:39:34,980
If they're pooling
risk like this,

683
00:39:34,980 --> 00:39:38,430
it shouldn't once you control
for, say, village level

684
00:39:38,430 --> 00:39:40,250
fixed effects.

685
00:39:40,250 --> 00:39:42,110
And, in fact, you reject.

686
00:39:42,110 --> 00:39:46,710
But the coefficients are tiny.

687
00:39:46,710 --> 00:39:50,960
They really are sharing a lot
of this risk, not all of it,

688
00:39:50,960 --> 00:39:52,030
but a lot of it.

689
00:39:55,530 --> 00:40:00,790
OK, one last thing just to
tie some of this literature

690
00:40:00,790 --> 00:40:03,420
to-- oh, god.

691
00:40:03,420 --> 00:40:09,810
So this thing, this is
like a famous parameter,

692
00:40:09,810 --> 00:40:12,500
because it's called
the Frisch elasticity.

693
00:40:12,500 --> 00:40:19,500
And that brings to the next
part of this lecture, which

694
00:40:19,500 --> 00:40:24,930
is this tension between the
small estimates, micro level

695
00:40:24,930 --> 00:40:30,690
estimates, of labor elasticities
compared to what we seem

696
00:40:30,690 --> 00:40:36,840
to need in a model to
get aggregate hours

697
00:40:36,840 --> 00:40:40,890
and participation and
move around so much.

698
00:40:40,890 --> 00:40:45,895
Now, this literature spans
micro and macro and development.

699
00:40:48,570 --> 00:40:57,540
From the macro side,
Chetty reviewed all of it.

700
00:40:57,540 --> 00:41:00,400
But, you know,
Sargent, for example,

701
00:41:00,400 --> 00:41:05,320
but way back to Rogerson
and Hansen talk about a way

702
00:41:05,320 --> 00:41:09,020
I'm going to show you
an indivisibility that

703
00:41:09,020 --> 00:41:14,270
makes the overall aggregate
elasticities a lot

704
00:41:14,270 --> 00:41:19,190
higher than you would get if
you looked at those individuals

705
00:41:19,190 --> 00:41:19,990
in the data.

706
00:41:19,990 --> 00:41:22,190
So again, just to
anticipate, this

707
00:41:22,190 --> 00:41:26,040
is a bit like zooming
out and zooming in.

708
00:41:26,040 --> 00:41:27,440
You zoom in to the individuals.

709
00:41:27,440 --> 00:41:29,120
If you have labor
supply and so on,

710
00:41:29,120 --> 00:41:32,270
you can estimate their
micro level elasticities.

711
00:41:32,270 --> 00:41:34,430
And there's just
tons of literature,

712
00:41:34,430 --> 00:41:36,530
which I suppressed
from the slides,

713
00:41:36,530 --> 00:41:40,220
because it will overwhelm
you of the interest

714
00:41:40,220 --> 00:41:42,770
in estimating
those elasticities.

715
00:41:42,770 --> 00:41:44,990
But you could say all
these individuals reside

716
00:41:44,990 --> 00:41:46,910
in a village or district.

717
00:41:46,910 --> 00:41:50,240
Let's look at what happens
when there's a rainfall shock.

718
00:41:50,240 --> 00:41:52,580
Let's look at aggregate--

719
00:41:52,580 --> 00:41:56,120
how much the wage is moving
as a function of the rainfall

720
00:41:56,120 --> 00:41:59,750
shock and try to
estimate from there

721
00:41:59,750 --> 00:42:03,260
how inelastic labor supply is.

722
00:42:03,260 --> 00:42:05,720
And it's natural that
aggregate up over individuals

723
00:42:05,720 --> 00:42:08,060
to get an aggregate
labor supply if you're

724
00:42:08,060 --> 00:42:14,000
talking about how district
wages move around with rainfall.

725
00:42:14,000 --> 00:42:15,930
Well, that's Seema's
paper at the end.

726
00:42:15,930 --> 00:42:20,230
She's seemingly not aware,
and she didn't make a mistake.

727
00:42:20,230 --> 00:42:23,680
But it would be quite easy
to make a mistake if you were

728
00:42:23,680 --> 00:42:28,160
quantitatively trying to
reconcile the data that you

729
00:42:28,160 --> 00:42:31,400
use in the aggregate with
the micro level elasticities.

730
00:42:31,400 --> 00:42:34,310
And so that's, as
it turns out, been

731
00:42:34,310 --> 00:42:41,510
the focus of this of
this macro literature, so

732
00:42:41,510 --> 00:42:42,770
many elasticities.

733
00:42:45,467 --> 00:42:47,300
And partly because the
different literatures

734
00:42:47,300 --> 00:42:49,270
developed different jargon.

735
00:42:49,270 --> 00:42:53,320
There is the
Hisksian elasticities

736
00:42:53,320 --> 00:42:56,350
and the Frisch elasticities.

737
00:42:56,350 --> 00:43:01,210
A Hisksian elasticity is
basically compensated.

738
00:43:01,210 --> 00:43:03,490
So when you have
a wage change, you

739
00:43:03,490 --> 00:43:04,840
don't just look at total hours.

740
00:43:04,840 --> 00:43:08,320
You push people back to the
original indifference curve

741
00:43:08,320 --> 00:43:10,480
and slide along it.

742
00:43:10,480 --> 00:43:12,910
That's Hicks.

743
00:43:12,910 --> 00:43:16,510
Frisch has to do with
variation that you

744
00:43:16,510 --> 00:43:19,330
would see if you held the
marginal utility of wealth

745
00:43:19,330 --> 00:43:22,000
constant.

746
00:43:22,000 --> 00:43:23,920
Why does that come up for us?

747
00:43:23,920 --> 00:43:25,690
Oh, risk sharing.

748
00:43:25,690 --> 00:43:28,960
The marginal utility
of wealth is basically

749
00:43:28,960 --> 00:43:32,980
at each date and state constant
over all the individuals.

750
00:43:32,980 --> 00:43:37,300
It's a very natural object and
related to the risk sharing

751
00:43:37,300 --> 00:43:38,560
literature.

752
00:43:38,560 --> 00:43:41,410
And then macro, they call--

753
00:43:41,410 --> 00:43:43,450
just to add Frisch elasticity.

754
00:43:43,450 --> 00:43:46,390
But it's of aggregate hours.

755
00:43:46,390 --> 00:43:49,660
And every one of these things
has extensive and intensive.

756
00:43:49,660 --> 00:43:53,800
And that's because there's a
big variation in participation,

757
00:43:53,800 --> 00:43:58,060
not just hours in any data
that people have looked at.

758
00:43:58,060 --> 00:44:00,370
I'm not sure Chetty did this.

759
00:44:00,370 --> 00:44:02,800
I'm not so sure
how helpful it is.

760
00:44:05,510 --> 00:44:11,450
But the idea is the Hicks
effect is like shifting a wage.

761
00:44:11,450 --> 00:44:14,420
And you're going to shift it--

762
00:44:14,420 --> 00:44:15,890
if you're going to
distinguish age,

763
00:44:15,890 --> 00:44:19,220
you shift it over the
entire age profile.

764
00:44:19,220 --> 00:44:21,410
That's the way he
thinks about it.

765
00:44:21,410 --> 00:44:23,660
Whereas he thinks about
the Frisch elasticity

766
00:44:23,660 --> 00:44:25,835
as just one little instant.

767
00:44:28,670 --> 00:44:31,280
I have trouble relating,
but anyway this is--

768
00:44:35,240 --> 00:44:37,570
and then there's this table.

769
00:44:37,570 --> 00:44:50,460
These are micro estimates of
the intensive means conditioned

770
00:44:50,460 --> 00:44:54,180
on working, how much your hours
are varying with your wage.

771
00:44:54,180 --> 00:44:56,790
These are, quote unquote,
"relatively low" numbers,

772
00:44:56,790 --> 00:45:01,840
0.3, 0.5.

773
00:45:01,840 --> 00:45:08,040
The extensive margin is in the
micro data also not big, 0.1,

774
00:45:08,040 --> 00:45:12,980
0.2.

775
00:45:12,980 --> 00:45:17,960
What this literature is
about is aggregate hours.

776
00:45:17,960 --> 00:45:22,220
Aggregate hours have
higher elasticities

777
00:45:22,220 --> 00:45:23,590
than any of those numbers.

778
00:45:23,590 --> 00:45:29,570
And in particular this
thing, this macro level,

779
00:45:29,570 --> 00:45:31,325
Frisch elasticity is huge.

780
00:45:36,850 --> 00:45:44,440
So in order to generate this
number that we see in the data,

781
00:45:44,440 --> 00:45:47,620
and keep the micro
parameters that you're

782
00:45:47,620 --> 00:45:51,630
going to fit in, to be
consistent with that you have

783
00:45:51,630 --> 00:45:57,630
to generate a huge elasticity
on the extensive margin somehow.

784
00:46:00,330 --> 00:46:05,255
So this is about trying to
reconcile micro with macro.

785
00:46:05,255 --> 00:46:07,710
AUDIENCE: Is that
bottom line, they're

786
00:46:07,710 --> 00:46:09,940
not estimates from the data?

787
00:46:09,940 --> 00:46:11,410
ROBERT TOWNSEND: Which one?

788
00:46:11,410 --> 00:46:12,532
AUDIENCE: The bottom line.

789
00:46:12,532 --> 00:46:13,990
ROBERT TOWNSEND:
Yeah, they're not.

790
00:46:13,990 --> 00:46:17,000
Although this is like that.

791
00:46:17,000 --> 00:46:19,890
So let's respect the
data on that dimension.

792
00:46:19,890 --> 00:46:23,230
And somehow create a very
big number here somehow--

793
00:46:23,230 --> 00:46:25,360
and I'll tell you
how in a minute--

794
00:46:25,360 --> 00:46:28,000
in order to get what we
see in the data over here.

795
00:46:31,120 --> 00:46:32,900
Now, in the original slides--

796
00:46:32,900 --> 00:46:34,220
I'll give them to you--

797
00:46:34,220 --> 00:46:36,310
there's like six
slides that now come

798
00:46:36,310 --> 00:46:39,860
you know with all these other
who did this and which author

799
00:46:39,860 --> 00:46:40,430
and so on.

800
00:46:40,430 --> 00:46:45,440
And I just felt that was going
to be bewildering and certainly

801
00:46:45,440 --> 00:46:46,310
time consuming.

802
00:46:46,310 --> 00:46:47,220
So I'm skipping it.

803
00:46:50,050 --> 00:46:52,930
But here's what I
already went through.

804
00:46:59,670 --> 00:47:01,140
And I guess I said this too.

805
00:47:10,390 --> 00:47:13,030
So the punch line
is going to be if we

806
00:47:13,030 --> 00:47:17,680
have a non-convexity
in participation,

807
00:47:17,680 --> 00:47:20,170
like you either have to work
full-time or not at all,

808
00:47:20,170 --> 00:47:20,800
is an extreme.

809
00:47:23,830 --> 00:47:27,190
And then we go through the
mechanics of how people

810
00:47:27,190 --> 00:47:29,260
would behave in that economy.

811
00:47:29,260 --> 00:47:33,580
We can generate a sort of
pseudo representative consumer.

812
00:47:33,580 --> 00:47:35,830
So that it looks as
if the aggregate are

813
00:47:35,830 --> 00:47:41,543
generated as a solution to
that representative household.

814
00:47:41,543 --> 00:47:43,460
And the trick is that
representative household

815
00:47:43,460 --> 00:47:47,830
is going to end up with
a Frisch elasticity

816
00:47:47,830 --> 00:47:54,100
in the pseudo function, which
will allow huge variability

817
00:47:54,100 --> 00:47:55,555
in aggregate hours.

818
00:47:55,555 --> 00:47:58,016
It responds to wages.

819
00:47:58,016 --> 00:47:59,690
Yep.

820
00:47:59,690 --> 00:48:05,810
AUDIENCE: This doesn't match
the extensive margin micro data.

821
00:48:05,810 --> 00:48:07,809
ROBERT TOWNSEND: Because
it doesn't have to.

822
00:48:07,809 --> 00:48:08,392
AUDIENCE: Why?

823
00:48:11,287 --> 00:48:13,620
ROBERT TOWNSEND: Because you
can populate the underlying

824
00:48:13,620 --> 00:48:18,180
individuals with actual observed
micro estimates and then say

825
00:48:18,180 --> 00:48:21,360
there is this inelasticity.

826
00:48:21,360 --> 00:48:23,370
I'll show you.

827
00:48:23,370 --> 00:48:23,970
I'll show you.

828
00:48:23,970 --> 00:48:31,060
OK, so we have labor,
capital, and output.

829
00:48:31,060 --> 00:48:34,160
Let's just do a single period.

830
00:48:34,160 --> 00:48:36,650
Imagine for a minute it's
not stochastic either.

831
00:48:36,650 --> 00:48:42,150
We have an aggregate
production function,

832
00:48:42,150 --> 00:48:48,737
as a function of aggregate
capital and aggregate hours

833
00:48:48,737 --> 00:48:49,570
Why are we doing it?

834
00:48:49,570 --> 00:48:52,600
This is basically
just going to allow

835
00:48:52,600 --> 00:48:55,210
us to close the model
instead of having

836
00:48:55,210 --> 00:48:56,720
it open partial equilibrium.

837
00:48:56,720 --> 00:48:59,050
So we're going to get the
wage and interest rate off

838
00:48:59,050 --> 00:49:00,800
of this thing.

839
00:49:00,800 --> 00:49:03,990
It's convenient.

840
00:49:03,990 --> 00:49:10,750
There are a continuum of
individuals indexed by i.

841
00:49:10,750 --> 00:49:14,125
Each individual has one unit
of time, one unit of capital.

842
00:49:18,040 --> 00:49:19,277
Time is indivisible.

843
00:49:19,277 --> 00:49:21,610
And you got to choose, you're
either going to do leisure

844
00:49:21,610 --> 00:49:23,300
or you're going to do work.

845
00:49:23,300 --> 00:49:27,358
So that's an extreme version
of the indivisibility.

846
00:49:30,350 --> 00:49:33,560
They care about
consumption and leisure,

847
00:49:33,560 --> 00:49:35,200
or the disutility of work.

848
00:49:38,420 --> 00:49:43,380
But this work level is
either 0 or 1 by assumption.

849
00:49:43,380 --> 00:49:47,350
So let's just normalize
and say v of 0 is 0.

850
00:49:47,350 --> 00:49:49,800
But v of 1 is whatever it is.

851
00:49:49,800 --> 00:49:50,600
Let's call it m.

852
00:49:50,600 --> 00:49:52,740
It's just a real number,
because you're never

853
00:49:52,740 --> 00:49:54,240
going to see at the
individual level

854
00:49:54,240 --> 00:49:57,920
any other levels of hours.

855
00:49:57,920 --> 00:50:02,320
So remember m is the
disutility of working 1--

856
00:50:02,320 --> 00:50:03,850
putting all your time into work.

857
00:50:06,410 --> 00:50:09,560
What's the consumption
set in this economy?

858
00:50:09,560 --> 00:50:12,740
Consumption is non-negative.

859
00:50:12,740 --> 00:50:16,220
You can't supply more
capital than what you own.

860
00:50:16,220 --> 00:50:23,060
But you have this non-convexity
in the labor decision.

861
00:50:23,060 --> 00:50:23,810
It's 0, 1.

862
00:50:23,810 --> 00:50:25,580
This is not an interval 0 to 1.

863
00:50:25,580 --> 00:50:27,000
It's either 0 or 1.

864
00:50:27,000 --> 00:50:29,655
It's the set consisting
of just the two objects.

865
00:50:33,170 --> 00:50:36,920
So it's a non-convex
consumption set.

866
00:50:36,920 --> 00:50:40,895
Linear combinations of
0 and 1 are not allowed.

867
00:50:47,410 --> 00:50:53,520
OK, so what would be a
competitive equilibrium?

868
00:50:53,520 --> 00:50:57,720
Households max, firms
max, markets clear.

869
00:50:57,720 --> 00:51:00,240
Here's the household
max problem.

870
00:51:00,240 --> 00:51:05,010
For any individual i, solve
for the consumption decision

871
00:51:05,010 --> 00:51:09,300
about participation and the
capital respecting the budget

872
00:51:09,300 --> 00:51:09,870
constraint.

873
00:51:09,870 --> 00:51:13,600
Namely, you can work or not--

874
00:51:13,600 --> 00:51:17,610
so this is either 0 or something
positive-- at a given wage.

875
00:51:17,610 --> 00:51:21,810
And then you get the rental
rate on your capital stock.

876
00:51:21,810 --> 00:51:25,190
And that's that.

877
00:51:25,190 --> 00:51:30,020
Firms maximize profits,
paying factors of production.

878
00:51:30,020 --> 00:51:32,030
Clearly for a given
f, this is going

879
00:51:32,030 --> 00:51:35,970
to generate marginal
products of capital and labor

880
00:51:35,970 --> 00:51:37,050
in the whole economy.

881
00:51:45,330 --> 00:51:52,190
And if we just solve
the problem this way,

882
00:51:52,190 --> 00:51:53,540
we could get an equilibrium.

883
00:51:57,940 --> 00:51:59,560
But it might not be optimal.

884
00:52:02,100 --> 00:52:07,420
We can make people better
off by sort of randomizing

885
00:52:07,420 --> 00:52:09,830
who gets to work.

886
00:52:09,830 --> 00:52:13,120
How are we going to do that?

887
00:52:13,120 --> 00:52:17,960
We're going to basically create
this probability object phi.

888
00:52:17,960 --> 00:52:24,780
So here would be the
expected utility given a phi.

889
00:52:24,780 --> 00:52:27,470
The realization with
probability phi,

890
00:52:27,470 --> 00:52:30,190
you're going to be the
worker in this economy.

891
00:52:30,190 --> 00:52:34,700
With probability 1 minus phi,
you're just not going to work.

892
00:52:34,700 --> 00:52:38,930
And at the moment, you can
get different consumption

893
00:52:38,930 --> 00:52:40,780
for the two different branches.

894
00:52:40,780 --> 00:52:41,280
Yep?

895
00:52:43,862 --> 00:52:45,320
AUDIENCE: I sort
of understand this

896
00:52:45,320 --> 00:52:48,785
is getting at which the
market might be flexible.

897
00:52:48,785 --> 00:52:49,910
There's not much part-time.

898
00:52:49,910 --> 00:52:51,620
There's not much
intensive margin.

899
00:52:51,620 --> 00:52:54,800
So people turn to things like
household and they specialize.

900
00:52:54,800 --> 00:52:59,020
But how do you interpret
the lottery as--

901
00:52:59,020 --> 00:53:03,060
like we imposed
lumpiness of working.

902
00:53:03,060 --> 00:53:05,090
And then we use the
lottery to break it.

903
00:53:05,090 --> 00:53:10,930
Why not have some other way
of having other types of labor

904
00:53:10,930 --> 00:53:11,430
worked?

905
00:53:14,394 --> 00:53:24,200
ROBERT TOWNSEND: Um--
well, I'm about to show you

906
00:53:24,200 --> 00:53:29,390
on the next slide the decision
problem for an insurance

907
00:53:29,390 --> 00:53:30,700
company.

908
00:53:30,700 --> 00:53:33,720
And the insurance
companies like pooling over

909
00:53:33,720 --> 00:53:35,630
all the individuals.

910
00:53:35,630 --> 00:53:38,570
And it's going to maximize, and
there's going to be free entry.

911
00:53:38,570 --> 00:53:40,970
And that's going to drive
the prices of certain things

912
00:53:40,970 --> 00:53:43,790
to be equal to their
actuarial values.

913
00:53:43,790 --> 00:53:49,920
So it's as if people were
buying and selling insurance.

914
00:53:49,920 --> 00:53:51,825
And they get to choose--

915
00:53:56,330 --> 00:54:00,620
they're not forced
as if by a planner

916
00:54:00,620 --> 00:54:04,550
to work or not work depending
on how the roulette wheel turns.

917
00:54:04,550 --> 00:54:08,470
They actually get to
choose voluntarily

918
00:54:08,470 --> 00:54:14,450
whether to decide to work
full time, decide to work zero

919
00:54:14,450 --> 00:54:15,890
or be on call.

920
00:54:15,890 --> 00:54:18,470
They'd be on call, they're
going to get a wage that's

921
00:54:18,470 --> 00:54:21,410
proportional to the
probability, because that's

922
00:54:21,410 --> 00:54:23,248
the pooling aspect of it.

923
00:54:23,248 --> 00:54:24,960
AUDIENCE: Like [INAUDIBLE]

924
00:54:24,960 --> 00:54:26,830
ROBERT TOWNSEND: Yeah, so far.

925
00:54:26,830 --> 00:54:27,330
Yeah.

926
00:54:27,330 --> 00:54:30,470
AUDIENCE: I don't know that the
lottery is so great, because it

927
00:54:30,470 --> 00:54:35,240
could be we have like some of us
turn up for the job interviews,

928
00:54:35,240 --> 00:54:38,513
and the people doing the
job interviews like just--

929
00:54:38,513 --> 00:54:40,430
you know, they're picking
based on some stuff.

930
00:54:40,430 --> 00:54:42,200
Maybe it's just as
good as random, right?

931
00:54:42,200 --> 00:54:43,972
Like so--

932
00:54:43,972 --> 00:54:44,930
AUDIENCE: That's crazy.

933
00:54:44,930 --> 00:54:45,888
AUDIENCE: No, it's not.

934
00:54:48,830 --> 00:54:51,055
ROBERT TOWNSEND:
Disconcerting maybe.

935
00:54:51,055 --> 00:54:54,440
AUDIENCE: Suppose there are like
100 unskilled manual laborers

936
00:54:54,440 --> 00:54:56,930
turned up for a job that
requires 30 laborers.

937
00:54:56,930 --> 00:54:59,520
I cannot observe past
project of these laborers.

938
00:54:59,520 --> 00:55:01,360
Like I can maybe
observe cleanliness

939
00:55:01,360 --> 00:55:02,810
and they're ability to speak.

940
00:55:02,810 --> 00:55:04,133
Like that's about it.

941
00:55:04,133 --> 00:55:05,300
So what am I supposed to do.

942
00:55:05,300 --> 00:55:06,675
ROBERT TOWNSEND:
So that would be

943
00:55:06,675 --> 00:55:09,590
an institutional arrangement,
where the markets seem

944
00:55:09,590 --> 00:55:13,820
not to clear, because not
everyone gets the allocation,

945
00:55:13,820 --> 00:55:16,150
not everyone supplies labor.

946
00:55:16,150 --> 00:55:18,560
You know, they're lined
up on the street corner,

947
00:55:18,560 --> 00:55:21,200
and the truck comes by, and
there are only so many jobs.

948
00:55:21,200 --> 00:55:22,950
AUDIENCE: But what if
we were indifferent,

949
00:55:22,950 --> 00:55:24,005
then it would still be--

950
00:55:24,005 --> 00:55:26,130
ROBERT TOWNSEND: But there
are other ways to do it.

951
00:55:26,130 --> 00:55:31,540
I mean you could have sunspot.

952
00:55:31,540 --> 00:55:35,020
You can even index off
of observable things

953
00:55:35,020 --> 00:55:37,990
like rainfall.

954
00:55:37,990 --> 00:55:39,970
And then it would look
like a state contingent

955
00:55:39,970 --> 00:55:45,998
contract, where the labor supply
is a function of the rainfall.

956
00:55:45,998 --> 00:55:48,040
But that doesn't mean
everyone's supplying labor.

957
00:55:48,040 --> 00:55:49,180
They have some arrangement.

958
00:55:49,180 --> 00:55:50,350
If it's sunny, I'm working.

959
00:55:50,350 --> 00:55:53,220
If it's raining, you're working.

960
00:55:53,220 --> 00:55:56,080
And it would be
very hard to say,

961
00:55:56,080 --> 00:55:58,480
unless you were
very clear headed,

962
00:55:58,480 --> 00:56:02,410
whether it's this sort
of extraneous risk

963
00:56:02,410 --> 00:56:04,990
or some intrinsic risk,
some underlying state

964
00:56:04,990 --> 00:56:08,500
that they were trying
to insure against.

965
00:56:08,500 --> 00:56:11,355
This last interpretation
is Arrow's interpretation

966
00:56:11,355 --> 00:56:11,980
of the lottery.

967
00:56:14,650 --> 00:56:16,020
And he's entitled.

968
00:56:20,150 --> 00:56:26,810
OK, so we're going to have these
prices, rental rates and wage

969
00:56:26,810 --> 00:56:31,010
rates, normalizing
the consumption good.

970
00:56:31,010 --> 00:56:33,800
OK, here's the
insurance company thing.

971
00:56:38,620 --> 00:56:46,540
Basically, if you're working,
you pay a premium ex post.

972
00:56:46,540 --> 00:56:52,530
And if you're not working,
you get this indemnity.

973
00:56:56,180 --> 00:56:58,220
Well, I mean intuitively
if you're working,

974
00:56:58,220 --> 00:57:03,440
you have more income, you
kind contribute some back

975
00:57:03,440 --> 00:57:06,520
to the pool.

976
00:57:06,520 --> 00:57:10,760
But if an insurance company
we're offering this,

977
00:57:10,760 --> 00:57:13,880
they would potentially
make profits

978
00:57:13,880 --> 00:57:16,250
depending on the fraction
of the people from whom they

979
00:57:16,250 --> 00:57:19,520
get the premium and
the residual fraction

980
00:57:19,520 --> 00:57:21,920
to whom they pay the indemnity.

981
00:57:21,920 --> 00:57:24,140
If you have free
entry into insurance,

982
00:57:24,140 --> 00:57:28,140
then this is going to drive
this equation to zero.

983
00:57:28,140 --> 00:57:28,640
OK?

984
00:57:28,640 --> 00:57:32,030
So that's going to tie the
premia and the indemnity

985
00:57:32,030 --> 00:57:40,005
to the probability or
fraction of people working.

986
00:57:44,170 --> 00:57:47,172
So now, we're going
to decentralize it.

987
00:57:47,172 --> 00:57:49,380
We're going to say look it's
as if the household were

988
00:57:49,380 --> 00:57:54,860
maximizing expected utility, but
as I said, voluntarily deciding

989
00:57:54,860 --> 00:57:57,800
whether to work for sure or
not work at all or something

990
00:57:57,800 --> 00:57:59,240
in between.

991
00:57:59,240 --> 00:58:04,498
And then they're going to
have this resource constraint.

992
00:58:07,250 --> 00:58:09,700
But I've now substituted in--

993
00:58:19,330 --> 00:58:22,240
the x's entered
into consumption.

994
00:58:22,240 --> 00:58:25,570
But the x's we're constrained
by that zero profit condition.

995
00:58:25,570 --> 00:58:28,690
So we've substituted in
the zero profit condition

996
00:58:28,690 --> 00:58:30,700
back into the household
budget constraint

997
00:58:30,700 --> 00:58:33,790
and gotten this equation.

998
00:58:33,790 --> 00:58:35,410
Capital is equal to 1.

999
00:58:35,410 --> 00:58:38,250
So this thing's just going
to go to 1 no matter what.

1000
00:58:38,250 --> 00:58:39,670
You're going to get a wage.

1001
00:58:39,670 --> 00:58:42,310
But you're basically
going to be paid

1002
00:58:42,310 --> 00:58:45,670
the expected value of
your pre-committed labor

1003
00:58:45,670 --> 00:58:47,950
participation.

1004
00:58:47,950 --> 00:58:49,240
I agree to be on call.

1005
00:58:49,240 --> 00:58:53,030
A hospital may call me.

1006
00:58:53,030 --> 00:58:55,280
I'm going to go to
work if I get the call.

1007
00:58:55,280 --> 00:59:00,190
But my salary is based
on whether I'm on call

1008
00:59:00,190 --> 00:59:02,200
all the time or not at all.

1009
00:59:02,200 --> 00:59:03,280
And I get to choose that.

1010
00:59:03,280 --> 00:59:06,940
Somehow that's the
flexibility in the contract.

1011
00:59:06,940 --> 00:59:13,590
And this is the decentralized
version with the household

1012
00:59:13,590 --> 00:59:14,460
optimization.

1013
00:59:14,460 --> 00:59:18,520
Firms maximize profits
and markets clear.

1014
00:59:18,520 --> 00:59:21,173
That's where we're going to
get the wage, the overall wage

1015
00:59:21,173 --> 00:59:22,090
and the interest rate.

1016
00:59:26,600 --> 00:59:28,600
So if you looked at the
first order condition

1017
00:59:28,600 --> 00:59:33,230
to the consumer problem, you
would see some obvious things

1018
00:59:33,230 --> 00:59:34,610
happening.

1019
00:59:34,610 --> 00:59:40,490
You're going to have a
theta price, shadow price,

1020
00:59:40,490 --> 00:59:43,190
on the budget constraint
that I just showed you.

1021
00:59:43,190 --> 00:59:48,300
And these probabilities
are entering symmetrically

1022
00:59:48,300 --> 00:59:50,842
in utility and in consumption.

1023
00:59:50,842 --> 00:59:52,300
So when you take
a derivative, they

1024
00:59:52,300 --> 00:59:54,850
are kind of like cancel out.

1025
00:59:54,850 --> 00:59:56,710
And so you get this.

1026
00:59:56,710 --> 01:00:00,220
But, look, here, they're
going to cancel on both sides.

1027
01:00:00,220 --> 01:00:02,220
So theta is common.

1028
01:00:02,220 --> 01:00:04,807
So consumption has to be common.

1029
01:00:04,807 --> 01:00:07,140
So you're going to get full
insurance on the consumption

1030
01:00:07,140 --> 01:00:08,100
side.

1031
01:00:08,100 --> 01:00:11,040
There's no reason to
let consumption vary.

1032
01:00:11,040 --> 01:00:14,830
Remember, this was a
separable utility function.

1033
01:00:14,830 --> 01:00:17,760
It was the utility
of consumption plus--

1034
01:00:17,760 --> 01:00:22,170
so why screw up the marginal
utility of consumption?

1035
01:00:22,170 --> 01:00:25,830
You can get full smoothing out
of on the consumption side.

1036
01:00:28,680 --> 01:00:31,920
So consumption is equal to each
other, whether you work or not.

1037
01:00:35,790 --> 01:00:40,780
But that theme may not
be at the boundary.

1038
01:00:40,780 --> 01:00:43,980
It could be interior.

1039
01:00:43,980 --> 01:00:48,430
So substituting a
common consumption,

1040
01:00:48,430 --> 01:00:50,170
remembering
disutility of work is

1041
01:00:50,170 --> 01:00:56,890
just m, you have this
new optimization problem.

1042
01:00:56,890 --> 01:01:00,350
And this is equivalent
with the other one.

1043
01:01:00,350 --> 01:01:02,360
Now, where's the
aggregate elasticity?

1044
01:01:08,140 --> 01:01:10,180
What's the disutility of work?

1045
01:01:13,589 --> 01:01:14,563
AUDIENCE: Phi, right?

1046
01:01:17,490 --> 01:01:20,300
ROBERT TOWNSEND: It's linear.

1047
01:01:20,300 --> 01:01:26,170
It doesn't get any more easy
to substitute than that.

1048
01:01:26,170 --> 01:01:30,470
See here it looks like
you're choosing phi.

1049
01:01:30,470 --> 01:01:34,750
But the coefficient in front
of it is just a number, m.

1050
01:01:34,750 --> 01:01:41,720
So phi is-- you know, why
is phi look like hours?

1051
01:01:41,720 --> 01:01:44,570
Phi is the fraction of
time you're working.

1052
01:01:44,570 --> 01:01:47,810
And if you work,
you're working 1.

1053
01:01:47,810 --> 01:01:51,980
So total hours is like the
fraction of the population

1054
01:01:51,980 --> 01:01:53,630
working.

1055
01:01:53,630 --> 01:01:57,560
That varies continuously-- this
sort of pseudo representative

1056
01:01:57,560 --> 01:02:01,620
consumer thinks that it can
choose any fraction it wants,

1057
01:02:01,620 --> 01:02:04,520
even though the underlying
population cannot.

1058
01:02:04,520 --> 01:02:06,390
This is a stand in household.

1059
01:02:06,390 --> 01:02:07,640
It's not a real household.

1060
01:02:07,640 --> 01:02:11,720
But the aggregates that
are predicted from this

1061
01:02:11,720 --> 01:02:16,460
will correspond to the
underlying problem.

1062
01:02:16,460 --> 01:02:19,060
So that was the insight
that Rogerson had.

1063
01:02:29,720 --> 01:02:32,240
And actually, I'm pretty
sure it's also 1 to 1

1064
01:02:32,240 --> 01:02:34,040
with the right welfare.

1065
01:02:34,040 --> 01:02:37,700
You can use that
aggregated utility

1066
01:02:37,700 --> 01:02:40,370
as a representation of
the underlying welfare.

1067
01:02:40,370 --> 01:02:41,603
Why is that?

1068
01:02:41,603 --> 01:02:43,520
Well, even though you
have all that diversity,

1069
01:02:43,520 --> 01:02:46,560
households are
really alike in terms

1070
01:02:46,560 --> 01:02:47,820
of their utility functions.

1071
01:02:47,820 --> 01:02:50,460
So expected utility
is the right metric.

1072
01:02:50,460 --> 01:02:52,020
And that's what that thing is.

1073
01:02:52,020 --> 01:02:54,720
It just looks funny
when you write it down.

1074
01:02:58,720 --> 01:02:59,220
OK.

1075
01:03:15,860 --> 01:03:19,750
And then quickly, here's
Hansen's version of it

1076
01:03:19,750 --> 01:03:27,080
with business cycles, going
to the actual aggregate data.

1077
01:03:27,080 --> 01:03:33,400
As I said, total
hours worked is hours

1078
01:03:33,400 --> 01:03:36,190
per person of
those participating

1079
01:03:36,190 --> 01:03:39,870
times the number participating.

1080
01:03:39,870 --> 01:03:42,060
And then you take
logs of it, basically.

1081
01:03:42,060 --> 01:03:44,350
And then you want to
look at a variance.

1082
01:03:44,350 --> 01:03:47,310
You say the variance of
a log of aggregate hours

1083
01:03:47,310 --> 01:03:49,420
has three components.

1084
01:03:49,420 --> 01:03:52,770
It's the variance of
individual hours, the variance

1085
01:03:52,770 --> 01:03:56,970
of participation on
the extensive margin

1086
01:03:56,970 --> 01:03:59,520
and this covariance term.

1087
01:03:59,520 --> 01:04:03,420
And in the US data,
those are the numbers

1088
01:04:03,420 --> 01:04:07,690
consistent with that table
that I showed you before.

1089
01:04:07,690 --> 01:04:09,780
So a lot of the
variability in hours,

1090
01:04:09,780 --> 01:04:11,730
say over a business
cycle or something,

1091
01:04:11,730 --> 01:04:13,688
has to do with
varying participation

1092
01:04:13,688 --> 01:04:14,730
among individual members.

1093
01:04:20,200 --> 01:04:28,810
OK, so this is sort of
familiar ingredients.

1094
01:04:28,810 --> 01:04:30,280
You'd have an
aggregate production

1095
01:04:30,280 --> 01:04:32,330
function over hours in capital.

1096
01:04:32,330 --> 01:04:36,630
You have some Markov
process on TFP.

1097
01:04:36,630 --> 01:04:43,050
You've seen this in the first
part of this class as well.

1098
01:04:43,050 --> 01:04:44,370
Capital depreciates.

1099
01:04:44,370 --> 01:04:46,740
You can add to it
with investment.

1100
01:04:46,740 --> 01:04:50,430
And you've got this
utility function.

1101
01:04:50,430 --> 01:04:54,510
Again, this is like a
Cobb-Douglas version, well,

1102
01:04:54,510 --> 01:04:56,610
logs off of a
Cobb-Douglas version.

1103
01:04:59,770 --> 01:05:02,770
And the overall
problem is to maximize

1104
01:05:02,770 --> 01:05:04,600
discounted expected utility.

1105
01:05:11,540 --> 01:05:15,410
Except we introduce
this indivisibility.

1106
01:05:15,410 --> 01:05:18,950
So you can either
work at h sub 0 hours

1107
01:05:18,950 --> 01:05:22,520
with some probability or 0.

1108
01:05:22,520 --> 01:05:29,070
Just like Rogerson with
slightly different notation,

1109
01:05:29,070 --> 01:05:36,710
you'd get alpha T, say the
probability of working or not

1110
01:05:36,710 --> 01:05:38,460
working.

1111
01:05:38,460 --> 01:05:41,660
You know, the way they've
normalized it, the log of 1

1112
01:05:41,660 --> 01:05:42,440
is 0.

1113
01:05:42,440 --> 01:05:44,600
This thing is going to drop out.

1114
01:05:44,600 --> 01:05:46,490
And you'll get a
utility function

1115
01:05:46,490 --> 01:05:50,795
over consumption and hours.

1116
01:05:54,970 --> 01:05:58,660
But remember, again,
just to reiterate,

1117
01:05:58,660 --> 01:06:02,770
h0 is the fixed number
of hours you have

1118
01:06:02,770 --> 01:06:05,490
to work if you work at all.

1119
01:06:05,490 --> 01:06:08,430
And alpha t is the fraction
of the population that's

1120
01:06:08,430 --> 01:06:09,810
being assigned to work word.

1121
01:06:09,810 --> 01:06:13,530
Alpha, this coefficient
in front of hours,

1122
01:06:13,530 --> 01:06:16,894
is the thing that's getting
determined in equilibrium.

1123
01:06:25,430 --> 01:06:27,830
So after you make all
those substitutions,

1124
01:06:27,830 --> 01:06:30,260
just like we did before
with Rogerson's paper,

1125
01:06:30,260 --> 01:06:34,650
we end up with
this pseudo utility

1126
01:06:34,650 --> 01:06:37,010
for the representative consumer.

1127
01:06:37,010 --> 01:06:42,550
And, yeah, again,
it's linear in hours.

1128
01:06:42,550 --> 01:06:44,860
So depending on the
wage relative to b,

1129
01:06:44,860 --> 01:06:49,570
it would look as if there's
just enormous willingness

1130
01:06:49,570 --> 01:06:54,820
to supply labor, or elasticity,
as you go from low wages

1131
01:06:54,820 --> 01:06:59,200
to high wages, you know,
an enormous response.

1132
01:06:59,200 --> 01:07:01,240
It's just a review
of what Rogerson did,

1133
01:07:01,240 --> 01:07:08,910
but it's in the context of
this business cycle model.

1134
01:07:08,910 --> 01:07:12,210
So solve the first
order conditions.

1135
01:07:12,210 --> 01:07:14,550
Compute the steady state.

1136
01:07:14,550 --> 01:07:18,210
Compute approximation
to the steady state.

1137
01:07:18,210 --> 01:07:21,630
Solve for the law of motion
of the endogenous variables.

1138
01:07:21,630 --> 01:07:23,940
And compute the moments.

1139
01:07:23,940 --> 01:07:27,240
Now, I don't want
to belabor this,

1140
01:07:27,240 --> 01:07:30,870
because it would just
take us too much time.

1141
01:07:30,870 --> 01:07:35,640
But we did models at
the beginning of class

1142
01:07:35,640 --> 01:07:39,780
that featured occupation choice
and development and so on.

1143
01:07:39,780 --> 01:07:42,300
This is like that
in the sense that we

1144
01:07:42,300 --> 01:07:46,215
have an attempt to match data.

1145
01:07:46,215 --> 01:07:47,590
In this case, in
this literature,

1146
01:07:47,590 --> 01:07:49,300
it happens to be
business cycle data.

1147
01:07:49,300 --> 01:07:54,260
But you're going to see Seema's
version of it in a moment.

1148
01:07:54,260 --> 01:07:56,830
So the cool thing is
you've got this machine.

1149
01:07:56,830 --> 01:07:58,470
You've got this model.

1150
01:07:58,470 --> 01:08:01,690
And if you're willing to take
a stand on parameters that you

1151
01:08:01,690 --> 01:08:04,990
observe in data or
calibrate other parameters,

1152
01:08:04,990 --> 01:08:08,090
you can generate time
series from the model.

1153
01:08:08,090 --> 01:08:10,900
And you can compare
the time series

1154
01:08:10,900 --> 01:08:13,710
to what you see in the data.

1155
01:08:13,710 --> 01:08:17,290
And this is the algorithm
for solving the model.

1156
01:08:17,290 --> 01:08:20,500
Again, I'm not going
to belabor this.

1157
01:08:20,500 --> 01:08:24,819
You know, they pick capital
share and depreciation rates

1158
01:08:24,819 --> 01:08:38,470
and discount rates and so on and
so forth and generate the data.

1159
01:08:38,470 --> 01:08:40,689
Now, they're two economies here.

1160
01:08:40,689 --> 01:08:44,470
This one with divisible
labor, this one

1161
01:08:44,470 --> 01:08:47,229
that we featured with
indivisible labor.

1162
01:08:47,229 --> 01:08:53,330
And you might want to look
at the variability of output

1163
01:08:53,330 --> 01:08:54,710
consumption and hours.

1164
01:08:54,710 --> 01:08:57,080
This is the variability
of ours in the data.

1165
01:09:00,350 --> 01:09:05,109
This is the variability of hours
without the indivisibility.

1166
01:09:05,109 --> 01:09:07,600
This is the variability
of hours with it.

1167
01:09:07,600 --> 01:09:10,750
Now, they didn't get it
all the way up to 1.66.

1168
01:09:10,750 --> 01:09:13,479
But they're certainly
dominating 0.70.

1169
01:09:13,479 --> 01:09:16,149
So that, you know,
as I keep saying,

1170
01:09:16,149 --> 01:09:19,029
these economies with certain
kinds of indivisibilities

1171
01:09:19,029 --> 01:09:21,939
are able to increase
the responsiveness

1172
01:09:21,939 --> 01:09:26,944
of aggregate hours to wages,
more consistent with what

1173
01:09:26,944 --> 01:09:27,444
we see.

1174
01:09:27,444 --> 01:09:28,320
Yeah, Matt.

1175
01:09:28,320 --> 01:09:32,069
AUDIENCE: Is this economy
with individual labor,

1176
01:09:32,069 --> 01:09:33,080
is it actually--

1177
01:09:33,080 --> 01:09:34,622
like one of your
student simulation--

1178
01:09:34,622 --> 01:09:37,290
is it identical to the RBC model
except the utility function?

1179
01:09:37,290 --> 01:09:38,560
ROBERT TOWNSEND: Yeah.

1180
01:09:38,560 --> 01:09:40,310
Well, except for the
indivisibility, yeah.

1181
01:09:44,880 --> 01:09:47,340
AUDIENCE: When you have
the utility function with

1182
01:09:47,340 --> 01:09:50,125
indivisibility, when you
have like the reduced form,

1183
01:09:50,125 --> 01:09:50,750
because don't--

1184
01:09:50,750 --> 01:09:55,712
I mean, in fact, again you've
got phi or the proportion

1185
01:09:55,712 --> 01:09:59,603
of time work--

1186
01:09:59,603 --> 01:10:01,270
ROBERT TOWNSEND:
That's what's going on.

1187
01:10:01,270 --> 01:10:03,020
See, there are two
economies here and kind

1188
01:10:03,020 --> 01:10:04,450
of two ways of doing business.

1189
01:10:04,450 --> 01:10:07,990
You could go to the micro
data, look at individual labor

1190
01:10:07,990 --> 01:10:11,080
supply, try to
populate the economy

1191
01:10:11,080 --> 01:10:14,650
with some fresh elasticity
that you observe in the data

1192
01:10:14,650 --> 01:10:17,290
and then go through
all the steps.

1193
01:10:17,290 --> 01:10:19,720
Or you could say,
no, no, no, no,

1194
01:10:19,720 --> 01:10:22,460
you either work
or you don't work,

1195
01:10:22,460 --> 01:10:26,230
get that pseudo household.

1196
01:10:26,230 --> 01:10:31,350
And it's going to generate
its own elasticity

1197
01:10:31,350 --> 01:10:33,660
and redo the simulations.

1198
01:10:33,660 --> 01:10:36,885
And that's what's
getting compared here.

1199
01:10:36,885 --> 01:10:37,795
Yep.

1200
01:10:37,795 --> 01:10:41,380
AUDIENCE: Compared to the
response of working hours

1201
01:10:41,380 --> 01:10:44,020
is totally driven by
the extensive margin.

1202
01:10:44,020 --> 01:10:46,090
ROBERT TOWNSEND: Yes.

1203
01:10:46,090 --> 01:10:47,800
AUDIENCE: If you
also want to match

1204
01:10:47,800 --> 01:10:52,320
the elasticity of this
extensive margin, maybe--

1205
01:10:52,320 --> 01:10:56,010
ROBERT TOWNSEND: Yes, I'll
show you another paper where--

1206
01:10:56,010 --> 01:10:58,330
so this is too extreme.

1207
01:10:58,330 --> 01:11:01,780
But it shows how to get--

1208
01:11:01,780 --> 01:11:06,600
OK, so this is Kim.

1209
01:11:06,600 --> 01:11:09,250
I'll take you through
this really quickly.

1210
01:11:09,250 --> 01:11:12,750
But it does link up to the
incomplete market literature

1211
01:11:12,750 --> 01:11:15,180
that we talked about last time.

1212
01:11:15,180 --> 01:11:19,340
And the point is you
can generate a higher

1213
01:11:19,340 --> 01:11:22,610
aggregate elasticity than
what you see in the data

1214
01:11:22,610 --> 01:11:26,690
without assuming all this
stuff with complete markets

1215
01:11:26,690 --> 01:11:28,650
and the lotteries.

1216
01:11:28,650 --> 01:11:30,480
So in other words,
there's something

1217
01:11:30,480 --> 01:11:34,420
crucial about the aggregation,
which is more limited here.

1218
01:11:34,420 --> 01:11:36,420
We're not going to
get as high a number.

1219
01:11:36,420 --> 01:11:43,920
But we are going to get a higher
aggregate elasticity than what

1220
01:11:43,920 --> 01:11:45,750
you would think if
you just looked,

1221
01:11:45,750 --> 01:11:48,330
as Yeng was saying
really, if you just

1222
01:11:48,330 --> 01:11:51,180
looked at the micro data.

1223
01:11:51,180 --> 01:11:52,950
This is a nice paper--

1224
01:11:52,950 --> 01:11:56,220
I wasn't aware of it till
a few months ago actually--

1225
01:11:56,220 --> 01:11:58,380
because they're really
into the details

1226
01:11:58,380 --> 01:12:00,960
of the cross-sectional earnings
and wealth distribution.

1227
01:12:00,960 --> 01:12:04,050
They're really looking
at the micro data.

1228
01:12:04,050 --> 01:12:07,760
Whereas those macro
guys, at most,

1229
01:12:07,760 --> 01:12:09,410
are borrowing a few parameters.

1230
01:12:14,340 --> 01:12:19,770
You've got males and females,
like a two-member household,

1231
01:12:19,770 --> 01:12:22,920
maximizing expected
discounted utility.

1232
01:12:22,920 --> 01:12:27,400
This is again this Cobb-Douglass
type specification.

1233
01:12:27,400 --> 01:12:28,950
This is at the
aggregate household

1234
01:12:28,950 --> 01:12:35,180
level for some reason already
aggregated with these weights.

1235
01:12:35,180 --> 01:12:37,470
So it's male hours
and female hours.

1236
01:12:42,280 --> 01:12:47,200
Gamma is the sort of degree of
inter-temporal sub-elasticity

1237
01:12:47,200 --> 01:12:49,660
that we care so much about.

1238
01:12:49,660 --> 01:12:53,357
And now, they're going
to be serious micro guys,

1239
01:12:53,357 --> 01:12:55,315
and they're going to
actually look in the data,

1240
01:12:55,315 --> 01:13:00,640
at male and female wages
and impute productivity.

1241
01:13:00,640 --> 01:13:03,340
So these x things--

1242
01:13:03,340 --> 01:13:06,730
there's going to be one for
males and one for females--

1243
01:13:06,730 --> 01:13:10,360
take hours and multiply
by productivities.

1244
01:13:10,360 --> 01:13:14,630
And these productivities
can move around over time.

1245
01:13:14,630 --> 01:13:17,440
So it's not just a firm having
TFP that's moving around.

1246
01:13:17,440 --> 01:13:21,380
It's households having
labor productivity

1247
01:13:21,380 --> 01:13:22,300
that's moving around.

1248
01:13:25,173 --> 01:13:26,590
And the household
is going to have

1249
01:13:26,590 --> 01:13:30,630
this sort of standard incomplete
markets budget constraints.

1250
01:13:30,630 --> 01:13:34,720
So this looks a lot like what
we had last time sort of.

1251
01:13:34,720 --> 01:13:36,970
They're going to have assets
in the previous period

1252
01:13:36,970 --> 01:13:38,440
and interest on that.

1253
01:13:38,440 --> 01:13:42,490
They can save to take
into the following period.

1254
01:13:42,490 --> 01:13:44,770
And the household,
as a whole, is

1255
01:13:44,770 --> 01:13:48,130
going to have wage earnings
as a function of hours.

1256
01:13:48,130 --> 01:13:49,480
Now, we have labor supply.

1257
01:13:49,480 --> 01:13:53,590
We didn't have
that last Tuesday.

1258
01:13:53,590 --> 01:13:56,140
And there's, say,
a bound on assets.

1259
01:13:56,140 --> 01:13:58,810
This could be
zero, no borrowing.

1260
01:13:58,810 --> 01:14:02,170
It could be a negative
number, a limit on borrowing.

1261
01:14:02,170 --> 01:14:06,100
Just like the standard
incomplete markets lecture

1262
01:14:06,100 --> 01:14:09,580
from last time, we give them
an aggregate Cobb-Douglas

1263
01:14:09,580 --> 01:14:11,780
technology.

1264
01:14:11,780 --> 01:14:16,535
And we solve this
forward-looking value function.

1265
01:14:20,800 --> 01:14:22,420
So it's a bit bewildering.

1266
01:14:22,420 --> 01:14:26,780
But they're both working.

1267
01:14:26,780 --> 01:14:28,180
This e is not we.

1268
01:14:28,180 --> 01:14:29,290
It's like wee.

1269
01:14:29,290 --> 01:14:33,100
No, ee means employed
employed, employed not

1270
01:14:33,100 --> 01:14:37,810
employed, et cetera, depending
on the female and male members.

1271
01:14:37,810 --> 01:14:40,810
And this is the case
where currently they're

1272
01:14:40,810 --> 01:14:43,660
both employed.

1273
01:14:43,660 --> 01:14:47,020
Solving for assets,
you know, anticipating

1274
01:14:47,020 --> 01:14:50,530
that they'll decide on
their work status next time.

1275
01:14:50,530 --> 01:14:54,850
Now, again, you can see
this extensive margin here

1276
01:14:54,850 --> 01:14:59,050
is playing a big role,
deciding member by member

1277
01:14:59,050 --> 01:15:02,280
whether it will
be working or not.

1278
01:15:02,280 --> 01:15:06,700
And at least in the '50s,
this was a big issue,

1279
01:15:06,700 --> 01:15:10,030
increasing female labor
force participation.

1280
01:15:10,030 --> 01:15:11,740
Things have settled down now.

1281
01:15:11,740 --> 01:15:13,060
It's much more symmetric.

1282
01:15:16,280 --> 01:15:18,200
But the wife did not
always stay home.

1283
01:15:18,200 --> 01:15:21,140
Sometimes they
participated, depending

1284
01:15:21,140 --> 01:15:22,560
on the household situation.

1285
01:15:22,560 --> 01:15:30,600
So family labor supply, and
what do I want to say here?

1286
01:15:30,600 --> 01:15:33,270
So not much.

1287
01:15:33,270 --> 01:15:37,840
It looks more
complicated than it is.

1288
01:15:37,840 --> 01:15:45,320
You have this all for a steady
state, which is complicated.

1289
01:15:45,320 --> 01:15:47,310
Someone asked me-- who was it?

1290
01:15:47,310 --> 01:15:51,433
Can't remember-- about
adding state variables.

1291
01:15:51,433 --> 01:15:53,600
So here, you know, we have
a lot of state variables.

1292
01:15:53,600 --> 01:15:55,140
We've got male productivity.

1293
01:15:55,140 --> 01:15:56,930
We've got female productivity.

1294
01:15:56,930 --> 01:15:59,000
We have the
distribution of wealth.

1295
01:15:59,000 --> 01:16:00,500
All these things
have distributions

1296
01:16:00,500 --> 01:16:03,190
in the population.

1297
01:16:03,190 --> 01:16:06,890
And we have the sort of shock
in the aggregate production

1298
01:16:06,890 --> 01:16:07,390
function.

1299
01:16:11,040 --> 01:16:14,060
So they're going to do
what those macro guys do,

1300
01:16:14,060 --> 01:16:16,020
which is just look
in the steady state.

1301
01:16:16,020 --> 01:16:18,080
They have to compute it.

1302
01:16:18,080 --> 01:16:20,420
And then they're going
to look at very--

1303
01:16:20,420 --> 01:16:23,390
there's no time subscripts on
the wages and interest rates,

1304
01:16:23,390 --> 01:16:25,730
because you're in a
steady state, where

1305
01:16:25,730 --> 01:16:28,790
all this distribution
stuff has settled down.

1306
01:16:28,790 --> 01:16:31,970
You have a lot of
churning from one status

1307
01:16:31,970 --> 01:16:35,060
to another within
the steady state.

1308
01:16:35,060 --> 01:16:37,760
So individuals and
households are experiencing

1309
01:16:37,760 --> 01:16:40,670
different things over time.

1310
01:16:40,670 --> 01:16:44,300
But the economy wide
wages and interest rates

1311
01:16:44,300 --> 01:16:45,980
have settled down to constants.

1312
01:16:49,080 --> 01:16:55,370
And I'll skip-- they actually
start estimating parameters

1313
01:16:55,370 --> 01:16:57,500
with the micro data.

1314
01:16:57,500 --> 01:17:05,930
And, you know,
their goal here is

1315
01:17:05,930 --> 01:17:09,830
to get this aggregate
elasticity high,

1316
01:17:09,830 --> 01:17:13,100
higher than you
would see as they

1317
01:17:13,100 --> 01:17:14,900
estimate from the micro data.

1318
01:17:14,900 --> 01:17:18,320
So, Yeng, that's a
little better example

1319
01:17:18,320 --> 01:17:21,890
of actually looking at
the underlying micro data

1320
01:17:21,890 --> 01:17:26,940
and still using this sort
of macro aggregation.

1321
01:17:26,940 --> 01:17:28,670
Yep?

1322
01:17:28,670 --> 01:17:32,170
AUDIENCE: Still saying
that from micro data

1323
01:17:32,170 --> 01:17:36,243
the elasticity, so it's
made to be smaller.

1324
01:17:36,243 --> 01:17:37,160
ROBERT TOWNSEND: Yeah.

1325
01:17:37,160 --> 01:17:38,290
Yeah.

1326
01:17:38,290 --> 01:17:40,550
AUDIENCE: Like even though
you have model, it just--

1327
01:17:40,550 --> 01:17:41,800
ROBERT TOWNSEND: That's right.

1328
01:17:41,800 --> 01:17:44,180
So why is it still happening?

1329
01:17:44,180 --> 01:17:46,100
Because, again, there's
this indivisibility

1330
01:17:46,100 --> 01:17:50,164
going on about whether
or not to work.

1331
01:17:50,164 --> 01:17:55,440
AUDIENCE: So if you incorporate
indivisibility, then--

1332
01:17:55,440 --> 01:17:57,830
because I thought
that point of Cheng

1333
01:17:57,830 --> 01:18:03,180
is that if you look at
the extensive margin,

1334
01:18:03,180 --> 01:18:05,130
that's still like
very small compared

1335
01:18:05,130 --> 01:18:09,964
to what the macro model
would be in individual labor.

1336
01:18:13,700 --> 01:18:16,550
ROBERT TOWNSEND: Yeah, again,
at the individual level,

1337
01:18:16,550 --> 01:18:18,867
you'll back out certain
low elasticities.

1338
01:18:18,867 --> 01:18:20,450
But the point of
this whole literature

1339
01:18:20,450 --> 01:18:23,540
is when you
aggregate up and look

1340
01:18:23,540 --> 01:18:27,230
at how aggregated hours
are moving around,

1341
01:18:27,230 --> 01:18:28,910
you can get bigger numbers.

1342
01:18:28,910 --> 01:18:30,370
It's not inconsistent.

1343
01:18:34,190 --> 01:18:38,790
Which brings me very briefly,
although fortunately, I've

1344
01:18:38,790 --> 01:18:44,560
said a few words about
this, Seema's paper

1345
01:18:44,560 --> 01:18:49,460
looks at productivity shots in
agriculture And, looks at what

1346
01:18:49,460 --> 01:18:50,405
happens to wages.

1347
01:18:54,510 --> 01:18:57,100
Now, you know, think
about a supply curve

1348
01:18:57,100 --> 01:18:58,390
and a varying demand curve.

1349
01:18:58,390 --> 01:19:01,350
So it rains a lot or
a little, that kind

1350
01:19:01,350 --> 01:19:03,150
of affects productivity.

1351
01:19:03,150 --> 01:19:05,880
That's moving the
demand for labor.

1352
01:19:05,880 --> 01:19:08,760
If the supply
curve is inelastic,

1353
01:19:08,760 --> 01:19:11,970
you're going to get a high
variability in the wage.

1354
01:19:11,970 --> 01:19:13,470
Her point is she cares.

1355
01:19:13,470 --> 01:19:17,370
The workers get hurt a lot
when there are droughts

1356
01:19:17,370 --> 01:19:20,160
and it doesn't rain
because the wage drops.

1357
01:19:20,160 --> 01:19:22,530
Landowners love it.

1358
01:19:22,530 --> 01:19:27,510
Their output is lower, but
they have less of a wage bill.

1359
01:19:27,510 --> 01:19:32,430
And then she couples that with
assumptions about how well

1360
01:19:32,430 --> 01:19:36,990
or how poorly the
banking system is.

1361
01:19:36,990 --> 01:19:40,950
So I'll just tell
you in words, she

1362
01:19:40,950 --> 01:19:45,650
has a 2-period problem, which
she thinks is enough, rightly,

1363
01:19:45,650 --> 01:19:47,870
to get what she wants.

1364
01:19:47,870 --> 01:19:49,530
She has people forward looking.

1365
01:19:49,530 --> 01:19:52,100
So they're solving
2-period problems.

1366
01:19:52,100 --> 01:19:55,220
And then you decide
whether or not to save

1367
01:19:55,220 --> 01:19:59,330
or to borrow based on
your current situation.

1368
01:19:59,330 --> 01:20:03,590
She's saying that if
they can borrow and save

1369
01:20:03,590 --> 01:20:05,630
in the banking system
is good, they're

1370
01:20:05,630 --> 01:20:13,430
going to be less needing to
work in a bad agricultural year,

1371
01:20:13,430 --> 01:20:16,970
because they can borrow
instead of working harder.

1372
01:20:16,970 --> 01:20:19,730
So this is an
incomplete markets model

1373
01:20:19,730 --> 01:20:25,310
with limitations and regional
variation in the banking system

1374
01:20:25,310 --> 01:20:30,080
that is focused on how
much wages move around

1375
01:20:30,080 --> 01:20:32,795
with productivity agricultural
productivity shocks.

1376
01:20:35,800 --> 01:20:41,160
Now, her paper is qualitative
in the sense that she shows that

1377
01:20:41,160 --> 01:20:44,370
there is with less what she
called wage elasticity--

1378
01:20:44,370 --> 01:20:47,670
there's less responsiveness
of wage to rainfall shocks

1379
01:20:47,670 --> 01:20:50,930
or instrumented
productivity shocks--

1380
01:20:50,930 --> 01:20:52,520
the better the
banking system is.

1381
01:20:52,520 --> 01:20:54,360
That's what she cares about.

1382
01:20:54,360 --> 01:20:57,650
Now, interestingly, she
doesn't get into the details

1383
01:20:57,650 --> 01:20:59,700
of the numbers.

1384
01:20:59,700 --> 01:21:03,970
How quantitatively big or
small is this response?

1385
01:21:03,970 --> 01:21:05,580
So she doesn't do
anything wrong.

1386
01:21:05,580 --> 01:21:07,860
It's actually a very nice paper.

1387
01:21:07,860 --> 01:21:12,630
But if she were seriously trying
to estimate the elasticities,

1388
01:21:12,630 --> 01:21:15,630
she would need to
take into account

1389
01:21:15,630 --> 01:21:19,410
whether there is this
extensive margin participation.

1390
01:21:19,410 --> 01:21:21,450
And the aggregate
level elasticities

1391
01:21:21,450 --> 01:21:28,560
could be lower than they might
seem to be as in the two papers

1392
01:21:28,560 --> 01:21:31,450
that we just covered.

1393
01:21:31,450 --> 01:21:33,060
So again, nothing wrong.

1394
01:21:33,060 --> 01:21:35,970
But, you know, when you're in
practice and development work

1395
01:21:35,970 --> 01:21:38,340
and you're thinking not just
about individual responses,

1396
01:21:38,340 --> 01:21:41,340
but sort of collective
responses and whole regions,

1397
01:21:41,340 --> 01:21:44,970
and so on then these sort
of tricks that we get,

1398
01:21:44,970 --> 01:21:49,930
lessons learned from
the macro literature

1399
01:21:49,930 --> 01:21:53,080
are very helpful in combining
the macro and micro data.

1400
01:21:56,650 --> 01:21:59,380
So you can read the
details of the slides.

1401
01:21:59,380 --> 01:22:02,460
But I basically covered it.

1402
01:22:02,460 --> 01:22:04,340
Thank you.