so we're going to talk about optimal social insurance today
and let me just remind me where we left off in the last lecture with the setup
love this
model that we're gonna discuss today so
we're considering an environment where we have an unemployment insurance system
that
is a constant benefit be to agents when they're unemployed
and those benefits are financed by one some tax tebe
in the state where you're employed the government provides insurance
subject to the balance budget constraint that the expected
amount of tax revenue to like sequels expected my benefits it pays
and a with the problem is that we're going to try to maximize the agents
expected utility
which is given by this expression here so the weighted average
utility in the two states minus the disk utility apps search effort
which is given by this point since I V which is a convex
increasing function okay so that's the basic setup up the problem
and we're gonna spend most disliked her talking about various solutions
to this problem so as a benchmark as in all these
analyses we usually just start by thinking about what happens in the first
blast
so first best is the case where there's no moral hazard problem in this case
because there's no
information problem I or another way to think about it is that the government
can choose both the benefit level and the agents effort
so there's no issue will be agent choosing effort sub-optimal E
given the government incentives are because
the government's controlling everything a
and so I when the government does this have let's consider the case for the
government's using puny jointly to maximize the agents
welfare and that
so you know you're picking be any to maximize that expected utility function
that we had in the previous slide
subject to this balanced budget requirement and if you just work out the
algebra
you'll see that the solution a
requires having the marginal utility consumption in the two states
equated and thats intuitive because this is a case for your actuarially fair
insurance it's fairly priced insurance
and so the agent we know because a diminishing marginal utility is going to
want to equate
marginal utility across this 22 states or if you have the same utility
the two states what that means is you want to equate consumption across the
two states in other words
there's no reason not to completely eliminate risk
in this environment in the first so that's kinda the
target that you're trying to achieve if there's no
our moral hazard problem okay now in practice
you're not going to be able to eliminate the moral hazard problem because
effort is unobserved by the government so fundamentally
all of these are you know the reason we're not able to implement the first
base best always boils down to some kind of informational problem and in this
case the informational problem
is that you can't see effort just like in the movies model in the case of taxes
you couldn't see the agents effort when you're designing the tax system into
your attacks on the base is a
earnings rather than skill levels right it's an analogous problem
your okay so what
a what why does this generator problem and why does this make us want reduced
optimal benefit level
its because agents only consider private marginal costs and benefits when
choosing effort
and there's a wedge between the private and social marginal product as soon as
you have a social insurance system
so the social marginal product at work
in this environment is your fixed wage W you're able to do something like produce
some good which someone is willing to pay
eww units W dollars for so that must be your marginal product
but your private marginal product is w mine is be because you know if you go
back to work
you give up the unemployment benefit so if I'm
earning five hundred dollars a week and I have a UI benefit of two hundred
dollars a week
I feel as if my net wages only three hundred dollars right
answered as a result I'm gonna search too little for new job
or I'm gonna be too selective in the set have job offers that I accept
and that's going to drive up unemployment rates in equilibrium
and dust generated Boston inefficiency
okay so now you can see our trade of the murders were trying to get to the first
best
level insurance but it while we're doing that we're gonna be creating this
problem
people working to little right
okay so how are we gonna soul for the optimal love a benefit level given that
trade-off
so this is the second best problem this is really what the
focus of this that occurs about a
so here effort is not going to be chosen by the government rather it's gonna be
chosen by the agent
to maximize is expected utility taking be
NTSB is getting so that's the critical thing
I don't internalize the fact that when I search less hard this has an impact on
the government budget constraint
and as a result I end up making a suboptimal choice from a social
it's exactly the same reason that you have that would cost models /a
optimal income taxation same principle
so the agent is solving this problem here taking tea is
a fixed number and be as a fixed number right so
why does that make sense as single agent
my behaviors gonna have a trivial impact on
government's budget constraint it so it makes sense that I take
T&B is essentially fixed right I
and so I just maximize my expected utility dividend government system
let's like the indirect utility function that comes out of that problem
be denoted by the FBT so give it a
benefit attack system I can
calculate what optimizing level about for days and that I can calculate what
the
level in rec utility that would be generated as for any given system
and then the government's problem is to maximize the agent expected utility
taking into account the fact that his behaviors gonna respond to the system
so is the planner what we're trying to do is Matt choose BNT to max
the FBT subject to are
balance budget constraint right that
you know the expected revenue St called expected cost
so we can write that more precisely as follows: this is really the optimal
social insurance problem
its to maximize the agency to liddy
so choose the parameters the system expose the agency Tory
subject to a resource-constrained which is the first constraint
here and an incentive compatibility constraint which is the second
constraint
that the choices effort or whatever choices the agent speaking more
generally
have to be optimal from the agents point of view to notice that
direct parallel to the optimal taxation problem
it has exactly the same structure we maximize
benevolent social planner maximizing some objective function the agency
toadies
subject to a resource-constrained compatibility constrain
informally you can actually think about this problem
as an optimal ramzi tax problem with state contingent taxes was a Ramsey tax
problem because
were restricting ourselves to be near instruments here
were not I you know solving in them more general release him from work
for the optimal nonlinear scheduled between the two
okay so let's think about how we go about solving this problem
so we've talked many times in this class about different approaches that you can
take me just remind you
as in all the other literatures there are two high-level
a sort of approach is that people have taken
wind is you can make it was a structural approach which is to specify complete
model economic behavior
estimate the primitives that model so
in this case we would identify the optimal benefit great as a function of
various primitives
other model search behavior so for instance
discount rates appear in a dynamic environment the nature borrowing
constraints which are very important
for determining agents consumption smoothing ability the degree of informal
insurance such let you know when you get unemployed you might be able to borrow
from family members
and things like that so you would estimate all those primitives and then
descent you would understand how he is chosen as a function to
various parameters and then want to fully understand that model you can
simulate what happens for different levels adoptable benefit
and just directly calculate what the benefit max
the what the level the benefit is that maximizes the agents expected
a second approach which is more along the lines of what we've been discussing
here
is to derive formulas for adoptable benefit as a function of the reduced
form elasticity is that people are estimating the program evaluation
literature
on social insurance and so the formula that i'm gonna talk about now is one
example of this and basically this lecture will talk about
various ways to solve that formula and connected
empirical estimates without putting too much structure on what the underlying
models
so a so let's talk about how we
us all for the optimal so we know that
at interior optimum the optimal benefit level has dissatisfied this first order
condition where
if I perturb the benefit slightly I don't change the agents and directed
so we want to calculate the derivative are the indirect utility function with
respect to the benefit level
so to calculate that derivative let's write BFB
as follows: so it's the maximum level
utility that the agent can attain when he chooses
optimally given the government's a
tax in benefits system in here I'm writing
really the government only has one choice burial rites a previously I was
reading BFT comedy
subject to the resource-constrained but we know that the resource-constrained is
gonna one-for-one
pin down tax rates when I have a benefit level
so I can just write you be there's a given to be there is going to be a
T that's implied by that by the government budget constraint
and then I can reduce the government's problem to picking this one
benefit level because the tax is going to be automatically applied
are and so what the government I'm wants to do
is picked the level would be that maximizes this function
where this you know the this party function here
he has been chosen to maximize it you know based on age and utility
maximization right
so basically this expression your
incorporates the objective we had initially
and both the constraints that are
now since this function has been optimized Overy
we know that we can apply the envelope theorem which
then implies that we can ignore
the impacts of changes any on
utility right because they're it's not so good have a first-order affect on the
agency
early it so we can write DVD be
as follows: it's one minus E times the marginal utility
consumption in the low state so that's the derivative
this expression sorry backwards
so yet it's the derivative this expression here on the
it's the second term on and then
when I raise be by one unit I'm gonna have to raise the taxes by DT DB
and then thats multiplied by he terms the marginal utility consumption in a
high state
okay so that I so mathematically
what's useful but this is beginning or the Peter talk about this many times
before
it also has a very intuitive on
interpretation right what's the impact raising benefits by one dollar on the
agency to it there are two terms
first I give you an extra dollar benefits when you're unemployed
what's the value of that to you it's the probability that you're unemployed one
minus E
times the value of an extra dollar when you're unemployed which is you prime
CL marginal utility conception boasted what is the cost of giving you that
dollar
I have to take extra tax revenue extra tax
away from you in a high state DTD be in order to finance that dollar benefits
I wait that by the amount of time you're in the high state
and the marginal utility consumption state okay so it makes perfect sense
that this is
the typeof expression you get for the marginal value chain
benefit by dollar okay so
now let's think about how we actually calculate that
I so before I do that as well
as show you a slight segue why this is valuable so we're considering a very
simple model here for you only choosing one thing your search effort
right but is the literatures developed since bailey's 1978 paper I think
what was viewed as one of the witnesses at the daily paper is that it didn't
really allow for many margins
choice out beyond this very simple
effort choice but what I want to show you next
is that this formula is going to be valid even if you have many other
considerations in terms of what people might do when they get unemployed
sir take one example I in a recent paper Gregg Kaplan who is it
you plan his job market paper he documents that many
people when they get a point especially younger workers
tend to move back into their parents house
you know your during the unemployment as a way to smooth consumption
right it so he correctly argues that this is potentially an important channel
have insurance
that should be taken into account when we're thinking about how to set benefits
optimally
now what I want to show here is that
while that's correct that will affect the optimal level of unemployment
benefits
it's not actually can affect the formula that which is derived for the optimal
level
unemployment benefits and so in a sense you don't actually need to change
anything
to take this mechanism into account so let's see how this works
so the general point to bring it was straight years exploiting this first
order conditions from
optimization particularly useful in more complex models than the simple one
work we were considering before so a
take the cap on paper what he does is he takes a structural approach in estimates
a dynamic model
the decision to move back in with your parents when you can get
unemployed and that he shows how that's going to affect up to 1m
so let me characterize that in a in a simpler way here just
that it into our notation the same logical apply in a more general dynamic
model like the one he considers
so to make this very simple suppose moving home
raises your consumption by each because you don't have to pay rent or whatever
right
and it has some cost geoBeats there has to be some cost
moving back or mother what you would always by government
this would be an interesting problem so a
okay so now you the agent is optimizing over two variables right
is optimizing over search effort and sorta how much time to spend it
which is the age variable there's a what another dimension like
you can think about it either in a continuous way like I'm doing here
or you can think about it as a discrete decision do I moved back over not
and then you're going to be an extensive margin model over there's going to be a
marginal guy
who's making that decision and its gonna basically work out the same way
in the aggregate okay
so when you just look at this problem you should see immediately
that there's really no difference between he and
H in the sense that both of them were gonna drop out
changes in either of those are not gonna have a first-order affect on the agency
to woody
and so because I agent optimization the formula for DB DB is actually going to
be just
identical it's totally irrelevant that you added this each dimension where
the you getting added H for consumption in the low state at a cost
GFH that makes no difference it
it doesn't affect this formula in any way
but it's important to be clear that this does not mean that
the ability to move back home has no impact
on the optimal level unemployment that's not what we're saying
what we're saying is this formula
doesn't change but the inputs into that formula are gonna change as a result the
ability to move back
right so in particular your consumption in the low state
is now going to include this part that comes from the ability to move back home
you need to measure consumption in the right way and that's gonna end up
affecting the optimal level benefits
but if you were measuring consumption the right way to begin with
you can totally ignore the fact that part of it came from moving back over
doing
girls question okay
so that's really the power looking at the problem in this way because as you
can see moving back home is like
my view 10 many many things that you might or might not
consider when thinking about this problem
okay so I
this formula here is not get something that we can relate it take to the data
with a couple more stops
so one problem is that we have these marginal utilities here
not totally obvious how you measure those we'll come back to that it's been
a long time and that the other problem is that we have DTD
here which is something that we want to translate into
empirically adjustable parameters that that's easy so let's work on that next
so that just comes from differentiating the government's budget constraint right
how much do I need to raise taxes when I raise benefits by dollar
so in order to calculate that you go back to the government's budget
constraint differentiate with respect to be
and if you just do the algebra
you get one-term that's a mechanical a fact
that's going to be determined by the ratio of the probability of the two
states right that determines
how much extra have to raise taxes to find an extra dollar benefits
and then there's a second term which is also familiar from the ramsey problem
which is related to a behavioral response and this is the fact that when
I raise benefits by dollar the guys gonna start working last
its ok to raise taxes by more than an equivalent amount
and that embodies the moral hazard cost of providing insurance
so you work out the algebra and you get an expression that looks like this
which as you might expect involves an elasticity
the probability of being unemployed one minus the
with the respective unemployment benefit level
and then you plug that back into the expression we had before I'm
not going through the details map it you should do that afterward
are DVD be you like that in and then factor and you get an expression
looks like this okay so I
this is the marginal value raising benefits by dollar
and then remember a optimum we want to go to the point where dddd
equal 0 and that's gonna give you then
and optimality condition which is now
used widely in this literature I
which is the following so you wanna set that gap
in marginal utilities between blowin' I States
that's the left hand side equal to
the elasticity of the unemployment rate with respect to the benefit
so what do I mean by gap in marginal utilities so
the way you can think about the left hand side this expression is that it's
that benefit
up transferring a dollar from the high state
to the low state so what do you get outta that you get an extra you primacy
L
in the low state lose your prime appropriation I state
and you can think about it loosely as calculating this in percentage terms
so that's kinda value of providing extra insurance
and then thats offset by the fact that you have this distortion
very cost because you're changing the amount that people work
I as you provide this insurance
on and so the right hand side you can bring covers
correspondent cost transferring that dollar the budgetary cost
above and beyond the you know the direct transfer
like what sort of the leakage from the system so coming back to a question
about the connection to the ramsey model
you know the point I was making is that here you have this positive
left inside so even though you can that be complete set up state could
taxes you still want to use them in order to
achieve this kid okay so
are this is the formula that you can see is sort of a starting point for various
trends
I love this letter for a way to organize the literature
so the way I think about it is this is an exact formula for the optimal benefit
rate
you know fairly wide awesome models but it's not an empirically implementable
for me
because while I know how to measure what's on the right hand side
I you can see how you might measure the lastest the unemployment rate with
respect
level his role empirically measurable concepts
the left hand side it's not obvious how you measure
right how do you get marginal utilities so the way I see it
you know sorta the way to think about literature
is how do you identify that perimeter on the left inside the value
transferring money across states in there are three ways
so far that people have developed to think about
how you identify that gap in utilities
the first is using data on consumption
the second is using day down reservation wages
in the third is a decomposition into moral hazard and liquidity affects
which is kinda analogous to a Slutsky decomposition come to that
in a little bit so let's start by talking first about the consumption
based approach
it's a normalization know you're right that so there's actually no deep reason
for that it's the
point at which you take the Taylor mission driving
you Primacy HP as opposed to you so
where that's going to matter is the way in which you'd
find drop in consumption the what you put in the denominator
second you're right there's no strong reason
okay so a
let's rate the gap
in marginal utilities using
a Sosa first restart a consumption-based formula
we're gonna first thing we do is write the gap in marginal utilities using a
Taylor expansion
I so you primacy on mine issue primacy %eh
can be written as you prime primer ch this
the point I was making the point Richard approximation
on times that change in consumption across the two states rights
just a different and more Utilities is given by the slope times the change in
expiry the change in consumption so
next what's define the coefficient relative risk aversion that
measure the curvature of the utility function right as
minus you prime privacy divided by your privacy
time see so just so people are clear on this
that is the last St the marginal
utility consumption
with respect to consumption
all these things are the same the formula that's on the slide
and this deal argue primacy the log see
or in other words how elastic is the marginal utility consumption
with respect to consumption so when I raise your consumption by one percent
in percentage terms how much does your marginal utility consumption
Paul that's a measure poker utility functions
that's intuitive rates and its usual us to stay concept
on and you you know you can also write it
like that okay so
a given that definition because
relative risk aversion and the Taylor approximation we can write this gap in
March utilities that we're trying to recover
as you prime primer for you prime-time ch again we're just about winning this
thing its %eh
on times Delta see oversee and what
Delta see oversees the percentage drop in consumption
from the high state to the low state and so that means
coming back to the point where you evaluate the still approximation
you want to measure this percentage drop a
as a percentage of consumption in the high state given the way that this has
been derived
and so a
with that notation the important thing to focus on year
is that I can write the gap in marginal utilities
as being a function of the curvature
utility function and the size of the consumption dropped between the low and
high states
just to make sure everybody's following this
me graphic so we've got
see each year out here
what are we trying to recover differences slopes
utility function in the high state the low state rate the percentage difference
between the slopes
those two states and watery saying basically if I tell you
how her dispersion that's what gamers and tell you how far apart CLNC
HR the product those two things tells you the difference between those two
slopes
make sense right it's how wide apart are you on the x-axis
and how much is basically it okay
so if I take Gamma Delta see oversee
then I can recover the gap and more utilities in a min
that I'm interested in it so what that means that is if I measure the curvature
of the utility
which is risk aversion there's a huge literature
in economics on estimating risk aversion and then measure the gap
a drop in consumption from the employed to the unemployed state
then I can say something empirically
about adopt a malevolent
so this is the first idea and this literature
I've how to use that service official statistic approached
estimate optimal
and so this just summarizes that
the optimal level of unemployment benefits satisfies this expression
here where Delta see oversees the consumption dropped gamers the cove
relative risk aversion and this is the last two Steve
the unemployment rate with respect so the point is
if I estimate these three parameters in a fairly general class
models I can say something about the right level
unemployment benefits to maximize expected utility
okay so
salute little bit more intuition this formula
as I was talking about earlier the left hand side you can think about is being
the model social benefit
you are it's basically a function I've two things you think I've it's been
quite intuitive
how volatile I was consumption across states
so how much risk for people actually facing and how much they care about that
risk
suppose the utility function as many right damn equal 0
I don't care if my consumption is high in one state blow another all I care
about is average consumption
been quite intuitively optimal level insurance 0 it's irrelevant what else
on is going on the left inside a zero right
I and the right hand side relates to the government revenue
leakage basically as i've
been stressing that extends to a model where people choose
and other behaviors and face and other constraints that is fairly general
environment
subject to some what I see is basically technical regularity conditions that are
not
substantively that important now there are
I you know what are we assuming that is actually quite important
the conditions that are required for the envelope conditions to old
which in particular are include agent optimization problem
central so one way you can think about
empirical work on the subject which I'm gonna talk about now for a little while
on is that it provides estimates
up three key parameters in this form and then we're going to see what the
problems are with these empirical estimates
and use that to motivate other sufficient statistic formulas for social
insurance
which is sorted where the frontier workers
alright so let's start with I think the thing that's actually the best
measured which is the parameter on the right hand side here
so as in many other parts of Economics where I think always quite good at
measuring to store sure it cost but not very good at measuring the benefits
a programs unfortunately so
on the early literature on the subject going back to the seventies
use cross-sectional variation replacement rates it is that simple idea
that
some people you remember the shape the UI schedule that I showed you where
its apart stopping and then there's a cap so that's going to generate
cross-sectional variation replacement rates right because
some people are gonna have hire replacement rates than
on than others
and so that you can imagine running into last regression comparing higher-wage
earners and lower wage earners but that's going to be totally confounded
by the fact that your variation is called linear
with wage rates and so you don't get in unbiased estimate of the impact
the UI benefit level on durations so modern studies use exaggeration from
policy changes
I I'm going to talk about the estimates from Bruce Myers econometric a paper
1990
which is very widely cited and then there's a lot of subsequent literature
I'll say a word about so here's the basic setup that's very commonly used in
this order
so it turns out that the way in which you I benefits are typically changed
provides really ideal variation for a quasi-experimental defender approach
so usually the way the change benefits is to keep the replacement rate
constants are remembered this is earnings on the x-axis
the size of benefits on the y-axis
so usually what people do is you've got this fifty percent or so replacement
rate up to some cap
and what States tend to play around with is that size that cap
so they'll raise the cap so what does that end up doing
it ends up giving you a set of workers whose benefits are totally
unchanged and then it ends up giving you this others that have workers
who are treated in the sense that they now
higher benefits and so now you have was quite nice about this variation is
you have within the same labor market at the same time
one set of workers whose benefits have gone up of a nontrivial E
and another side workers whose benefits really not change and they're relatively
close to each other
so as long as you're willing to make common trend assumption about guys
making like
thirty thousand dollars a year verses forty thousand dollars a year which you
can try to evaluate
you're in pretty good shape your and so this actually turns out to be one case
where
we have pretty decent and fairly consistent
on empirical estimates the size it was lost
that we could use those is like a partially treated right
I mean there's a variable treatment so it's just gonna be a little bit messier
to quantify the size of the treatment
use them on
okay so I because let's talk about how this estimation typically works
I wanna say a little bit but hazard models
so that people know how to interpret these estimates because that standard in
this area
so a rather than to be in
well as regression the way people usually think about
estimating the impact serve these benefit systems on behavior
is using hazard so what's the logic hazard model
its to look at the time varying fax a policy
on behavior at various points at the unemployment also so what's the idea
are if I raise unemployment benefits
on from say three hundred dollars a week to four hundred dollars a week
typically UI benefits last for only 26 weeks right
so you know naive model you might think that their the benefits are
only gonna affect behavior during those twenty six weeks
like if you've been unemployed for a year maybe it's irrelevant
that you've got higher benefits from weeks 12 26
or maybe it's not irrelevant because you might have saved some of that money in
it continues to a factor
the point is you don't wanna on summarize
behavior with the single statistic like the duration of unemployment
the better way to think about it is how is you I affecting search behavior
throughout the point and the effects could be very different
in different parts that the unemployment alright so maybe at the beginning
raising UI benefits doesn't change behavior very much
but as you're coming close to exhausting their benefits maybe it has a much
bigger fact
the point is there could be all these time during a fax you want to capture
those and that's exactly what a hazard
so let's define the hazard rate
HT as the number of people that find a job
in week T divided by what's called the risks at
number people who were unemployed at time t
so the way to think about HT is that it's just are
the conditional probability of finding a job at time t
conditional on being unemployed for at least weeks
right so if they're 2000 guys in your sample unemployed
week 10 and a hundred of them find a job in week 10 which had a hazardous 5
percent
and it's always defined as a fraction of the people who are still unemployed
and you can see because HT is gonna very over the spell and we're going to model
how
HD varies would-be we can get a full picture
how the unemployment benefit system is affecting search behavior
at these different weeks
so there are many ways in which you might think about modeling
HD the most common specification that's used
literature is what's called a Cox proportional hazards model
which is a semiparametric hazard
so here's the functional form that's used in the Cox model
remodel HT as some baseline hazard
out thirty office at: multiplied by
power be tax were axes a set of covariates
whose influence I'm interested in characterizing on the hazard
for instance the unemployment benefit okay so the reason we call this
a semiparametric specification is because the baseline
hazards are totally nonparametric
so the probability that you find a job in week one might be seven percent
23 percent week for might be whatever else
there's no functional form on that so that's what differentiates the Cox model
from
other hazard models like the libel or other type so
parametric models where you assume the functional form
for the Alpha sub T okay
you're just letting that be totally flexible which is I think the best thing
to do
misconduct there's no reason to impose some form
I and then the reason it's not totally nonparametric
is because be are posing parametric structure
in how the axes a fact the hazard rate
in particular that this YouTube a tax form right
so a we allow the hazards very freely across weeks and identify coefficients
of variation
across spells so the way in which we identify the effect
UI benefits by comparing two different people have different UI benefits
at the same week so take two guys
in week 4 one guy has a UI benefits three hundred and other guys you have
benefited 200
how to their probabilities are finding a job in week 4 very
that's what's gonna help you i'd
so let's talk about what Bruce Meyer actually does
so he a the key covariate that he's interested in
is the UI benefit level
which I'm gonna call be here so he models log
HD is August 31st May 21 B-plus various covariates
in this specification deal Aug HTD log be
which is better one is the elasticity
hazard great with respect to the UI been right
so if I included aggressor log be
then the way I can interpret beta1
is the elasticity are
the hazard rate in any given period with respect and love you
benefits and if you want to map this back to the thing that matters in the
model
that is the fraction of time that you stay unemployed or in this context
your unemployment duration I
den its you know as a rough approximation it's useful to note
that if you have a constant hazard model so model where
there's a it's like memoryless in every period
you have a given probability failure like five percent or something
that it's easy to show that in that model the
I'm expected duration:
is one over the hazard rate and so it follows immediately then
that the elasticity the duration of unemployment
with respect to the benefit level is just negative the elasticity
hazard with respect to benefit
does that make sense though what i'm saying is that
expected unemployment duration
is able to one over age rate is not very
across periods so that's just an approximation because in fact
hazard is going to be very cross period and so because it that
deal Aug H de la be
is going to be minus the log expected duration
this is just a useful group
but you can see a more generally if i characterize how you I benefits affect
the force hazards
I can recover the impact on the expected duration just by adding up all the
impacts
so Meyer a uses administrative data for you I claimants which we actually had
even
you know quite some time ago for small samples show you how this looks
so he runs a bunch of hazard models of the type that i described
where he's got log you out benefit on the right hand side you can see is
estimating elasticities in the vicinity
but pointed
so a that's what he gets
and then their a tiny subsequent studies
and I would say the consensus estimate is now somewhat smaller than what Meyer
initially estimated to probably about .5
on there's a little bit better on that but it's much much better estimated
labor supply list okay so I want to mention one other thing which
unfortunately
not bring put in the slides
a which is there's another methodology that you can use to estimate
are duration: elasticities which i think has been done nicely in a recent pic
paper by Camille Monday which is to exploit this fact
even without a reform there are sharp kinks
I in the schedule right like when you cross this
earnings level the slope are the relationship between benefits and
earnings
changes discussed discontinue and so
you can use a basically a very into
regression discontinuity designs regression King design
which exploits the fact that this
discontinuous change in the derivative is useful for identification
I and so that the logic
is if we expect the relationship between
durations of unemployment earnings to bolts which seems natural
why would there be sharp kinks in that relationship so
make the assumption that it would evolve smoothly than any evidence of a sharp
turn in that relationship at this point
can be attributed to the effective you i'd system
its ok im you essentially implements that estimator
shows that there are in fact sharp turns
in the relationship between durations and
earnings at that point and he gets I think very credible estimates
the duration: elasticity unemployment benefits which turned out to be pretty
much in line with the Meijer in subsequent us
okay so you can see how you estimate the
the right hand side thing the elasticity and there's a rich literature on that
which
good looking to further if you're interested the best surveys the Kruger &
Myers
so the consumption smoothing benefits you I
so thinking about how to measure the left hand side the Bailey
so in a very nice paper in the ER 1997
John Gruber takes the Bailey formula to the data
by estimating the consumption smoothing response
he uses the same methodology essentially is Bruce Meyer
are so that its cross-state in time variation in UI benefits
and uses the drop in food consumption as the left hand side variable
instead of the unemployment duration right so you can save basically the same
setup but consumption problem
inside not a problem is that data on consumption is
notoriously bad it so he makes do with the best
you know we can because data from the PSID on
food consumption PSID asks you and annual level
how much food did you can zoom in you know recent past
that's the measure that he's using
so he estimates an equation that looks like this so drop in consumption on the
left hand side
as a function of the UI benefits replacement rate
and then he's controlling for alls a
source is a variation in the replacement rate except for the part that comes from
law changes
okay so he's isolating watching this as a source of variation
in benefit replacement and he ends up estimating a beta 10.2 for in this
equation better to have minus point to it
so what's the right but interpret that
on what this means is without unemployment insurance if you
extrapolate
out-of-sample okay to a case for you don't have you
which is not actually true in the data but assume it's a linear function
then the consumption dropped would be 24 percent
okay so he's saying given the variation we see in the data
if we did not have you why consumption would fall by 24 percent when people
lose their jobs
in practice with the current benefit level
the mean drop the mean observed drop is 10 percent
so what does that make sense because when beta when B is .5
we get back point five times minus point 28 so minus point 14
on and subtract that of a 24 percent drop that we would have without you I
and you get a 10 percent
those are essentially the two facts you can the covers
identifying that equation
neither would think about it is that a 10 percentage-point increase in the UI
replacement great
causes a 2.8 percent reduction and
consumption so what should you take away from the study
first I think it supports the view that
insurance markets are not perfect which is actually a debate that
is has been going on for some time should we think I've insurance markets
as being essentially
perfect because people can always borrow from family or do some other
private sector arrangement that we may not directly measure in the data
so this rejects the view with perfect private insurance markets the fact that
you I benefit seem to matter
and it's just that you i'd does in fact player consumption smoothing role
but unfortunately the estimates are pretty imprecise so I think this has
like roughly 80 start up to so the confidence interval spends anywhere from
zero to like
50 percent and moreover for
you know people who worked with this type of data subsequently you find that
these results
are pretty or not super robust
because there's a tremendous amount of noise here you don't have that many
observations it a couple dozen
rations and so and having a lot of fertility so I actually think
wanna the highest potential areas for further
empirical work in this area is to do
this type of analysis or other related analyses using administrative
consumption
so wanted a I frontiers in terms of minister did we really have not made a
major breakthrough
yet is in consumption we talked about great
data sources for taxable income
labor supply savings all this other stuff but on
consumption remains quite limited and you can't think about
potentially they're getting data from credit card companies or value-added tax
registers I think those are two
potential sources have a lot of data on consumption which have been
really been totally nailed down
so interesting
yeah that's a good a question so another issue here
is like basically stockpiling affects right that I might
spend a lot I and
that doesn't directly reflect the amount that I'm consuming
on I would you know what people try to do is focus on particular types of goods
where you think it's perishables or non-durables or whatever you come closer
but absolutely there's a distinction between consumption expenditure
whichever occurs does emphasize
his work and he argues that that's what drives
summer that fluctuations in what we think of as fluctuations and conductor
actually just operations and expenditure a
so that's a potential further data concern but in my view even getting
decent
expenditure great stuff so
a an interesting question to think about
is you notice that people are
a are in fact consuming more when you get the more you buy benefits but one
thing I want to emphasize here sorry should have stressed
here is that when you raise you I benefits by a dollar
consumption is going up by much less than a dollar
okay so it's much less than one-for-one
assorted passed through to consumption and so that raises the question
what is substituting for certain getting crowded out by the increase in UI
benefits
that's relevant because if you think about the sufficient statistic formula
the only thing that enters that for me between consumption
and so the bigger the impact event one dollar increase in your benefits on
consumption is
the more valuable the entrances if all the insurance is doing is
offsetting other insurance arrangements that you had any way
it's not as valuable so it's interesting
you notice from intellectual point of view to understand what are people
actually doing such that when you raise
benefits by dollar they're all there is a consumption by dollar
bye bye like to 20 cents or something so
a John Gruber as a series of papers kinda filling in the rest of that gap
between the dollar and thus a twenty-five cents
are and there are various channels that he's
thought about one I'll talk about here is spousal labor supply
as a potential entrants mechanism so in this paper with Julie Colin
day study the behaviour wives unemployed husbands
index examine wives labor supply
as a function at the level of the husband's UI benefits
so when the husband has a higher UI benefits does the wife were class when
he's unemployed disorder fill the gap
right so or different way to think about it is if we did not have UI benefits the
secondary her
would come into the labor force and act as an insurance mechanism
and so if that's the main thing that's going on
then when we provide you I benefits we're not actually entering people
better we're just
substituting one entrance another potential channel which i think we
touched on
in earlier lecture is that when you have hired you benefits
people also seem to build up smaller buffer stocks to a small degree
are so they save a little bit less xn T and again you're gonna end up seeing
smaller consumption okay
so on so what Gruber does
is takes are his estimates
up the consumption drop and calibrates bailey's model
so this you know I think reason this paper is important is it's one of the
first
papers to connect the reduced form in
Oracle evidence to a formula derived from theory
not really at that time it thinking about it from the sufficient statistic
perspective that is
realizing I general that result was but it's an
early illustration on combining the reduced form evidence would
the model to say something about optimal policy so
a you saw remember this is the formula
this is hit the way he specifies the consumption drop
okay so gamma times the consumption dropped was the last st
you can see that how you can solve for the optimal replacement rate to solving
the equation
I and so that gives him this formula for the optimal replacement rate where he is
making actually an important assumption that i think is not technically
correct as we know now from further analysis
a which is that when you write the formula in this way
you're suing the elasticity and again don't change
with respect to the benefit rate it's as if everything on the right side is a
constant
with respect a great that actually cannot be true
which will talk about later this elasticity years here
hasta very with mint you can show that on
in these models but anyway he assumes that that's a constant number
are so then he up plugs in this elasticity which comes from
Meyer typeof estimate I and then
assumes a five percent unemployment rate which corresponds to your point 95
internet Asian
and then you get here plugging in all those numbers you get this formula here
so the one parameter I've left is the gamma right we're gonna
look at various values again I think about what that implies now for the
optimal benefit
okay so the result which is not surprising
is that the optimal unemployment benefit rate varies considerably with them
so of course if gamma is
on one which is the linear a
saris are gamers zeros the linear case if gamers one you're still at a quarter
where the optimal benefit level is 0 so what that means is
the marginal value of providing a dollar benefits
falls below the marginal cost in terms of government revenue
leakage and so the optimal benefit level is still 0
and that is damn I gets high you're going to get very high replacement rates
so Gruber says in his paper you know introspection an existing evidence
address damn
less than two and I think that's plausible so you remember when I was
talking about the curvature
and labor supply paper on
I argued that given labor supply patterns you can't really generate risk
aversion much about one
that would suggest that you're in this end of the spectrum you're right
like very close to an 0 optimal you i'd
and that's pretty surprising because it basically says
you know in this context those those elasticities are actually barge in the
sense that you do not want to have any you
at all given the size of the elasticities
we estimate which seems very surprising given that we actually spend like
hundreds of billions of dollars on these income security programs
so I think the grouper view in the view based on this type of evidence is that
the consumption smoothing benefits
are just not big enough relative to the distortion very cost
to make these types of large social safety nets
that desirable but that of course
in just fundamentally on what you think this camera parameters
and a that gamma parameter is I think quite poorly identified in general
so I want to talk about this at a couple of levels first
empirically and then thinking about the radically why gamma might be so unstable
so empirically there a couple of nice papers that are just
coming out in the ER by Percy at all and
the running of pickles go authors that show you the instability of first
preferences in
means that we actually care about so they estimate
risk preferences across different domains for the same
I said individuals so they look at a set of workers at save
big companies like Alcoa where people are choosing health insurance
deductibles prescription drug coverage
disability insurance 401 K portfolio allocations eccentrics you're making a
bunch risky choices
and then basically ask does it seem like people have stable
risk preferences across these domains and the answer which is consistent with
much evidence from the beach ever let
butter in the past using experiments is basically no
their is some there is definitely
a positive correlation risk preferences across the means which they document
clearly
but there's tremendous amount of heterogeneity across to me so lotsa guys
who are making
picking very conservative health insurance plans but investing in very
risky ways
for case and so right and so
that then suggest that the appropriate value gamma
is highly context-dependent which is result that behavior like
economists emphasizing for some time what I want to point out here is it's
not just behavioral factors and free
that might affect what the level of risk aversion
people exhibit is but also just standard neoclassical factors
so I'll let me talk briefly about this paper woods I told which gives you
one on intuition for why risk aversion can vary a lot across settings
so on inserts this idea of consumption commitments
so any a standard expected utility model like the one we're working
we always assume that there's just one consumption good see
so obviously people consume many different things so what is that
assumption
actually mean in practice it's the assumption that there is a composite
commodity
or you know what that and what that requires
is that people can optimize and all the different margins consumption
at all times so in particular what I have in mind here
is that I if you face in unemployment rock
the assumption that there's a single composite commodity
requires that you can read optimizer entire consumption bundle when you face
this unemployment rock so
when you get unemployed you cut consumption of food which was what
grouper focuses on
but the formula requires that you are also able to
optimally cut consumption housing cut consumption of cars
furniture and so forth now in practice
is you can see quite intuitively it's difficult to adjust many elements of
consumption in the short run
because a fixed adjustment costs right so if you're unemployed for
weeks which is the typical spell normal times
a you're not going to sell your house and move to a different place you're not
going to sell your car
you're not going to change your kids school and all the stuff you think a lot
of
things are very rigid and so you in fact see that in the data services from
the paper without his idol I where we do an event study
of consumption around the data on employment
using this is the same date as grouper uses yes I did
received both food and housing so in the year
are a on
in the year the unemployment rate to this is plotting food and housing
consumption growth rates
in the year the unemployment right there's a sharp debt
in food in food consumption
but if you look at housing consumption its a bit noisy but its
relatively flat compared to food consumption in this is drawn for
homeowners
okay so this is consistent with the view but there's heavy adjustment on this
margin
but hardly any adjustment on that
in contrast if you look at renters there's pretty significant adjustment on
both margins
so it's easier to move for renters and homeowners
and renters do in fact cut the amount to Brent they're being pretty significant
unemployed okay so that a
you know this is just one what we're able to look at the data but you might
think more generically there's lots of stuff
that people are not able to adjust very easily it so what does that do
to risk aversion and what does that do to the optimal unemployment
so think is it is what the papers really but think about a model with
to goods safe food and housing for the utility function
looks like this additive over food and housing
and the key property of the motto is that adjusting
housing requires payment to the fixed cost K
and so as a result the agent is gonna follow an SS policy
where if he looks at his current well
and as if his wealth goes up a lot he's gonna move
up to a better house and if as well falls a lot
is gonna pay the switching cost and move down torsos
so those types assess models have a long history in
economics particularly in macro on
but what we want to emphasize in this paper is thinking about
what that type adjustment pattern
does too risk aversion that is does to the curvature of the value function over
well
which is what determines our risk attitudes and let me just summarize the
main result but this graph
so I if you had no commitments
your utility function with just look like the red like some gradual
concave thing because you are adjusting on both food and housing
throughout know the key thing that the commitment model does
is when you're in the stranger moderate shocks
like small unemployment trucks that are not going to make him about your house
your utility function is going to book much more curved in that region
and the intuition is that every dollar you lose
has to be born completely on a subset at the margins
so it's like if I lose a dollar income
and I have I let's take this example suppose I spent half my money on food
and have my money on housing
I in suppose when I'm unemployed and forced to cut expenditure by 10 percent
data have to cut food consumption by 20 percent
in order to balance my budget right because the fifty percent housing is
totally stock
and cutting food consumption by 20 percent raises my marginal utility by
much more it's like I'm
not able to diversify the shock I'm concentrating at on a small set of goods
which is driving upward and so
that makes the value function much more curved in this region as you see more
he's going up very sharply and then you reach a point where you say you know
I've had enough
I can't cut back food consumption anymore I'm gonna pay this much it cost
to move to a smaller house
and then the marginal loss no longer has
as steeper than the fact on the word it so as a result
risk aversion over shocks like unemployment could potentially be very
high because you're not able to divers
shocks where's risk aversion over much bigger shocks like disability
which might make a move outta your house could actually be much lower
so you can see risk aversion can be super context-dependent once you have
this type of adjustment battered and you know what we are doing this paper is
you know maybe the relevant beloved amor for things like unemployment
might actually be something like four rather than the lower numbers
and then you can see the optimal replacement rates potentially much
higher
similar to what we have today solve the problem
in more general problem is that it's very hard to know what the right level
of damages
and so how can we think about optimal insurance without having to rely on the
system's
gamers will talk about next