13of19 - The New Economics of Mortality - Investments in health

last time we were being without the

effect of destiny can help creating that

human capital investor and we had a

simple super utility function I was

right here so s is the probability of

surviving to the second pervaded the

usual discount rate and then we add the

budget constraint so expected

consumption is equal to discounted there

may retain when we one of the doors open

go get some air x 1 plus s X 2 1 plus r

+ u of h is equal to W 1 and 1 minus l1

+ s 1 plus R times right

we had and if we differentiate this with

a specter of different variability

assuming that s is concave and age age

is quality purchase of some help

improving activity like number of

downloads drugs and so on geez

expenditures okay then a first-order

condition I didn't write couple these

down to write them down now you x 1

equal lambda u X such a beta 2 X 2 equal

to lambda put us here s s and over 1

plus R right so eyes and because we have

full annuity insurances the S is

canceled out we don't have to worry

about them and notice in the special

case where beta is equal 1 over 1 plus R

let's say as an equilibrium condition

the interest rate adjusts to the rate of

preference simple models of a great

amount of investment that would be your

delivery condition what complicates

miles if you look at the growth rate but

it just I mean not necessary for us to

just simplifies or we write down you see

that implies that you have one is

marginal utility this is the margin

utility of you in first for respect

there's one other utility made by Ray

Rotem of the st. will generally not be

the same and that's going to be equal

the beta + + + 10 1 plus R since they're

equal to each other that ratio is one

right so we just have

equals you x2 so well known condition

and these circumstances if interest

rates are equal to the utility discount

rate and usually we say in a certainty

that the margin in equilibrium the large

utility of consumption and first we r is

equal marginal utility consumption all

other periods that applies even is this

foam of unserved you get full full

insurance okay so we show that before

and we can do the same thing with

respect to time so to get you l 1 is

equal to lambda times ww1 right of this

condition here w 1l one is the

foreground earnings so from leisure time

and then we would have the same thing

for bay that's s mu l 2 is equal to

lambda w2s over 1 plus r and the s drops

out SMS drops out of both sides and a

beta is equal one over one plus or we

get you l 2 is equal to w 2 so that UL

one in that circumstance / UL to it's

simply through the ratio of the way and

we show that all right i'm not going to

use all that but something i'm going to

use again now

we differentiate with respect to H what

do we get we get s prime look at how our

utility function which is up here s

prime derivative with respect to H times

beta do and I'm just going to say you

too now designate hips in the second

period is going to be equal to what

loads going to equal lambda times G

prime that's a marginal cost of taking

more hell plus a term s prime of 1 plus

R times x2 minus w2 + 4 minus L to the

ones of the educated we're assuming help

was dumb l plus hours of work recall

that m is equal and h is in goods not in

time otherwise we'd have to modify that

so we went through at the end of last

time with this was this is clearly the

marginal benefits the benefits you're

picking up from improving your health is

the fact that you have a higher

probability of getting the utility the

second story not the utility the

marginal utility but the utility

somebody asked me a question you asked

me that question didn't you about what

part of the utilities due to health what

part isn't maybe wasn't you blaming I do

okay I thought it was you and if we

don't have to make that distinction this

automatically comes out it's you I can

any self protection problem you're

changing a probability of an event to

utility level that you're picking up not

the part do do that that kind of act

that's a board director okay maybe they

asked that question yeah okay so I mean

like a I could be sitting and I mean I

think of us the same thing to Professor

Murphy but I mean like he did it in an

aggregate sitting so he said sir made

much more sense but she let's say that

when a given level so you would have a

problem if like x2 is bigger than the

amount that you're earning right because

you could obey you could have problem

where the aging like dies and then

there's really no want to repay that

debt no I think you missed I'll just

really misunderstanding or what we're

assuming it Murphy Murphy believe me

knows the answer that question okay

that's it's a difficult question yeah

and it's fundamental why is not a

problem here I know what you're saying

yes your lighter than the problem here

because if buying insurance think of the

person the beginning of the first spear

saying buy insurance and annuity

insurance you pay certain amount and

they say yes you pay that certain amount

right and you're going to give you

certain amount in terms of expected

wealth over your lifetime that you can

spend either in either period if you

don't lift the second period we get it

and if you lifted a second fig you're

getting more than you paid but it's it

there a selection problem when the

people divide it because you could

affect

we like we're all a lot from the future

right well it might be you know it might

be heterogeneity in across individuals

and a question is do you know you know

their probabilities that if you could

offer them fair insurance I'm assuming

you offering describe fair insurance so

it may be different than everybody else

but as long as you know what his

probabilities are competitive market

will drive you and no cost of running

this visible driving that direction all

right so it's all stuff we did in 301

yeah so you're you're assuming you have

the framework suggested so again I mean

if I didn't have that framework then

your issues would come up you'd have to

say well how am I constraint right I

have that maybe you have a constraint I

can't leave negative debt after

everything i consumed in the first

period has to be subject to the fact

that i can end up leaving negative debt

if I lift the leading debt if I lived

through the second that might be one

concern right it might be a constraint

in this case where we have it if you die

beginning the first period you're

leaving an asset to everybody else yeah

tributing to their welfare and you live

definite riveting to you the guys who

died earlier contributing to you that's

yourself seems like a good starting

point it's not necessarily the only way

you want do in fact we just at a

dissertation defense yesterday in which

he asks why don't people buy annuities

even though the loading factor is very

low fact is nobody much wiser I mean we

have social security and people have out

its up but a lot of there's a lot of not

into annuitized wealth that people i

have a particularly a higher income

levels not so much the low-income oh

they don't mind

but they may i'm not going to go into

everything is answer is answered and i

think it's the right answer is that they

leave it in the leaving bequest to their

children so the have any children ensure

there's a cheaper way of doing it than

the ten percent that's it is ten percent

discount off of fair insurance and it's

cheap it will have your children take

care of you if you live longer than to

buy that unfair insurance and you know

any show it under various utility of

sometimes you can do it it's not a big

effect on the kids well what you leave

them for the most part and so on but

well he's right and I is doesn't really

matter for we're doing good we have full

annuity insurance and even in his case

to us you could buy perfectly fair

insurance everybody would do it okay but

good note good question anything else

so tell professor Murphy I answered the

question so this is what we have and

this is our basic condition and this is

the cost is the marginal cost 88 and

young and most interesting part of this

neither of these terms for model but

that fact that marginal benefit tens on

utility levels on a marquetry tours

that's that's interest okay now para so

from that point of view you can think at

least in part as investing in your help

as form of self protection you may be

but I would buy and complete annuity

insurance remember again I keep going

back to what we did in 30 one thing

wasn't 30 whenever I discussed

uncertainty I don't know what's real 130

to remember we said even if they bought

complete insurance would say fair

insurance you still may want to engage

in self-protection but what self

protection does is maximize your

expected discounted well and what

insurance does is smooth that out and

turn it over to your consumption in

different states of the world so you had

get in generally on our boat and here we

have the self to Protection component

there's no reason because we're buying

full and nobody insurance that this

marginal benefit should be 0 less reason

than some of the insurance problems we

gave but it's a very much related

problem alright so so what if people is

doing

well not a way to look at that is we

know it lambda is lambda is simply mark

utility in period one right with this

assumption my utility in period 1 is

equal to marginal utility in period 2 so

we can replace lambda by you 2x here we

have again now it was converting

everything in monetary units you two

over you two x's is x2 really and the

elasticity of substitution but say over

time it or the concavity overall which

will show it later on anyway this is

what we have here okay so and then we're

equal to that cost that's another way to

look at it now there are two reasons

there to reason I'm is what I want to

show maybe Murphy did this as well but I

don't far as I opt years it goes into

this sort of question to reason why we

might as individual Murphy just

everything you say in the aggregate so

you're going at economy-wide of fact

switches and I'll get a little bit into

the economy wide something as an

individual level and then we'd go up the

economy level the two reasons why you

self you may spend on health as an

individual in a Maximizer one that the

self protection reason that you lower

the probability of the bad state namely

dying and so you have self you have some

self protection and the second reason

that's often more emphasized in the

literature on the value of life sitting

Rosen's class on or near it

classical paper on valuing life risk was

written that frozen emphasizes is the

difference between average and marginal

utility the concavity of the utility

function so forget about leisure now

being in the utility function and you

One X / u X will be a function of the

concavity of the utility function I'll

show you that's true generally what you

want majors it so if you have if you

have a different tree marginal average

utility then an advantage of loading the

rip lowering the risk of dying is that

when you lower the risk the guy you pick

up average utility when you simply

transfer across periods like with

insurance you just pension insurance

just picking up marginal utilities

you're creating margin utility the

simple taking equating margin utilities

in the different states nice but if

there's a difference between marginal an

average than by investing more in your

help you pick up the advantages of

average utility / margin to lure the

concavity of the utility function and

I'll show you that very explicitly

any questions at this point

all right now no question all right so

let's look at this a little bit suppose

I want to see what what this term looks

like oh it's you you two over you 2x

okay but what it was let's make it a

little more explicitly that makes one

assumption that the utility function is

homothetic in time and goods of degree

gamma that down will be less than one

less than equal of one in a strong case

right because of concavity so basically

okay so what do we mean so if we have

homo medic utility functions it means

that you the property if the function is

homogeneous let's a degree homogeneous

of degree gala then we can write utility

in period 2 is equal to 1 over gamma

times you 2x times X 2 plus u 2 L times

L to write with constant returns damn is

one and this is simply the usual Euler

adding up equation right product of

marginal productivities and quantities

summed up is equal to the output or

utility when you have them with it home

genius functions of degree gamma then

you have to adjust for one over gamma ok

so if gamma and that's where they differ

between marginal an average utility

number then if gamma is a hat and

utility is double what you get from the

marginal products effects alone crisis

I'll give you some numbers on that later

on so now if we take that up and look at

this thing divide this by you 2x you 2x

you you 2x what do we get well this

drops out so we get equal one over gamma

and x2 + l to write that out here now

what's this well let's go back that's

what i wrote these equations down if

that's it was really a dependent on that

assumption but let's say in general that

if we take this condition we have let's

write it down that's a beta into x

then equal to lambda over 1 plus R right

and then we know from l2 that they to

you l 2 is equal to lambda 1 plus R

times W right so lambda over 1 plus r is

equal to beta mu l 2 over w 2 or beta

stencil out and that means that you to

well over you 2x take X down here bring

this up is equal to W that's general so

the marginal rate is up to you let me

put it in word the margin rate of

substitution between goods and leisure

and any period really only depends upon

the cost of time versus goods in that

period doesn't defend the money

discounting or anything else are you

doing everything within the same period

that's what you're going to be looking

at ok why is that interesting well

because now we can say we can replace

this thing like w2 and so we only have a

very neat expression

we have this very neat expression what

we're saying is that you we look now

back to our condition here s Prime bathe

and then this turns out to be 1 over

gamma times x2 + l2 w the gong to be

will then be Prime

okay so what we have done is saying well

this value this benefit we now convert

as benefits and weak expression right

what it because the tournament brackets

is your full consumption in period two

to measure your full consumption that we

take yet actual consumption we wait you

lose your time by the wage rate so we

can add them up their own good units now

and it's your full consumption so we say

well the benefit is just full

consumption adjusted for the difference

between modern average utility 1 over

gamma I mean it's paid in there not

potential that's what we have no waiting

for you may not abate is there okay all

right and the cost or you didn't do

anything on it on the cost side any

questions on this

question now we can rewrite this let's

take here we have a w-2 l2 term over

here s prime okay let's assume now let's

assume that babe let's go back to our

something babies equal 1 over 1 plus R

okay so how many places by 1 over 1 plus

R ok so now I have s prime over 1 plus R

X prime of 1 plus off I'm going to bring

that to the other side and I'm going to

get on this side s prime of 1 plus R on

one over gamma minus 1 times X 2 plus l2

w 2 is equal to what is equal to s prime

over no it is equal to s prime over 1

plus R times right

it's equal to G prime of a plus s prime

of 1 plus R minus w2 that's what the

conditions all right now I said what

does that condition me well will you

understand what that condition means is

the following suppose you chose H to

simply not to maximize expected

discounted utility but to maximize

expected discounted well so what we're

going to maximize ok we're going to

maximize is the following maximize w

that's going to be equal to by

definition full well let's say W 1 over

beta sw2 minus G of a so don't choose H

to maximize net full wealth so if you

simply interested in getting as much

expected wealth as possible not worrying

about in terms of utility that's what

you'd be trying to do and we do that

what's the first-order conditions well

first order catalyst and we have aged

ages are only visible here w is fixed w1

government to is given first-order

conditions is what it's beta s Prime and

w 2 is equal to G prime of H

right and concavity with G double prime

negative right that's remember we

mentioned that would be perfectly good

maximization first order condition now

what does it mean what does it mean well

anybody

era what is it I mean so the g prime of

h is the marginal cost you pay now well

yeah you only hit that yeah he doesn't

want howitzer decided to run but it did

help now what you're getting out of that

is the chance that you marginal increase

in blooming well you get some is get

wealth yo man I mean what it means

literally if that's what you saying you

know by having a higher probability the

living you pick up the expected gain in

the wage rate I'm definitely that's what

you getting at your full way that's what

you might think yeah that's really loved

surfing and second what that may not I'm

trying to get it something a little

deeper than that and maybe I didn't make

myself clear but with me go back to

utility maximizing condition over here

all right right here one over gamma

minus wanted when will that condition be

the same as this condition

remember babies you go on over 1 plus R

in camera is one when gamma is one so

that's it as an important implication

really hasn't been recognized that much

in the Health Board of you it's implicit

obviously what it implies if it down to

equals 1 it implies max V quipment to

max w and the reason for that is well if

gamma is equal to 1 is constant returns

to scale really doesn't matter as long

as you can pull more time into the

leisure out of the markets that really

doesn't matter if you consume more in

one period or you can assume it broken

down by a couple of period I me to

something to beta is equal to 1 over 1

plus R we use the garage it doesn't

really matter so that's the case Michael

just maximize expected well so under

those conditions all you're doing is

self-protecting you're maximizing I'm

calling maximizing expected discounted

wealth self-protection okay but we know

that's not the whole problem okay

because we assume in general down less

than one script concavity right so we

have strict concavity then what do we

know well we know that this side here is

positive right good one over gamma it's

greater than ones that's positive okay

and so that would say that G prime H is

then greater than s prime over 1 plus

rw2 right

Yeah Yeah right turned out just right

and what does that mean what does that

mean well Eric you would say well it

means that the marginal cost is ephrata

Strickland the discounted value of the

weight right right it does me back but

what does it mean more than that

it means you're spending less on health

but at 8 g is concave is convex yeah

increasing cost of good question and

make sure yeah geez convex you ask the

cost so it's a convex corresponds

absolutely G is convex so what does it

mean you spending more because you

picking up an additional benefit you're

picking up this differently marginal

average utility which is what gamma less

than 1 means you're picking up that

extra benefit figure that extra benefit

and so you're going to go further so the

optimization that we see is a company

with gamma sudoku that's the one is a

combination of the self protection got

as close to one that's pretty much all

you have and it happens a lot away from

one and a lot of what you have is that

is very traditionally emphasis in the

value life literature there's it would

be marginal average utility right the

concavity the utility okay everybody

follow that

you follow yeah okay why is it happening

oh because on utility you have

diminishing return so what you like what

you know what you can get in terms of a

better utility string counts for a lot

that counts for a lot even though you

know your margin utilities are gone be

equal in both periods in equilibrium all

right so maybe this may seem well this

is just a lot of formal nonsense but it

gets interesting and tell you what

you're really getting in your theory of

the optimal investment in health which

is the theory that I think is very

relevant because I stress right at the

beginning of the analysis of the health

area the personal input into health is

important sure we're all born with

genetic predetonate extras so we're not

looking for which predisposes you toward

various healthiness o disease and

there's no doubt people different

honestly in that but in addition to that

people can do a lot to affect the health

and then I said in the modern world they

can do more and more do you smoke and I

probably very few people in this room

smoke the educated people don't smoke in

in in pretty much pretty much all

countries not all used to be truly

educated people smoke more than others

because they were wealthier and like

everything else smoking was a superior

good now yet Katie people smoked glass

good question why partly this reason

here now that's in a way of investing in

their health and that and will say that

their lives are more valuable now you

say that

politician will see red but all that

we're saying is from their calculation

it's more valuable what you want to do

with public policy may be another story

allure you won't ignore that public

policy let me say but we're not saying

that the politicians can relax no

there's going to be politically correct

and still say this that people see

themselves is more valuable they have

more utility utility levels that are

important and more utility that's

crucial in the value life literature

okay the record recognize I'm going to

use that property but it's a lot of

personal input and I think that's really

important that's really important it's a

lot of personal i mentioned smoker Magra

mentioned a hundred thousands another

thing that we all do and I mentioned a

lot would be Florida worth repeating

what kind of diet do I have do I

exercise or not do i do I drink you know

heavily but do I don't not drink at all

do I you know when I have a symptom do I

know what my symptoms are too i go to a

doctor quickly i say certain type of

cancer center but if i go quickly I can

get control or do I go quickly attack

you go on and on with things that people

have to do there's nobody there it was

telling you do this you do that you do

that we're free individuals that's the

great value for free society but it also

means response you know what you do is

kind of on what you do you have you have

to make those type of decisions so

they're very important in a modern world

and that therefore an individual

maximization problem even though this

genetics and even though there is

Madison and the state of medical

knowledge and there's drugs and so on

what the individual does give it all

that that's an RG function given all

that is still very important you're

sensitivity levels to matter so much how

do we think about the cardinality of

utility for me we know that any of many

our transformation services there any

monotonic transformation is also utility

function it is is it's true now how do i

always think about it in this particular

context there the monotonic

transformation will be on the v function

first of all you gotta remember not on

the new function but may mean once you

take a monotonic transformation and I

think some additive utility frame that's

the way you have to think about so then

when you cook your first order

conditions you have to look at what

those first-order conditions are and

they did not add a commute ility

you may be nothing the amount of kind of

transformation on the huge taking upbeat

that's which that's crucial so we're

dealing with a particular form of that

but we're putting in monetary mainly in

monetary units they make full income the

only property of the utility phone we

say well we're going to estimate an

additively separable utility function i

assumed among the class of utility

functions the explaining people's

behaviors an additive week separable

utility function and we're going to ask

me there for a parameter the gamma of

that attitude which separate that's the

only problem that we need to know if we

assume that also there exists all the

fetid among that class infinite classic

you're so much so perfectly is perfectly

legitimate what we're doing it does a

challenge what galleries and i'll come

i'm going to say a lot about that but

that is a parameter that's been used in

a number of studies okay anything else

yeah another way to put it let me put it

away something you may like if if you

have damage equal one all you have to do

is estimate fokin something wrong let's

duel would be great having from a

political point of view that'd be great

guesstimate the wage rate you measure

the value of leisure time by the way

Drake perfectly legitimate not fully

legitimate data to consumption good you

have it your home have everything

the problem is we've got much less than

one what you have is well I said

different marginal average Julie another

way it says you have consumer surplus

that's what that's another as which

consumer surplus concert so you also

have to have a way of measuring the

consumer surplus now I Congress have

ways of trying to measure i put in a

particular form here you may measure by

some area under some demand function h

for example that you change the course

of age and so on alright so in addition

to the full egg huh in fact that your

time and good in utility terms is more

than may be greater than these marginal

values that's consumer service

alright so now let me go into him unless

you're a question until more explicitly

into what's called the statistical value

of life the many studies that have been

trying to measure in a sense what we're

getting at how much are people willing

to pay to award certain risk oh how much

are people do I have to get paid to make

them willing to take on various risk so

for example the classical studies

started with looking at jobs some jobs

were risky you could die on the job

another's we're not how much more if

anything to the people who had taken

their risky jobs get pay compared to

those who are taking a less risky jobs

so if you knew what the difference in

risk you can tease out from that and

assuming you have to worry about

selectivity of people taking the risky

jobs may be less concerned about risk

than others so it's not a simple and

econometrica problems I'm saying

assuming you can do that the first study

didn't try to do anything much on that

so many more recent study of course try

to do that but what he meant you know

whatever there is you may get a biased

estimate but what you're estimating when

you has to even just look at that

difference is some measure what the

marginal person is willing to pay in

order to take on this greater risk of

dying and that's a monetary value so

let's say if the greater risk of dying

is 110 thousand in period 2 e-business

very small additional risk in our

problem how much would you be willing

have to be paid to take on a 110

thousand additional risks well we have

that from our first first-order

conditions

we would say well what they have to be

paid is if we change as prime 2 x 1

10000 how much more would we have to

change the right hand side in order to

measure how much they have to be paid to

be compensated for that it's one way you

can think about that or an indirect

utility function where probability of

dying edges is how much more wealth they

have to have to compensate that's a

better way I like to think of eternity

the indirect utility function functional

wealth makes prices and probabilities of

survival how much higher wealth make

sure it just as well off with a little

bit higher probability we get that out

of it should come out of some

maximization problem you can get out of

just out of utility function but the

utility is adjusted to equilibrium so

it's really comes out of your

maximization and they the sort of

estimate the literature you find is for

if the increase in the risk is 110

thousand per year now we have only one

additional yips mother was lifetime

you'd have to be paid an additional five

million dollars I'm sorry what I'm

saying is right but I did say it right

so well thing is wrong when people say

what I'm saying alright what I know what

I'm writing them that's not right here

right I usually say you don't know what

you're right I know what you're right

usually get it right so I didn't get it

right so let me go back and say look at

what's going on let's say the risk is

that if they're going to increase their

risk by 1 10000 in each year how much

they have to be paid each year and

answers that is around five hundred

dollars now from some studies they find

three hundred dollars of two hundred

thousand seven hundred dollars or so on

but if you take five hundred dollars

five hundred dollars divided by the

change in the probability equal one over

one and pound that's equal to five

million dollars right so it's like

taking s Prime changing it by 1 10000

dividing it by this term on the right

and that's a numerators five hundred

dollars and the that / in verse 1 10000

is 5 million dollars so if you don't

know that's what people how people get

what they talk about when I say we're

talking about this physical value life I

mean the notation I'm going to use for

that is b.s you know and that's let's

say anywhere from 3 let's say seven in

us and not to everybody in the US for

the typical person in us a lot of

qualifications you have to make there's

not a single a poor person will pay less

will have a lower value than a rich

person why well thats a utility function

same gammas then the fully fully thumb

is lower so my flowing time is half your

full income and my gown is the same then

basically I'm I'm going to be willing to

spend proximately half of what you're

willing to my value life yours is Phi

mine would be two and a half so we'll go

to a poor country well the value of life

in India is 162 that in the United

States now some people complain while I

wait a minute wait a minute you're

saying people in India are worth as much

as people in the United States well

depends on what

the metric you using if you ask is

people as the government of india

willing to spend as much to reduce the

probability of a road accident a depth

of a road actions of government of the

United States the answer would be

clearly no just travel in India and you

can see that the roads are terrible

invitations accident so poor countries

are willing to spend a lot less they

recognize this it's obvious cuz you know

instead of getting all upset and on one

side let's think of the following I'm a

government of limited resources how much

do I want to put into reducing the

probability of dying put into education

put into some other form of activity and

the Indian government is limited

resources are they going to put listen

to this so they recognize this no way

they recognize it proportionately or not

you are you about that in some sense

they reckon it i'll show you some

evidence they recognize it in a sense

more than proportional not less than

proportionately more but they recognize

it so I mean it's just an aspect of

reality that the individual has to

recognize and the society has to

recognize without you're not talking

about ethical works or different

individuals you're not talking about

that and you may say we should try to

equalize expenditures on health provider

II that's fine Oh guys what's the value

that people in a place on this what's

the value of the society places on

that's going to vary with individual

income and with incomes of a country and

it does vary not just so you know a

theoretical and because it does vary you

see it all the time so now the US spends

on health is a lot different than the

amount that say Nigeria spends on health

or Pakistan or India or China or Brazil

that doesn't mean we have the highest

life expectancy in the world that's a

good question maybe we'll get into that

but there's a clear strong relationship

between expenditure on health and for

company coming up country both private

expenditures and public expenditures any

questions empirically how does the value

of life change the word person is you

increase the probability of that well we

don't have a lot of good estimates on

big changes in the probability of less a

jerk for the United States yeah you can

say in I'm glad you asked this question

you can distinguish various concepts you

can say given what I know what s is what

are we willing to pay for a vehicle

small variations around us but another

question you can add is given the same s

what am I willing to spend if my option

is s goes to 0 to avoid s going to 0

let's say I'm faced with a major disease

and am I spent enough resources of my

own I cannot say eliminate that risk of

that disease at least for a while

suppose that was true how much mine

wasn't suspend to do it right well the

same analysis is relevant but now you

have to think of this not as a marginal

change in this for some integral over

all these marginal changes right it may

be that integral is going to happy it's

going to be all your well you can spend

more than you have all right see if

upper limit is you well so let me put in

the context here I mean it philipson

Murphy and I have a paper on exactly

this problem people at older ages how

much how people won't expand for modest

improvements in your life expectancy

let's say they can take a drug that can

increase their life expectancy by six

months how much is that worth

say their 85 and they could and have a

disease have a cancer the new drug just

came out for prostate cancer vaccine

it's a revolution I think what it does

so far they can estimate on the average

increases the is for people with serious

prostate cancer everything else doesn't

work and it increases their life

expectancy on the average in them here

in their sample group advice for months

it costs hundred thousand dollars a year

take that drug people willing to spend

it alright so some people say well

what's four months I mean on the other

hand if it's four months or nothing and

you're not leaving money to anybody a

lot people say well else let me throw

everything I mean more or less in it

these seems something with me to consume

but throw everything in now I'm will you

spend a lot so it's it's it can vary

with the type of risk you're facing and

to the extent that your willingness to

pay and we would expect that from this

analysis more or less you expect your

willingness to pay very with as assess

various to be a function of s

particularly if you think a percentage

change the nest this analysis it doesn't

put sanitarian isn't it will will vary

so we conclude that people may be

willing to pay a lot for just small

extensions in their life expect no and

one reason is about it even that had as

it might be willing to pay a lot but we

get it out of this type of statistical

value of life enough it has to be more

sophisticated so it's a good question

afford right hadn't really been answered

in the literature very effectively any

other questions

yes parochial interest rate risk is the

same wrestling to order well for small

changes they have continuity yeah that

should be it should be the same right so

in first we haven't said you know s

prime could be positive s prime can be

negative and the first-order conditions

are the same but big change of course it

could make a difference I can make a big

difference from let's say my probability

survive me but I don't do anything is

point 8 and if i do a lot becomes point

95 but I don't do anything might be

point six five though it may not have

the same values it's like any function

you know dependent maybe one presumption

might be the avoidance maybe as a bigger

value but a little tricky but it may not

may not gentle on the same value right

marginally this analysis right is

continuity and others it should have the

same guy but generally not on the same

time okay you see why there isn't any

problem how much my willing to pay for a

cue club well that's i just bought three

cucumbers and a what 3650 sense of

cucumber I don't know what they would be

self will it's a good you go to Whole

Foods sitting there so you're going to

pay a lot can you say but ok this is how

much am I willing to pay for ten

cucumbers doesn't would mean I'm willing

to pay fifty cents each for the

additional seven right that's what we

say is an egg lean client demand

function you willing to pay less as you

go on your mario dylan's declining well

something very similar is going on here

and always not to come as its nose life

years or something

yes protec invest a accidents involve no

train people from Team yourself or what

do you mean from killing well we don't

we have laws against helping people

commit suicide yeah because there's some

things you can't I mean you should be a

land beyond question I mean we have had

laws that I don't forget people taking a

line I think also the laws of God now we

have that boil and you can't assist

people people have been sent to prison

for a system but they take the

individual how's the government going to

do it miss some things you can do more

easily some things you can't to a more

easy how's it going to now what we have

is you can have somebody committed

somebody who suicidal and a relatives

there is this person suicidal they can

be forcibly committed to a mental

institution now the laws have made it

much harder and harder to do that I

think in recognition of the fact that

you know individuals have a lot of

rights to say what they're doing and

being committed people against their

will you know maybe in case that's

justified you gotta be very cautious and

using that power good it can be and it

has been greatly abused at time so

there's certain things you can't affect

very much and for the P let let me go on

people want to commit suicide we'd say

this testicle you life is 0 or negative

they do they do but maybe let's answer

your question good question but there's

an answer to every good question has a

good at search should have a good answer

sometimes we don't know what it is but

it happened what's a good answer to your

question let's say I'm drive no not that

may be true I'm driving recklessly I'm

affecting everybody else I'm effective

externality issue is well then we dealt

with that one I think I mention that

when the drunk driving issue I'm

affecting everybody else yes there's an

interest in limiting the recklessness of

individual driver now we had a tort

system where you could be forced to pay

fully compensate the burden which

wouldn't do it but that's you know you

killed somebody's life and that's the 01

situation they may be putting all their

wealth on you don't have that well

what's called judgment proof so we have

other way we sent you the prison you

know we do other restraints on maybe

build a better Road preventive that's

why we do it okay anything else all

right

so basically what way I think about a

discounted value of life so here we're

going to be that's that's going to be

patient I just one year basis is come to

be now we have a whole series of years

so then we'd be summoning something we

had n periods or you spending utility

function of and periods you can take an

action I'm going to do that a little bit

later on let me hold that football and

you have to think of it over number it

but if I improve your think of this is a

very end it speeds the demography of

that of what life expectancy as is

interesting fundamental complementarity

if if I'm fighting an action that raised

my probability of surviving next year it

also raised my unconditional probability

of surviving the following year and the

following year and the following doesn't

necessarily affect a conditional

probability once i reach it but the

unconditional probability is effect and

see on conditional probabilities that

enter into the analysis not the

conditional problem i'll show you that

more explicitly it should be clear

because cap s let's say the conditional

problem is a little less the

unconditional a cat beds just cat s so

by three period y'all sure that it is

important these are utility function up

there nine three periods I Toby and

going to be equal to a loser exactly

everyone

let's call that

go on not paid oh whoa what is just us

to it as of the present what's my

probability vibing through the second

period s3 is as of the present what's my

probability surviving through the third

these are unconditional probability now

you can write s 3 is equal s 2 times

little s 2 n plus three little s 2 is

the condition conditional probability of

surviving the second group given that I

reached secondary well this example

everybody recent second beer but may not

be this three is given I've reached the

third career what's my probability

surviving through the third group and

capital s 3 is the product looks that

that's a compliment Harry type property

and so we'll use it to gain some

insights right but not yet

okay down there

all right yeah yeah this is where it's

doing no I said five million let's say

is in the central Tennessee is that a

reasonable no well for the United States

the typical person how we're going to

know let me give you a back of the

envelope type of calculation the charger

is well it seems like a lot of mine is

pretty reasonable so you take this is

only for young people I'll come back

protect young person

make the following calculation say the

typical income in the United States is

forty thousand dollars approximately

annually

now in the early literature really very

early literature on the value life like

there is a famous book written by an

actuary out Alfred black though we've

actually really important work

fundamental theory of demography what's

called steady-state theorems on age

distribution we wrote a book the money

value of man and you just look at

discounted earnings we don't want to do

that we want to look at fooling them

well so we have to add the amount spent

in the household well how much well we

know 40 hours a week is spent in the

marketplace then what 128 hours a week

is left over 168 hours a week so 120

hours a week but a lot of that we would

call maintenance maybe like sleeping

personal care so I take like 1.8 times

the hours at work and multiply that so

that means i would add to this 40,000

1.8 times that for the value of the

household time so what's 1.8 times

40,000 that 72,000 I think right so it

plus 72,000

4l ok this is earning so WM this is l

can I get like 100 I say 110,000 son

12,000 that's my annual value just a lot

of money you think of 40,000 how am I

ever going to get 25 million but I don't

have 40,000 I have 110,000 now discount

110,000 at

no not quite yet not quite yet I was

going to say it but there's a step

before the discount and what's the step

before the discount funny bro she okay

yeah maybe let's think of it as Spain

stagnant know what it could be if it

could be grown ok so you to grow that up

that's not what I'm thinking but you

could be I mean it make some sense it

could be grown oh yeah oh you know if I

know what to grow great wait for what in

our analysis how would I capture gamma

so you know I have to this is full what

I have now is a full income full

consumption approximately backing them

multiply that times 1 over gamma so in

study is the paper written by filson

myself and pence ends Rodrigo Suarez and

what we use for we use some

international comparisons full income

taking account of life expectancy and

compare inquiry and so on we use gamma

people based on literature gamma R equal

to a half so I mean that is the figure

is not fully pinned down by enemy let's

say gamma equal to a half all right so 1

over gamma is 2 so x 1 over gamma equal

to x this that gives 220,000 so this is

the full income now we have this is a

mess measure of the utility each year

220,000 so now we're talking about no

real money but that's going to be

discounted so it's going to be this an

annual game let's discount that at

four percent so we have to multiply that

times 25 right roughly put this is like

for perpetuity people living a long time

and roughly you have to multiply that

times 1 over R 1 over r is for

percentage 25 all right so now we take 1

over R 1 hold a little R that's equal to

25 times all that and then we get x 25

and we get what what do you get that's

right

finally the roughly by now right twenty

thousand one thousand two hundred

thousand times 25 will be 50 it's a

roughly five million so we're getting it

we're getting up there the right numbers

now of course these are rough

calculations John it may not be a half

interest rate maybe it's five percent

rather than four percent but still you

can get a number in the right ballpark

so it's back-of-the-envelope calculation

is very useful the convinces our skeptic

these numbers make a lot of sense 3 4

i'm not saying it says five million

three four five six seven either that

was the right doesn't write ranges so

Murphy does is aggregate stuff which was

very interesting you know it really

doesn't matter so much but the

individual value life would because

you're multiplying by all the people in

every age it make me know it makes a

difference it's going to be a huge

number instead of using five million

used three million two million still

going to be in the trillions tens of

true but the individual level makes the

difference and it makes a difference the

idea of a gun don't misunderstand what

I'm saying put everything you're going

to multiply up so I'm using two and a

half out of five everything is sort of

cut in half in his numbers as well as

these but so this shows that showed

convinced me when I trying to ask myself

with these are these reasonable nauman

Isaac really are they're pretty

reasonable if you think about it yeah

no no how much the individual values

their life that may be related to what

society values this is all based on the

individual is all comes out of

individual maximization now to extend

that society's values or reflecting

individual values and somehow

utilitarian view of what society does

yes not what also reflects society's

rather it's not directly reflecting

society's calculate societies that we

use web leaders again if we were to

calculate the societal value as a disco

life would be either still be including

would walk with the value of leisure

still be included with my castle well

why not well your society value less the

time you have fun in the household

rather than the good you eating why

should i why should we come home I don't

know if you measure like production asst

like what led to this household

production household production that's

the main part of the game in the house

already I'm only kidding about that but

it's an important component of what is

it I you should just look at what's in

the marketplace I mean that's what we're

trying to get away from just look north

in the market rate we're just looking to

market late we look at x4 why ignore

people's the tip phones why your peoples

leisure all right most people have

mainly leisure and little X so should we

give them a little weight if we

separated out

production department if we separated

out well depends why you if you can

measure the household for my abilities

then that would be fine but they would

include in them to have time input so

they're just different ways of measuring

the same thing you can measure national

income accounts from the out some of all

the outputs all the goods consumed all

consumables investment or you can sum up

all the value of all the influence and

those should give you the same things

you can't double count you don't want to

count the output and then also count the

employees that's double counting but you

could do it either way since we don't

have good measures of outputs for a lot

of household production we look at

inputs okay anything else yes so you can

doesn't rhyme an experience so if you

have several periods a half he should

have more incentive we would you know

too long we the total of the discounted

value of what we're doing we're taking a

set of actions that affects some

probability surviving only next period

but several periods and that's going to

be greater than just what the one period

effect would be assuming the one period

affection control yes now be great so

younger people to a higher value on it

on the life than you all of you you see

more young people teaching reasons and

all the people all the people shouldn't

mention Sigma wonder what I mean all the

people have a lot of risk throwing up

the younger people others seek out the

risk I mean older people have diseases

and so on they're facing a lot of risk

theirs are not facing race how much I

spent as an individual on various ponds

of medical treatment so I go for

preventive checkups I mean you know

should I have this procedure in that

procedure which I myself check for this

that these are all rest so they just

have different risks nobody could say

that all the people if you were risk

when it comes to this area to have more

risk but they should be you're saying

that I'm not not accounting for the ways

they have but or that

well they haven't I don't know how their

attitudes toward risk compared to

younger people if you mean in terms of

the preferences I mean you want to

separate out behavior from preferences

and prices I think I don't know if their

preferences are different they may look

as if they're taking fewer risks that

say in their investments that doesn't

mean the preference toward risk is any

different they're in a very different

situation let's say I'm the younger

person taking risky investments I've no

average that's partly insurance over 30

year period they averaged out their

return so the variance in the 30-year

return did Scott it may not be so big

well an older person my stock market

portfolio goes down maybe that's just a

bad break but you know this I don't have

that many more years I have to worry

about that so you can understand that

behavior toward risk as being different

doesn't mean their preference services

those a different issue yes okay good

question is it a general issue is what

determines their values in my

calculation I was able to get the value

of four to five million without putting

any emphasis on the value the fact that

you value being able to see your spouse

and your children your grandchildren and

all that that should if anything only

make the numbers higher our framework

doesn't capture that because all I'm

looking at is my

own consumption of my only I could

easily capture i put in other people's

consumption let's say my ex is not some

with my own consumption but it's my

children from subject and i have it so

in principle I hadn't incorporated the

way I did this sort of

back-of-the-envelope calculation I

didn't incorporate and that means I'm

being conservative on their estimating

will again what we put generally it's

better than err on the conservative side

when you're trying to say the number is

kind of reasonable

so last couple of things I want to say

today so you take a country like India a

summers 16 to the United States and

approximately speaking let's say they

had the same gamma their value life

should be like 160 United States it's

five million the United States it should

be like 800,000 not negligible for young

person is still a lot less so in terms

of what they will do they're gonna

they're not those are willing to spend

as much as somebody United States will

expect they know he's an individual oil

the government to come back to your

barrel in the individual of the go and

they don't spend it okay you can see

them what there's been a medical care of

what we what they spend roads or other

factors that can affect this fact if you

look at the ratio health spending to GDP

that rises with economic development

richer countries spend a lot on a large

amount got a larger fraction so let's

say we spent fifteen percent or so on

medical to India might spend maybe six

percent so there's so many other than

said they're not willing to put as much

resources into it it makes a lot doesn't

mean everything is exact but it makes a

lot of sense okay

this for what

if you think about this and I'll discuss

it next time suppose you know we just

had a swine flu scare government and

grossly overestimating the likelihood of

such a pandemic but we'll show reason

why it's rational to err on the side of

caution and the question you think about

for class on Tuesday is well what would

be the cost of a pandemic similar to the

flu pandemic that occurred in nineteen

eighteen nineteen ninety look up some

numbers on that i'll test john on

tuesday and then try to give a value for

the world tell me what would the you

estimate of what the cost of the world

would be of a pandemic equal in

magnitude to nineteen eighteen nineteen

nineteen flu pandemic which was the

greatest pandemic not in history the

worst ones but in the last century and

three-quarters

and then answer the private how much how

much should we willing to pay to try to

avoid that okay that will be your

assignment for tuesday

you