Master,2016-2017 F.Langot Sant´eetassurance

Sant´e et assurance

F. Langot

Univ. Le Mans (GAINS-TEPP & IRA) Paris School of Economics

Cepremap & IZA

Master, 2016-2017

The rise in the life expectation at age 65 - OECD

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

11 12 13 14 15 16 17 18 19

Years

Life expectancy, Males at age 65

US Jap Ger Fr

The rise in expenditure on health - OECD

19602 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

4 6 8 10 12 14 16 18

Years

% gross domestic product

Total expenditure on health. % gross domestic product

US Jap Ger Fr

Motivation : Long run trends

Table–Stylized facts : the US case

Years Heath Expectation of Productivity share life at birth index

1960-65 0.06 69.9 1.0000

1965-70 0.07 70.7 1.1000

1970-75 0.085 71.3 1.2100

1975-80 0.095 73.24 1.3310

1980-85 0.11 74.32 1.4641

1985-90 0.115 74.86 1.6105

1990-95 0.13 75.58 1.7716

1995-00 0.14 76.36 1.9487

2000-05 0.16 77.04 2.1436

2005-10 0.163 77.82 2.3579

Motivation : The use and the value of life.

Technological progress increases economic resources. How to use them ?

Hall & Jones (2008) show how this rise in the productivity is used to increase the health expenditures and thus the life expectancy.

Technological progress increases wealth ⇒“additional resources”.

The increase in wealth ⇒more health expenditures and “live longer”.

Assumptions 1

Let x(t) denote the person’s health status :x(t) can be interpreted as an health “capital”.

The mortality rate of an individual is the inverse of her health status x(t).

mortality rate = 1 x(t) ⇒

Prob(Survive in t) = e^{−x(t)1 t} Lifetime expectation = x(t)

Rappels sur la loi exponentielle

Une loi exponentielle mod´elise la dur´ee de vie d’un ph´enom`enesans m´emoire

⇒ La probabilit´e que le ph´enom`ene dure au moins s+t unit´es de temps sachant qu’il a d´ej`a dur´et p´eriodes heures est la mˆeme que la probabilit´e de durers p´eriodes `a partir de sa naissance.

⇔ Que le ph´enom`ene ait dur´e pendantt p´eriodes ne change rien `a son esp´erance de vie `a partir du tempst.

Rappels sur la loi exponentielle

SoitX une variable al´eatoire d´efinissant la dur´ee de vie, d’esp´erance math´ematiqueE(X).

Si on suppose que

∀(s,t)∈R²+, P(X >s+t|X >t) =P(X >s) alors, les fonctions de densit´e et de r´epartition deX sont : ( dF(t) = 0

dF(t) = ¹

E(X)e^−E(X)t

F(t) = 0 sit <0 F(t) = 1−e^−E(X)t sit ≥0 X suit une loi exponentielle de param`etreλ= ¹

E(X).

On aP(X >t) = 1−F(t) =e^−λt qui donne la probabilit´e que la dur´ee de vie soit sup´erieure `at ⇔Prob de survie.

Assumptions 2

Individual receives a flow of resourcesw, that he spends on consumptionc(t) and healthh(t).

Health expenditures govern the health status : x(t) =f(h(t)) avecf⁰>0 etf⁰⁰<0

Health and endogenous lifetime (HJ)

The intertemporal budgetary constraint is : Z ∞

0

e^−(1/x(t))t[c(t) +h(t)]dt = Z ∞

0

e^−(1/x(t))twdt The heath production is given by

x(t) =f(h(t)) withf⁰>0 andf⁰⁰<0 For the intertemporal utility function, we have :

max

c(t),h(t)

Z ∞

0

e^−(1/x(t))tU(c(t))dt

Health and endogenous lifetime (HJ)

ASSUMING STATIONARIY :z(t) =z, ∀t and∀z Z ∞

0

e^−(1/x)tydt = y Z ∞

0

e^−(1/x)tdt

= yh

−xe^−(1/x)ti∞ 0

= yh

−xe^−(1/x)∞+xe^−(1/x)0i

= y[0 +x]

Stationarity ⇒ Z ∞

0

e^−(1/x)tydt = xy

Health and endogenous lifetime (HJ)

ASSUMING STATIONARIY :z(t) =z, ∀t and∀z The intertemporal budgetary constraint becomes :

Z ∞

0

e^−(1/x)t[c+h]dt = Z ∞

0

e^−(1/x)twdt Stationarity ⇒ x[c+h] =xw ⇔ c+h=w The heath production is given by

x =f(h) withf⁰>0 andf⁰⁰<0 For the intertemporal utility function, we have :

max

c,h

Z ∞

0

e^−(1/x)tU(c)dt

⇔ max

c,h {xU(c)}

In this case, the problem becomes : max

h {f(h)U(w−h)}

The FOC with respect to medical cares h

f⁰(h)U(w−h)

| {z } Marginal returns to medical cares

= f(h)U⁰(w−h)

| {z } Marginal costs to medical cares f⁰(h) ⇒ ↑in the lifetime expectancy

U(c) ⇒ welfare of an additional year of living.

U⁰(c) ⇒ dh>0⇒dc <0⇒ ↑ cost of the medical cares f(h) ⇒ lifetime duration

Health expenditures are proportional to the value of a year of life : h=η_fV where

V = _U^U0

ηf = f⁰(h)_f(h)^h ⇔ xf(h) f⁰(h) =xV

The marginal cost of saving a life

Letdhbe the increase in health expenditure anddmthe reduction in the mortality rate. With our assumptions, we have :

x = f(h) ⇒ ^dh_dx = _f0¹(h)

m = ¹_x ⇒ ^dm_dx = _x¹2

⇒ dh dm = x²

f⁰(h)⇒xf(h) f⁰(h)=xV

| {z }

?

The idea of Hall and Jones (2008) : they observe x and h⇒This allows them to estimate f(·), and thus the marginal cost of one additional year of living _f^f0^(h)(h).

Health technology (x_data = fb_i(h_data)) by cohorts i

x= 1/(nonaccident mortality rate) ;h= health expenditures per capita

The marginal cost of saving a life

Table–c^dh

dm - 2000 Dollars - Hall and Jones (2008)

Age 1950 1980 2000 Per year of

life saved (2000)

0-4 10 000 160 000 590 000 8 000

10-14 270 000 2 320 000 9 830 000 152 000 20-24 1 170 000 3 840 000 8 520 000 155 000 30-34 500 000 2 120 000 4 910 000 108 000 40-44 160 000 740 000 1 890 000 52 000 50-54 70 000 330 000 1 050 000 39 000

60-64 50 000 280 000 880 000 47 000

70-74 40 000 280 000 790 000 67 000

80-84 40 000 340 000 750 000 125 000

90-94 50 000 420 000 820 000 379 000

Health and endogenous lifetime (HJ)

The health expenditures are proportional to the value of a year of life : h

c = ηf

V

c ⇒h=ηfV whereV =_U^U0

IfU(c) =A+^c_1−σ^1−σ andf(h) =Fh^α, then h

c = α

Ac^σ−1+ 1 1−σ

Ifσ >1, thus the health share increases in total expenditures when w ↑.

Health and endogenous lifetime : quantitative implications

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

50 100 150 200 250 300 350

Hall & Jones: w=1/w=1*gap

health

function

marginal benfit w=1*gap marginal cost w=1*gap marginal benfit w=1 marginal cost w=1

Left : Calibration :w= 1,α= 0.15,σ= 1.5,A= 5,F = 100

The gap between actual data and the optimal lifetime expectancy : h such that x^fb(h)

fb⁰(h)= xV

For different curvatures of the utility function ^c_1−γ^1−γ withγ∈ {1.01,1.5,2,2.5}, and different trends in the health technology sector.

Fact 1 : GDP Share of Health Spending

The US spend more on health care than other countries

051015GDP share of health expenditures

DE SE NL SP IT FR DK US

Source : OECD Health statistics. 2005.

Fact 2 : Health Status

The extra spending does not purchase any additional health

.6.65.7.75fraction in good health

DE SE NL SP IT FR DK US

percent in good health (corrected)

Source : Authors’ calculations

Comparable measure of health Status

Health varies across countries in part due to different reporting scales (Jurges, 2007 ; Kapteyn et al., 2007).

We adopt the strategy proposed by Jurges (2007) :

1 Using European data on Nrespondentsn= 1, ...,N, we first consider a logit for self-reported health Hn,i= (0,1) (1 = good to excellent, 0 = fair/poor) :

H_n,i = I(X_n,iγ+ψ_i−ε_n,i>0) Pr(Hn,i = 1) = Λ(Xn,iγ+ψi)

on chronic conditions, demographic variables Xi,t and country fixed effectsψi.

2 Arbitrarily take France as the reference country, we use this statistical model to predict the health status for an individual in other countries (including US) : p(Hb n,i= 1|Xn,i, γ, ψFR)

Corrected Fraction in Good Health

.4.5.6.7.8.91fraction in good health

DE SE NL SP IT FR DK US

percent in good health

corrected uncorrected

Source : Authors’ calculations

Health Status Transitions

We can use a similar strategy for transition rates :

1 The statistical model is estimated using European data for j= 0,1 : Pr(H_n,i,1=j,H_n,i,2=j) = Λ(X_n,i,1γ_j,1+X_n,i,2γ_j,2+δ_i,j)

2 We predict the health status transition for an individual in each countries (US and Europe), taking France as reference :

p(Hb n,i,1=j,Hn,i,2=j|Xn,i,1,Xn,i,2, γj,1, γj,2, δFR,j) (1)

3 Using Bayes Rule we have for eachj= 0,1

p(Hb n,i,2=j|H_n,i,1=j) = bp(Hn,i,1=j,Hn,i,2=j)

p(Hb n,i =j) (2)

Corrected Transition Rates

.4.5.6.7.8.91fraction

DE SE NL SP IT FR DK US

transition rates

bad−to−bad health good−to−good health

Source : Authors’ calculations

How can we Reconcile these 2 facts ?

Fact 1 : s=pm

y =pM(y,p)

y with M⁰_y>0 and M⁰_p<0 Fact 2 : π(m) =π(M(y,p)) with π_m⁰ >0

Where

M(y,p) are the health expenditures : they depend on incomey and pricesp.

π(m) : the probability to be in good health depends on the health expenditures.

How can we Reconcile these 2 facts ?

Observation 1.We haveyUS>yEU

⇒through income effects (M⁰_y >0), this can leads tomUS >mEU

Observation 2.But fraction in good health lower in the US π(m_US)< π(m_EU)

⇒This suggests thatm_US <m_EU Observation 3 : We havesUS =pUSmUS

y_US >sEU =pEUmEU

y_EU with

m_US

yUS << ^m_y^EU

EU :

⇒price effect must be large : p_US>>p_EU with lowm_US and largeyUS

Heterogeneity in Prices

Healthcare spending does not purchase much additional health.

Excess Medicare spending in the US represents up to 20% of total : Skinner et al. (2005).

Suggestive evidence of higher prices in the United States than in any other OECD country : Anderson et al. (2003) or Cutler and Ly (2011)

McKinsey Global Institute (1996) : for a specific set of health conditions, Americans paid 40 % more per capita than Germans did but received 15 % fewer real health care resources

The International Federation of Health Plans (2013) did a comparison of cost for various interventions/drugs.

Comparison of Prices (IFHP, 2013)

Diagnostics Drugs Hospital cost Surgery Angiogram Gleevec (Cancer) per day Bypass surgery

US 907$ 6214$ 4293$ 75345$

EU 290$ (SP) 3321$ (NL) 481$ (SP) 15742$ (SP)

price ratio 3.1 1.9 8.9 4.8

Specialist U.S. physicians earn 5.8 times what the average worker does, compared to the non-U.S. average of 4.3 times (Cutler & Ly (2011)) Measuring health prices across countries is a daunting task (Berndt et al.

2001, Koechlin et al. 2014)

Observations at the micro level

Inside a specific country, we observe

The fraction of persons with a good health increase with the income per capita.

If the probability to have a good health is an increasing function of the amount of health servicesm, this stylized fact implies thatm_y⁰ >0.

Health-Income Gradient

1 Compute quantiles of household income for each country using SHARE and HRS, age 50-75

2 Compute the mean of the predicted probabilities of being in good health (corrected measure) by quantiles in each country

3 We express this mean relative to the first quantile in each country

Health Income Gradient

11.11.21.31.41.5fraction good health

1 2 3 4

household income quartiles

DE SE NL SP

IT FR DK US

income gradient for pct good health

Source : Authors’ calculations

Demand of Health : Basic models

General case

maxm U(y−pm,m)

⇒ pU₁⁰(y−pm,m) =U₂⁰(y−pm,m) The solution gives the equilibrium values for

m = M(p,y) pm

y ≡s = S(p,y) The questions are : For each type of utility function,

What are the properties of the demand functions for health ? What are the properties of the functions that give the share of health expenditures in incomes ?

Demand of Health : the Hall and Jones case

The utility function is multiplicative,U =f(m)u(y−pm) : pf(m)u⁰(y−pm) = f⁰(m)u(y−pm) IfU(c) =A+^c_1−σ^1−σ andf(m) =Fm^α, then

s = α(1−s)

Ay^σ−1(1−s)^σ−1+ 1 1−σ

⇔s = S(y) Property

Ifσ >1 then m=M( p

|{z}−

, y

|{z}

+

). If A>0, thenS⁰ >0, but does not depend on p. Rejected by the data.

Demand of Health : separable preferences

IfU =A+^c_1−σ^1−σ +χ^m_1−ρ^1−ρ, where A≥0. The FOC is p(y−pm)^−σ = χm^−ρ ⇒m=M( p

|{z}−

, y

|{z}

+

)

p^1−ρy^ρ−σ = (1−s)^σ

s^ρ ⇒s=S( p

|{z}

−/+

, y

|{z}

+/−

)

Property

The derivative of the functionS(p,y)are such that σ > ρ σ < ρ

ρ <1 −,+ −,−

ρ >1 +,+ +,−

Demand of Health : separable preferences and lotteries

IfU =π(m)h_(y−pm)1−σ

1−σ +hi

+ (1−π(m))^(y−pm)_1−σ^1−σ, and givenπ(m) must satisfyπ(m)∈[0,1),∀m≥0, we assumeπ(m) = 1−exp(−αm), withα >0, then

αexp (−αm)h = p(y−pm)^−σ ⇒m=M( p

|{z}−

, y

|{z}

+

)

αexp

−αy ps

h = py^−σ(1−s)^−σ ⇒s=S( p

|{z}

−/+

, y

|{z}

+/−

)

Property

As previously, the signs of the derivatives ofS are undetermined, but now they depend onσ, α,h, but also on the levels of the variables.

Distortions induced by the Health Insurance System

Zero profit Health Insurance : τy = (1−%)pm

Out-Of-Pocket %(OOP)⇔the inverse of the ”generosity”.

Equilibrium

π⁰(m)h = %pu⁰(y(1−τ)−%pm) τ = (1−%)^pm_y

⇒ π⁰(m)h

%=pu⁰(y−pm) where pand%affect differently the demand function. We deduce

m = M( p

|{z}−

, y

|{z}+

, %

|{z}−

) s = S( p

|{z}

−/+

, y

|{z}

+

, %

|{z}

−/+

)

Equilibrium implications : not sufficient to match data

m^s(p) p

p(US) p(Fr)

m^d(p,y) m^d(p,y+dy)

m(US) m m(Fr) p(Fr)

m^s(p) p

p(US) p(Fr)

m(US) m m(Fr)

m^d(p,y+dy,OOP(h)) m^d(p,y,OPP(l))

Observations at the macro level : A solution for the puzzle

Price distortions⇔supply matters

The unit cost of health services is higher in the US than in Europe, e.g. in France :pUS>pF.

If the price effect dominates the income effect in the demand of health servicesm(y,p), then we deduce fromπUS< πF than mUS<mF even ifyUS>yF.

Policy : the out-of-pocket (OOP)

When OOP is larger, the health expenditures at individual level is reduced

OOPUS>OOPEU : this can also explain whymUS>mEU

Proposition

A synthetic indicator : the share of the health expenditures in the total income (s= ^pm_y )

⇒It can be higher in the US than in Europe (data), even if y , p and OOP have contrasting effects on individual behaviors.

Supply of health : How to explain the price gaps among countries ?

The supply of health services

Production function :m=z(e(w)s)^θx^1−θwhere z : TFP

s : number of physicians

e(wh) : effort at work not perfectly observed by the manager x combination of the capital input used in the health services the Profit of the health sector is

Π_H =pm−w_hs−κx

Supply of health : How to explain the price gaps among countries ?

The supply of health services FOC lead to

p(1−θ)^m_x = κ pθ^m_s = wh

⇒ (1−θ)^pm_κ = x θ^pm_w = s

⇒ m=z

e(w)θpm w

θ

(1−θ)pm κ

1−θ

⇒ p=1 z

1 θ

w e(w)

θ κ 1−θ

1−θ

Given that w_hs−κx ≡C(m,w_h), withp=C_m⁰, the cost function is : C(m,w_h) =¹_z

κ 1−θ

1−θ

1 θ

w_h e(wh)

θ

m

Supply of health : How to explain the price gaps among countries ?

The supply of health services

Cost function :C(m,wh) = ¹_z ^κ_θθ

1 1−θ

wh

e(w_h)

1−θ

m

How to control the effort of the physicians ? It is necessary to implement an incentive contract

Hospitals set wages such as the effort of the physicians is optimal.

Supply of health : How to explain the price gaps among countries ?

Optimal wage :

minw_h C(m, ,w_h)⇔min

w_h

wh

e(w_h) ⇒e⁰(w_h) wh

e(w_h) = 1

| {z } Solow condition Ife(w) = ^w−r_r β

, we have β ^w−r_r ^β−1w

r w−r

r

β = 1⇔ β^w_r

w

r −1 = 1⇔w = 1 1−βr w

e(w) =

1 1−βr _β

1−β

^β = 1

β^β(1−β)^1−βr = (1 +µ)r withµ >0 iffβ∈(0.5,1).

Supply of health : How to explain the price gaps among countries ?

The supply of health services

Cost function using the Solow condition C(m) =¹_z ^κ_θθ_(1+µ)r

1−θ

1−θ

m

Hence, the zero profit condition leads to p_US= 1

z κ

θ

^θ(1 +µ)r 1−θ

1−θ

> 1 z

κ θ

^θ r 1−θ

1−θ

=p_F p_US =1

z(1 +µ)^1−θA_p > 1

zA_p=p_F

Equilibrium implications : sufficient to match data

m^s(p,markup=0) m^s(p,markup>0) p

p(US) p(Fr)

m^d(p,y) m^d(p,y+dy)

m(US) m(Fr) m p(Fr)

m^s(p,markup=0) m^s(p,markup>0) p

p(US) p(Fr)

m(US) m(Fr) m

m^d(p,y+dy,OOP(h)) m^d(p,y,OPP(l))

General equilibrium

The total amount of resources isy.

A part of these resources is used to transform this endowment

”capital”x/”eduction”s, sold at the priceκ/wh

cost functions :Cx(x) =^ϕ_ϑ^xx^ϑandCs(s) = ^ϕ_ϑ^ss^ϑwithϑ >1 The household budgetary constraint becomes

(1−τ)y− Cx(x)− Cs(s) +κx+whs−%pm.

Definition

If% > ^ϑ−1_ϑ , the GE is defined by the system π⁰(m)h =

%−ϑ−1 ϑ

pu⁰

y− 1

ϑpm

p = 1

z(1 +µ)^1−θA_p

Equilibrium vs. first best allocation

(Eq) π⁰(m)h =

%−ϑ−1 ϑ

(1 +µ)^1−θAp

z u⁰

y−(1 +µ)^1−θ1 ϑ

Ap

z m

(FB) π⁰(m^?)h = 1 ϑ

Ap

z u⁰

y−1 ϑ

Ap

z m^?

(ϑ(%−1) + 1)(1 +µ)^1−θu⁰

y−(1 +µ)^1−θ1 ϑ

Ap

z m

= u⁰

y−1 ϑ

Ap

z m^?

% =

u⁰

y−_ϑ^1A_z^pm^?

ϑ(1 +µ)^1−θu⁰

y−(1 +µ)^1−θ_ϑ^1A_z^pm

+ϑ−1 ϑ

Equilibrium vs. first best allocation

Proposition

The optimal policy is

%(µ,y) = u⁰ y−_ϑ¹cmm^?

ϑ(1 +µ)^1−θu⁰ y−(1 +µ)^1−θ_ϑ¹cmm+ϑ−1

ϑ with cm=ϑ−1 ϑ

Ap

z

showing that%⁰_µ <0, with%= 1whenµ= 0, and

%⁰_y >0 ifεy <1and%⁰_y <0 iffεy >1

The generosity (OOP) must be high (low) when the price distortion is large.

⇔ Price distortion ⇒under-investment in health

⇒ Low OOP gives incentives to correct this under-investment.

Precautionary saving

How does the uncertainty affect these behaviors ?

m,a≥0max U = max

m,a≥0{u(y−a−pm) +βE[u(Ra+ey) +π(m)h]}

a>0⇒

pu⁰(y−a−pm) = βπ⁰(m)h

u⁰(y−a−pm) = βRE[u⁰(Ra+ey)]

a= 0⇒ pu⁰(y−pm) =βπ⁰(m)h

Financial arrangements (the second market imperfection) : One non-contingent asset and financial constraint,a≥0

Without risk : a= 0 andc1=y−pmandc2=y.

With risk : Lotteries s.t.ye∈ {0,_1−$^y }with probabilities ($; 1−$)

⇒E[ye] =y andσ²

ey = 2($y)²

Precautionary saving

Proposition

With uncertainty, and incomplete financial market, we have a>0and m<m.

Saving can be larger than in a economy without health choices.

forβ =R= 1, the FOC onmleads

(m−m) = − pu⁰⁰(y−pm) p²u⁰⁰(y−pm) +π⁰⁰(m)ha

For the precautionary saving, we havea|m>0≶a|m=0 :

1 +u⁰⁰(y−pm) u⁰⁰(y)

π⁰⁰(m)h p²u⁰⁰(y−pm) +π⁰⁰(m)h

| {z }

>1

a = σ²_y

whereas ifm=m= 0 2a = σ²_y

Precautionary saving

Example

Utility : u(x) = ^x_1−σ^1−σ withσ= 2.

Health : h= 1 andπ(x) = 1−exp(−αx+b) withα= 0.75 and b= 0.1

Income risk : $l = 0,$m= 0.5 or$h= 0.9.

Income levels :yl= 1.5,ym= 2.5 oryh= 3.5 µ= 1⇒β= 0.5 andr = 0⇒R= 1

Precautionary saving

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-1 -0.5 0 0.5 1

asset -_l

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 2 3 4

health -_l

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.4 0.6 0.8

share -_l

price

oop_l oop_m oop_h

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.29 0.3 0.31 0.32 0.33 0.34

asset -_m

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 2 3

health -_m

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.1 0.2 0.3 0.4

share -_m

price

oop_l oop_m oop_h

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.36 0.37 0.38 0.39 0.4 0.41

asset -_h

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.5 1 1.5 2 2.5

health -_h

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.1 0.2 0.3 0.4

share -_h

price

oop_l oop_m oop_h

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-1 -0.5 0 0.5 1

asset -_l

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 2 3 4

health -_l

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.4 0.6 0.8

share -_l

price

oop_l oop_m oop_h

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.29 0.3 0.31 0.32 0.33 0.34

asset -_m

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 2 3

health -_m

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.1 0.2 0.3 0.4

share -_m

price

oop_l oop_m oop_h

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.36 0.37 0.38 0.39 0.4 0.41

asset -_h

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.5 1 1.5 2 2.5

health -_h

price 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.1 0.2 0.3 0.4

share -_h

price

oop_l oop_m oop_h

Precautionary saving

What do we learn ?

With no uncertainty ($= 0) No saving

m=M(y,p) withM⁰_y>0 andM⁰_p<0

forp<p,e s(yh,p)<s(yl,p) : decreasing returns of health

fory =yl one can haves_p⁰(yl,p)<0 : substitution effect dominates (more consumption and less health)

With uncertainty ($ >0) a>0 : precautionary saving

fory =yl one can havem(yl,p) = 0 : substitution effect dominates (more consumption and saving, but less health)

fory =ym,m(ym,p)>0 butsp(yl,p)>0 forplow andsp(yl,p)<0 forp hight : substitution effect dominates only for hightp

∀p,y we havem|$>0<m|$=0.