Healthcare and Consumption with Aging

Transcription

1 Healthcare and Consumption with Aging Yu-Jui Huang University of Colorado Joint work with Paolo Guasoni Boston University and Dublin City University University of Colorado September 9, 2016

2 This is research on DEATH How to make life more pleasant? consumption: feels good at the moment. healthcare: defers death.

3 DO WE KNOW HEALTHCARE? Journal of American Medical Association (JAMA) On July 11, 2016, Barack Obama published Editorial summary: United States Health Care Reform Progress to Date and Next Steps 4 big surprises from ACA e.g. cost of healthcare while quality. All pundits of healthcare were WRONG about ACA

4 MORTALITY V.S. AGE cohort 1940 cohort Exponential increase in age [Gompertz law]: dm t = βm t dt (β 7.1%)

5 LITERATURE How Exogenous Mortality affects Consumption? Yarri (1965), Richard (1975), Davidoff et al (2005) Healthcare? Health as Capital, Healthcare as Investment Grossman (1972), Ehrlich and Chuma (1990) Health Capital observable? Mortality rates decline with health capital. Ehrlich (2000), Ehrlich and Yin (2005), Yogo (2009), Hugonnier et al. (2012) Gompertz law? include healthcare Gompertz law

6 THIS PAPER IDEA Household maximizes utility from lifetime consumption: [ τ ] sup E e δt U(c t X t )dt. c,h 0 Money can buy... consumption, which generates utility... healthcare, which reduces mortality growth... = buying time for more consumption. QUESTIONS Find optimal control processes {ĥt} t 0 {ĉ t } t 0, {ĥt} t 0. = endogenous mortality curve = follows Gompertz law?

7 THE VALUE FUNCTION Naïve approach: [ τ ] sup E e δt U(c t X t )dt. c,h 0 NOT invariant to utility translation! If U becomes U + k, { [ τ ] E e δt U(c t X t )dt sup c,h 0 [ ]} 1 e δτ + ke δ OBSERVE: τ is endogenous = NO translation invariance.

8 THE VALUE FUNCTION Our approach: After death, household carries on with the same optimization problem. 0 τ 1 τ 2 τ 3 time x ζx τ1 ζ 2 X τ2 ζ 3 X τ3 Death scales household wealth by factor ζ [0, 1]. (ζ: inheritance tax, annuity loss, foregone income...) The value function: [ V(x, m) = sup E c,h n=1 (Translation Invariant) τn τ n 1 e δt U(ζ n X t c t )dt ] with τ 0 := 0.

9 ASSUMPTIONS Simplifications: 0 τ 1 τ 2 τ 3 time x ζx τ1 ζ 2 X τ2 ζ 3 X τ3 m M τ1 M τ2 M τ3 Surviving spouse in similar age group. Most weight carried by first two lifetimes. Isoelastic utility: U(x) = x1 1 0 < 1

10 MORTALITY DYNAMICS Without healthcare, mortality grows exponentially [Gompertz law]: dm t = βm t dt. Healthcare slows down mortality growth dm t = (β g(h t ))M t dt ht : healthcare-wealth ratio g : R+ R + measures efficacy of healthcare g(0) = 0, g is increasing and concave. Example: g(h) = a q hq a > 0, q (0, 1)

11 ASSUMPTIONS Efficacy depends on healthcare-wealth ratio. Means-tested subsidies; Chetty et al. (2016, JAMA): life expectancy is significantly correlated with health behaviors but not with access to medical care. = h t reflects time and lost-income costs.

12 WEALTH DYNAMICS Household wealth grows at a constant interest rate r > 0, minus consumption and health spending: dx t = (r c t h t )X t dt.

13 A STOCHASTIC CONTROL PROBLEM The value function: State processes: V(x, m) = sup E c,h [ n=1 τn τ n 1 e δt U(ζ n X t c t )dt dx t = (r c t h t )X t dt X 0 = x, dm t = (β g(h t ))M t dt, M 0 = m. ] Distributions of death times: { P(τ n > t τ n 1 < t) = exp t τ n 1 M s ds }.

14 3. Verification argument V(x, m) 1. Dynamic Programming + Stochastic Calculus Construct a solution to PDE PDE for V(x, m) (HJB eqn.) 2.??? (Perron s method in our case)

15 DYNAMIC PROGRAMMING PRINCIPLE V(x, m) = sup E c,h [ n=1 τn τ n 1 e δt U(ζ n X t c t )dt ] 0 T time V(x, m) V(X T, M T ) DPP states: [ T V(x, m) = sup E e t 0 (δ+ms)ds [U(c t X t ) + M t V(ζX t, M t )]dt c,h,π 0 + e ] T 0 (δ+ms)ds V(X T, M T ).

16 ITÔ S FORMULA By Itô s formula in stochastic calculus, d (e ) t 0 (δ+ms)ds V(X t, M t ) = e t 0 (δ+ms)ds [ (M t + δ)v(x t, M t ) + V x (X t, M t )dx t + V m (X t, M t )dm t V xx(x t, M t )(dx t ) 2 ].

17 DPP + Itô s Formula yields [ T 0 = supe e t 0 (U(c (δ+ms)ds t X t ) + M t V(ζX t, M t ) c,h 0 (δ + M t )V(X t, M t ) + [r + µπ t c t h t ]X t V x (X t, M t ) +(β g(h t ))M t V m (X t, M t ) + 1 ) ] 2 σ2 π 2 Xt 2 V xx (X t, M t ) dt. A Big Guess: V(x, m) is a solution to the PDE { 0 = sup U(cx) + mv(ζx, m) (δ + m)v(x, m) c,h 0 +[r + µπ c h]xv x (x, m) + (β g(h))mv m (x, m) + 1 } 2 σ2 π 2 x 2 V xx (x, m).

18 HAMILTON-JACOBI-BELLMAN EQUATION The HJB equation for V: sup {U(cx) hxv x (x, m)} c 0 + sup { mg(h)v m (x, m) hxv x (x, m)} h 0 δv(x, m) + rxv x (x, m) + (V(ζx, m) V(x, m))m + βmv m (x, m) = 0. (pde) Optimal strategies: ĉ = V x(x, m) 1 ( ), ĥ = (g ) 1 xvx (x, m) x mv m (x, m)

19 REDUCTION TO ODE Taking V(x, m) = x1 1 u(m) ( δ + (1 ζ u(m) 2 1 )m ( +mu (m) sup h 0 { g(h) 1 gives ( ) ) r u(m) ) } u(m) mu (m) h β = 0. (ode) Optimal strategies: ĉ = V x(x, m) 1 = u(m), x ( ) ĥ = (g ) 1 xvx (x, m) mv m (x, m) = (g ) 1 ( 1 ) u(m) mu. (m)

20 THREE SETTINGS To understand effects of aging and healthcare, consider 1. Forever Young [Neither Aging nor Healthcare]: M t m Gompertz s law [Aging without Healthcare]: dm t = βm t dt, M 0 = m > The general case [Aging with Healthcare]: dm t = (β g(h t ))M t dt, M 0 = m > 0.

21 NEITHER AGING NOR HEALTHCARE M t m > 0 (forever young). (ode) reduces to ( δ + (1 ζ u(m) 2 1 ( )m ) ) r u(m) = 0. Optimal strategy: ĉ = u(m) = c 0 (m) := δ + (1 ζ1 )m ( ) r.

22 AGING WITHOUT HEALTHCARE dm t = βm t dt. This is non-standard, cf. Huang, Milevsky, & Salisbury (2012)) (ode) reduces to ( δ + (1 ζ u(m) 2 1 ( )m ) ) r u(m) βmu (m) = 0. Optimal strategy: ĉ = u(m) ( = c β (m) := 0 ( e (1 ζ 1 )my (βy + 1) Asymptotics for old age (large m): c β (m) = c 0 (m) + β + O( 1 m ). 1+ δ+( 1)r β ) ) 1 dy

23 40 Consumption-Wealth Ratio (%) Aging Forever Young Forever Young + β Immortal Mortality (%) green curve: c 0 orange curve: c β Mortality and aging have large impacts on ĉ.

24 AGING WITH HEALTHCARE dm t = (β g(h t ))M t dt. Assumption : healthcare... slows aging, does not stop aging. Intuitive idea: a solution u to (ode) should satisfy c 0 u c β, u is concave. Observe: Under the condition ( ( )) 1 g (g ) 1 < β, c β is a supersolution to (ode), c 0 is a subsolution to (ode).

25 CONSTRUCTION OF u Perron s method: u (m) := inf u(m) m > 0, u S where S is the collection of u : R + R + satisfying c 0 u c β, u is a viscosity supersolution to (ode). u is concave, increasing. Regularity: = u is a viscosity solution to (ode) viscosity solution property + concavity = u is a classical solution to (ode)

26 VERIFICATION u (m) is a classical solution to (ode). x1 1 (u (m)) is a classical solution to (pde). Using verification argument, V(x, m) = x1 1 (u (m)), (x, m) R 2 +.

27 Main Result Suppose 0 < < 1 and c := δ + (1 1 then ) r > 0. If ( ( )) 1 g (g ) 1 < β, (1) V(x, m) = x1 1 u (m) for all (x, m) R 2 +, where u : R + R + is the unique nonnegative, strictly increasing solution to ( δ + (1 ζ u(m) 2 1 )m ( +mu (m) sup h 0 { g(h) 1 ( ) ) r u(m) ) } u(m) mu (m) h β = 0.

28 Main Result (conti.) Furthermore, (ĉ, ĥ) defined by ĉ t := u (M t ), are optimal strategies. ĥ t := (g ) 1 ( 1 u ) (M t ) M t (u ), t 0, (M t )

29 CALIBRATION Take efficacy function as g(h) := a hq q, with a > 0, q (0, 1). Take r = 1%, δ = 1%, = 0.67, ζ = 50% from literature. Calibrate β, m 0, a, q to mortality rate data: cohort 1940 cohort = β = 7.7%, m 0 = 0.019%, q = 0.46, a = 0.1.

30 LONGER LIVES 5.0 Mortality H%L cohort without healthcare 1940 cohort with healthcare Model explains decline in mortality at old ages.

31 OPTIMAL STRATEGIES 5 Spending-Wealth Ratios (%) Consumption Healthcare Age (years) Healthcare negligible in youth. Increases faster than consumption (in log scale!)

32 HEALTHCARE AS FRACTION OF SPENDING Healthcare-Spending Ratio (%) Age (years) Convex, then concave; rises quickly to contain mortality. Slows down when cost-benefit declines.

33 THANK YOU!! Q & A Preprint ssrn.com/abstract= Healthcare and Consumption with Aging