Is Smoking a Fiscal Good? Shantanu Bagchi and James Feigenbaumy Preliminary and Incomplete February 15, 2010

Abstract The argument is often made that because smokers die more quickly the expected health expenditure on smokers is actually less than for nonsmokers. Thus society may actually be better o¤ if it encourages more people to smoke. We consider this argument in a general-equilibrium model where health expenditures are paid for by a single-payer healthcare system …nanced by taxes. The question then is whether the tax rate increases or decreases with the number of smokers. In a simpli…ed model where labor is supplied exogenously, we …nd that whether the health-care tax rate increases or decreases with the elimination of smokers is highly sensitive to risk aversion. JEL Classi…cation: E21, H51, I18 Keywords: general equilibrium, annuities, bequests, mortality risk, overlapping generations, smoking, health expenditures, single-payer healthcare system

The argument is often made that smokers die quicker than nonsmokers (Barendregt et al (1997)), so the expected health expenditure for a nonsmoker is larger than for a smoker. Consequently, it might actually be good if more people smoke. This is a purely partial equilibrium argument that ignores two factors that would arise in general equilibrium. First, an increase in survival probabilities will increase the labor supply: both because there will be more workers alive at any age and also because a longer lifespan may give workers an incentive to retire later. Second, if households cannot perfectly insure against mortality risk, then households that expect to live longer will have to save more. Thus the overall e¤ect of smoking in general equilibrium is ambiguous. Here we consider this question in a model with a single-payer health-care system and an exogenous separation of households into two types: smokers and nonsmokers. Our metric of how the economy fares is the tax rate that must be imposed to fund the health-care system. Assuming a single-payer system greatly simpli…es the problem since all health risk is borne by the government We would like to thank Frank Caliendo for providing many helpful suggestions. This work also received …nancial support from the Koch Foundation. y Corresponding author: [email protected]. http://huntsman.usu.edu/jfeigenbaum/

1

and has no e¤ect on the household problem. We look at what happens both with exogenous and endogenous labor supply and both with perfect annuities markets and without. The most straightforward cases to compute have exogenous labor supply. Both with bequests and with annuities, we …nd that the argument of Barendregt et al (1997) that smokers may actually be good for the economy holds in our baseline model with unit risk aversion. All agents retire at age 65, before which the divergence of survival probabilities between the two types remains small. Likewise, nonsmokers save more than smokers, but the di¤erence is mitigated by the ability to insure against mortality risk. Thus while capital and labor do increase with the fraction of nonsmokers, the e¤ect is not large enough to overcome the increase in health expenditures. The tax base, i.e. net domestic product, increases at a slower rate than health expenditures, and the tax rate must increase to fund the health-care system. For risk aversion of 3 or more, this result can be reversed. While it is intuitively appealing to conclude that higher risk aversion makes smoking more costly, the only risk in the model is mortality risk, and one has to be careful in applying the aversion to consumption risks to mortality risk. Intertemporal elasticity is presumably more relevant in the model. The lower elasticity of intertemporal substitution means a larger di¤erence in saving between nonsmokers and smokers as they put more value into smoothing their consumption.

1

The Model

To begin with, let us consider the simple case where labor is supplied inelastically. There are two types of workers: nonsmokers (n) and smokers (s). These workers di¤er only in terms of their mortality risk. Time is continuous. Let Qi (s) be the probability that a worker of type i survives to age s. Agents live to a maximum age T . We use the notation that a lifecycle variable xi (t; ) represents the value of x at time t for an agent of type i who is born at . An agent of type i born at time will choose a consumption path to maximize expected utility Z

T

Qi (s) exp(

s)u(ci (t + s; t) li (t + s; t)1

; )ds,

(1)

0

where aversion:

> 0,

2 [0; 1], and u(c; ) is the CRRA utility function with risk u(c; ) =

1 1

ln c c1

=1 : 6= 1

(2)

For simplicity (and also not to trigger criticisms of discrimination), we assume that both types of worker have the same productivity pro…le e(s) for 2

s 2 [0; T ]. Workers at time t and age s 2 [0; T ] earn (gross) labor income we(s)(1 li (t; t s)). For ages s > Tr , households also receive a Social Security bene…t b(t) at time t, which is funded through a payroll tax SS .1 We include Social Security because it is presently the largest redistribution mechanism administered by the federal government, and we are studying the impact of another proposed redistribution mechanism. Since the purpose of this study is not to understand distortions caused by the …ne details of Social Security, we make the simplest assumption that households receive bene…ts at age Tr regardless of whether they stop working, and labor income is taxed to pay for this bene…t at the same rate both before and after age Tr . Let d ln Qi (s) (3) zi (s) = ds denote the hazard rate of dying at age s for a type-i agent, i.e. zi (s)ds is the probability of dying between s and s + ds if the agent survives till age s. We consider two regimes for dealing with the wealth of the dead. In the annuities regime, we assume markets are complete and agents can purchase annuities than insure them against mortality risk. In the bequest regime, the wealth of the dead is spread uniformly across the surviving population, resulting in an inheritance B(t) at age t. In either regime, agents accrue wealth by investing in capital ki (t; ). In the bequest regime capital earns the (gross) interest rate r, whereas in the annuity regime capital earns r + zi (s) at age s. Let hi (t; t s) = hi (0; s) exp(gt) be the average health expenditure of an agent of type i at time t and age s. Health expenses are paid for by a singlepayer insurance plan run by the government. This plan is paid for by taxes y on labor income, k on capital income, B on bequests (in the bequest regime), and c on consumption. For now, let us suppose that the fraction fi of each type of agent is exogenous, where fn + fs = 1. The economy is endowed with a production sector governed by the CobbDouglas production function at time t Y (t) = K(t) (exp(gt)N )1

;

(4)

where K(t) is the capital stock at t, which depreciates at the rate > 0, N is the supply of labor, and g is the growth rate of labor-augmenting productivity. Firms behave competitively, and we assume the economy is on a balanced growth path so factor prices are given by w(t)

=

r

=

(1

) K(0) N

In equilibrium, we must have Z T K(t) = [fn Qn (s)kn (t; t

K(0) N

exp(gt)

(5)

:

(6)

1

s) + fs Qs (s)ks (t; t

s)] ds

0

1 We

do not require households to stop working to receive Social Security bene…ts.

3

(7)

and N=

Z

T

[fn Qn (s)(1

ln (t; t

s))en (s) + fs Qs (s)(1

ls (t; t

s))es (s)] ds:

0

(8)

Aggregate consumption is C(t) =

Z

T

[fn Qn (s)cn (t; t

s) + fs Qs (s)cs (t; t

s)]ds

(9)

s)]ds

(10)

0

while the total population is P =

Z

T

[fn Qn (s) + fs Qs (s)]ds:

0

Let H(t) =

Z

T

[fn Qn (s)hn (t; t

s) + fs Qs (s)hs (t; t

0

be total health expenditures at time t. The government must balance the budgets of both Social Security and the health-insurance program separately. The former constraint implies that b(t)

= =

w(t) SS

RT

Tr

RT 0

[fn Qn (s)(1

SS w(t)N

ln (t; t s))en (s) + fs Qs (s)(1 RT [f Q (s) + fs Qs (s)]ds Tr n n

ls (s; t

s))es (s)]ds

:

(11)

[fn Qn (s) + fs Qs (s)]ds

The health-insurance budget constraint depends on the regime.

1.1

Bequest Regime

This version of the model follows the example of Bullard and Feigenbaum (2007). In the bequest regime, a worker of type i born at time will solve the problem max

Z

T

Qi (s) exp(

s)u(ci ( + s; ) li ( + s; )1

(1

y )w(1

; )ds

(12)

0

subject to @ki (t; ) @t

=

SS

+(1

B )B(t)

0

li (t; ))e(t

(1 +

c )ci (t;

) + (1 )+

(t

k )rki (t;

) (13)

Tr )b(t)

ki (0; ) = ki (T; ) = 0;

(14)

li (t; )

(15)

1 8t 2 [ ; + T ] 4

where 1 x>0 0 x 0

(x) =

(16)

is the step function. The health-insurance budget constraint is H(t) =

y w(t)N

+

k rK(t)

+

c C(t)

+

B B(t)P:

(17)

We must also satisfy the bequest-balance equation B(t)P =

Z

T

[fn Qn (s)zn (s)kn (t; t

s) + fs Qs (s)zs (s)ks (t; t

s)]ds:

(18)

0

1.2

Annuities Regime

This version of the model follows the example of Hansen and Imrohoroglu (2008). In the annuities regime, a worker of type i born at will solve the problem max

Z

T

Qi (s) exp(

s)u(ci ( + s; ) li ( + s; )1

; )ds

0

subject to @ki ( + s; ) @t

=

(1

+[(1 (1 + 0

y )w(1

SS

li ( + s; ))e(s)

(19)

k )r

+ (1 a )zi (s)]ki ( + s; ) Tr )b( + s) c )ci ( + s; ) + (s

li ( + s; )

1 8s 2 [0; T ]

ki ( ; ) = ki ( + T; ) = 0: Before taxes, annuities pay the return r+zi (s), which compensates for mortality risk. We allow the government to tax the capital return r separately from the compensation for mortality risk zi . The former is taxed at the rate k while the latter is taxed at the rate a . Total tax revenue from the annuities tax will be Z T a R (t) = [fn Qn (s)zn (s)kn (t; t s) + fs Qs (s)zs (s)ks (t; t s)]ds(20) a 0

=

a

Z

0

T

fn

dQn (s) kn (t; t ds

5

s) + fs

dQs (s) ks (t; t ds

s) ds

Integrating by parts, we can rewrite this as Z

T

fn

0

=

[fn Qn (s)kn (t; t

= =

dQn (s) kn (t; t ds

Z

Z

fn Qn (s)

0

fn Qn (s)

0

dQs (s) ks (t; t ds T s)]0

s) + fs Qs (s)ks (t; t

T

T

s) + fs

@kn (t; t @

@kn (t; t @t

s) s)

+ fs Qs (s)

+ fs Qs (s)

s) ds Z

T

fn Qn (s)

0

@ks (t; t @

@ks (t; t @t

s) s)

dkn (t; t ds

s)

+ fs Qs (s)

ds dK(t) ; dt

ds

where we use (57) in the last step. Let us denote S(t) =

Z

T

fn Qn (s)

0

We can then write Z T dQn (s) kn (t; t fn ds 0

@kn (t; t @t

s) + fs

s)

+ fs Qs (s)

dQs (s) ks (t; t ds

@ks (t; t @t

s)

s) ds = S(t)

ds:

(21)

dK(t) dt

(22)

Then the health-expenditure budget constraint can be written H(t) =

2

y w(t)N

+

k rK(t)

+

c C(t)

+

a

S

dK(t) dt

:

(23)

Solving the Model

It is helpful to break the problem into an intertemporal and intratemporal problem. Let us de…ne total expenditures on goods, both leisure and consumption, as Ei (t; ) = (1

ss

y )w(t)li (t;

)e(t

) + (1 +

c )ci (t;

)

(24)

for an agent of type i at time t who is born at . We further de…ne V (E; pc ; pl ) = max u(c l1 c;t

; )

(25)

subject to pc c + pl l = E 0

l

6

1:

(26)

dks (t; t ds

s)

ds

2.1

The Annuities Regime

Under the above decomposition, we can rewrite the problem of a household of type i born at in the annuities regime as Z T Qi (s) exp( s)V (Ei ( +s; ); 1+ c ; (1 ss max y )w( +s)e(s))ds (27) 0

subject to @ki ( + s; ) +Ei ( +s; ) = (1 @t

y )w(

SS

+s)e(s)+[(1

k )r+(1

a )zi (s)]ki (

+s; )+ (s Tr )b

(28)

ki ( ; ) = ki ( + T; ) = 0: The intratemporal problem, which is standard, is solved in Appendix B. The intertemporal problem has the Hamiltonian Hia ( )

= Qi (s) exp( s)V (Ei ( + s; ); 1 + c ; (1 + a (s; )[(1 SS y )w( + s)e(s) + [(1 + (s Tr )b Ei ( + s; )]:

y )w( + s)e(s)) )r + (1 k a )zi (s)]ki ( + s; ) (29)

ss

Hamilton’s equations are then @Hia ( ) = Qi (s) exp( @Ei ( + s; )

s)VE (Ei ( +s; ); 1+ c ; (1

ss

y )w(

+s)e(s)) (30)

d

a (s; ) = ds

@Ha = @k( + s; s)

[(1

k )r

+ (1

a )zi (s)] a (s;

):

(31)

Since we have shown in Appendix B that, for a given pc and pl , VE is invertible with respect to E, let M (x; pc ; pl ) be the inverse of VE . Then we have from (30) Ei ( + s; ) = M

a (s; ) exp( s); 1 + Qi (s)

c ; (1

y )w(

ss

+ s)e(s) : (32)

Integrating (31), we get ln

a (s;

)

ln

a (0;

)

Z

=

s

[(1

k )r

+ (1

a )zi (s)]ds

0

a (s;

)=

=

(1

k )rs

+ (1

=

(1

k )rs

+ (1

a (0;

)

Qi (s) Qi (0) 7

1

Z

s

d ln Qi (s0 ) 0 ds ds 0 Qi (s) : a ) ln Qi (0) a)

a

exp ( (1

k )rs)

(33)

a (s;

)=0

Ei (t; ) = M

a (0;

)

Qi (0)

!

a

Qi (t ) Qi (0)

exp([

(1

k )r](t

)); 1 +

c ; (1

Note that we only get perfect insurance with respect to mortality risk if in which case Ei (t; ) does not depend on Qi (t ). For a household that is working, @ ln Ei ( + s; ) @ ln ci ( + s; ) = = @s @s

1

(1

)(1

)

d ln e(s) dt

a

a

y )w(t)e(t

ss

) :

(34) = 0,

d ln Qi (s) + dt

(1

k )r

:

Thus the growth rate of consumption is (1 @ ln ci ( + s; ) = @s

k )r

a zi (s)

+

(1

)(

1) d ln e(s) : dt

(35)

For a retired household, Ei ( +s; ) =

(1 +

c)

(1

)

a (0;

)

Qi (0)

ci ( + s; ) = 1 ci ( +s; ) = 1+

Ei ( + s; )

(1 +

exp([

c)

(1

)

a (0;

)

1

1 +

a

1

1 +

k )r]s)

+ s)e(s)

+(1

ss

y )w(s+

:

c

!

a

Qi (s) Qi (0)

Qi (0)

(1

y )w(

ss

1+

c

@ ln ci ( + s; ) = @s

(1

!

a

Qi (s) Qi (0)

exp([

d ln Qi (s) + ds

(1

(1

1

1 +

k )r]s)

k )r

The growth rate of consumption is @ ln ci ( + s; ) (1 = @s

k )r

1

a zi (s)

+

:

(36)

Then we have @ ln li ( + s; ) @ ln Ei ( + s; ) = @s @ @ ln li ( + s; ) (1 = @s

k )r

+

d ln ei (s) (1 = ds a zi (s)

k )r

1

+

+

a zi (s)

d ln e(s) : ds

+

(1

(37)

We also have Ei (0; ) = M

a (0; ) ;1 + Qi (0)

c ; (1

y )w(

ss

)e(0) :

Thus a (0;

) = Qi (0)VE (Ei (0; ); 1 + 8

c ; (1

ss

y )w(

)e(0)) :

(38)

)(

1)

d ln e(s) dt

)e(s):

Finally, we have to solve the intertemporal budget constraint (28). @ exp( (1 @s

1 k )rs)Qi (s)

a

ki ( + s; )

=

(1

k )r exp(

1 k )rs)Qi (s)

(1

a

ki ( + s; )

dQi (s) exp( (1 k )rs)ki ( + s; ) ds @ki ( + s; ) 1 a + exp( (1 k )rs)Qi (s) dt 1 a = exp( (1 )rs)Q (s) k i @ki ( + s; ) [(1 k )r + (1 a )zi (s)] ki ( + s; ) : @s +(1

a )Qi (s)

a

Thus we can rewrite (28) as @ 1 a exp( (1 ki ( + s; ) k )rs)Qi (s) @s 1 a exp( (1 [(1 k )rs)Qi (s) SS y )w( + s)e(s) +

=

(39) (s

Tr )b( + s)

Ei ( + s; )] :

Thus ki ( + s; )

ki ( + s; )

=

=

Z s exp((1 k )rs) 0 0 1 a exp( (1 k )rs )Qi (s ) Qi (s)1 a 0 0 0 0 [(1 Tr )b( + s0 ) SS y )w( + s )e(s ) + (s Z

1

s

Qi (s0 ) Qi (s) 0 0 0 y )w( + s )e(s ) + (s

exp((1

k )r(s

0

[(1

SS

Ei ( + s0 ; )] ds0

a

s0 ))

(40) Tr )b(t + s0 )

Ei ( + s0 ; )] dt:

The condition k(T ) = 0 gives rise to the lifetime budget constraint Z

Z

T

exp( (1

1

k )rs)Qi (s)

a

Ei ( + s; )ds

=

0

exp( (1

1 k )rs)Qi (s)

(41)

a

0

which we solve to determine factor exp(g ) to get Z

T

a (0).

1 k )rs)Qi (s)

a

ss

y )w(

+ s)e(s) +

(s

Tr )b( + s)]ds;

Note that we can factor out the growth

T

exp( (1

[(1

Ei (s; 0)ds

=

0

Z

T

exp( (1

1 k )rs)Qi (s)

a

0

[(1

ss

y )w(s)e(s)

+

(s

Since we can write the wage at t = 0 as w(0) = (1

)

9

r+

1

;

(42)

Tr )b(s)]ds;

all endogenous variables are functions of r and b(0). In equilibrium, we must choose r and b(0) such that (7) and (11) are jointly satis…ed. This is further simpli…ed for the special case where = 1 since in that case li (t; ) = 0 for all t and i = n; s, and (11) reduces to RT [fn Qn (s) + fs Qs (s)]e(s)ds ss w(t)N = RT b(t) = w(t) SS 0R T [f Q (s) + fs Qs (s)]ds [f Q (s) + fs Qs (s)]ds Tr n n Tr n n which is completely exogenous. In that case we need only solve for r.

2.2

The Bequest Regime

The intertemporal problem of a household in the bequest regime is Z T max Qi (s) exp( s)V (Ei ( + s; ); 1 + c ; (1 ss y )w( + s)e( ))ds 0

subject to @ki ( + s; ) + Ei ( + s; ) @s

=

(1

SS

+(1

y )w(

k )rki (

+ s)e(s)

(43)

+ s; ) + B( + s) +

(s

Tr )b( + s)

ki ( ; ) = ki ( + T; ) = 0: The Hamiltonian is Hb ( )

= Qi (s) exp( s)V (Ei ( + s; ); 1 + c ; (1 ss y )w( + s)e(s)) + b (s; )[(1 SS y )w( + s)e(s) + (1 k )rki ( + s; ) (44) +B( + s) + (s Tr )b( + s) Ei ( + s; )]:

Hamilton’s equations are @Hb ( ) = Qi (s) exp( @Ei ( + s; )

s)VE (Ei ( +s; ); 1+ c ; (1

ss

y )w(

+s)e(s))

b (s;

(45) and

d b (s; ) = ds

@Hb ( ) = @ki ( + s; )

(1

k )r b (s;

):

(46)

This has the simple solution b (s;

where

b (0;

)=

b (0;

) exp( (1

k )rs);

(47)

) is again an integration constant. From (45), we get

Ei ( +s; ) = M

b (0; ) exp([ Qi (s)

(1

k )r]s); 1

+

c ; (1

ss

y )w(

+ s)e(s) : (48)

10

)=0

For a household that is working, 0" 1+ Ei ( +s; ) = @

(1

c

1

y )w( + s)e(s)

ss

1

#1

b (0; ) exp([ Qi (s)

1 A

(1

k )r]s)

(1

k )r

Thus

@ ln Ei ( + s; ) @ ln ci ( + s; ) = = @s @s

1

(1

)(1

)

d ln e(s) ds

d ln Qi (s) + ds

;

so the growth rate of consumption is @ ln ci ( + s; ) (1 = @s

k )r

zi (s)

(1

+

)(

1) d ln e(s) : ds

(49)

Meanwhile the growth rate of leisure is @ ln Ei ( + s; ) @ ln li ( + s; ) = @s @s

d ln e(s) (1 = ds

k )r

zi (s)

1

+

d ln e(s) : ds

(50) For a retired household, ci ( + s; ) =

1 1+

(1 +

c)

(1

)

b (0;

)

Qi (s)

c

1

exp([

(1

k )r]s)

1 +

;

so the growth rate of consumption is (1 @ ln ci ( + s; ) = @s

k )r

1

zi (s) +

:

(51)

Finally, we determine ki ( + s; ) from the budget constraint, which can be rewritten @ki ( + s; ) +Ei ( +s; ) = (1 @s @ (exp( (1 @s

k )rs)ki (

SS

+ s; ))

y )w(

=

+s)e(s)+(1

(1

k )r exp(

k )rki (

(1

+s; )+B( +s)+ (s Tr )b( +s)

k )rs)ki (

+ s; )

@ki ( + s; ) @s = exp( (1 k )rs)[(1 SS y )w( + s)e(s) +B( + s) + (s Tr )b( + s) Ei ( + s; )]: + exp( (1

Integrating, we obtain Z s ki ( + s; ) = exp((1

k )r(s

s0 ))[(1

SS

k )rs)

y )w(

0

+ (s0

Tr )b( + s0 )

Ei ( + s0 ; )]ds0 : 11

+ s0 )e(s0 ) + B( +(52) s0 )

1

:

The constant b (0; ) is determined by the condition ki ( + T; ) = 0, which is the lifetime budget constraint: Z T Z T exp( (1 )rs)E ( + s; )ds = exp( (1 (53) k i k )rs)[(1 SS y )w( + s)e(s) 0

0

+B( + s) +

3

(s

Tr )b( + s)]dt:

Calibration

We begin with macroeconomic targets. We set the share of capital = 1=3; the capital-output ratio to K=Y = 3, and the growth rate of technology to g = 0:015. We target shares of expenditures according to H=Y = 0:15 and C=Y = 0:7. Then since I = (g + )K from the income-expenditure identity, we will have =

Y

H K

C

g=

H Y K Y

1

C Y

g=

:15 3

0:015 = :035:

In some regimes of the model, the preference parameters and are separately identi…ed. In the model with annuities and exogenous labor, they are not. In that case, we set = 1 and choose to match the K=Y target. For now let us consider the case where health expenditures are paid for by taxes on labor and capital income, which are equal: k = y and c = a = 0. In that case, H = y (wN + rK) = y (Y K): Thus the health-care budget balancing tax rate is the ratio of aggregate health expenditures to net domestic product: y

=

H Y

1

K Y

= 0:168:

(54)

Following Feigenbaum (2008), we set the Social Security tax rate to SS = 0:106. Health expenditure data was obtained by …tting a fourth-order polynomial to data from Barendregt et al (1997). Let hi (t; ) = nh e hi (t

) exp(gt);

(55)

where e hi (s) is measured in dollars and nh is a normalization factor chosen to match H=Y = 0:15. Health expenditures per capita, e hi (s), are plotted for both types of workers in Fig. 1. Survivor probabilities were obtained by averaging the probabilities for light and heavy smokers from Rogers and Powell-Griner (1991).and then …tting to a 12

~

~ Health Expenditures per capita (hi(t))

10000

9000

8000

7000

6000

NONSMOKING SMOKING

5000

4000

3000

2000

1000

0 25

40

55

70

85

100

Age (t + 25)

Figure 1: Health expenditures per capita measured in dollars e hi (t) for both smokers and nonsmokers.

13

1

0.9

Survivor Probability (Q(t))

0.8

0.7

0.6 NONSMOKERS SMOKERS F (2008)

0.5

0.4

0.3

0.2

0.1

0 25

40

55

70

85

100

Age (t + 25)

Figure 2: Survivor probabilities for smokers and nonsmokers. For comparison, the survivor probabilities in Feigenbaum (2008) are also given.

sixth-order polynomial (which must be extrapolated for ages above 75). We normalize Qi (0) = 1 for i = n; s. These are plotted for both types of smokers in Fig. 2. For comparison, we also include the survivor probabilities used for the representative agent in Feigenbaum (2008), which line up closely with the nonsmoker survivor probabilities. The argument of Barendregt et al (1997) can be summed up by Fig. 3, which depicts expected health expenditures Qi (t)e hi (t).as a function of age for both types of household. Total expected health expenditures over the lifecycle are represented by the area under each curve. Although expected health expenditures for smokers are higher at younger ages, they fall o¤ later because smokers die faster. Conversely, expected health expenditures for nonsmokers are much larger than for smokers late in life with a peak in the early 80s. Holding the labor supply and capital stock constant, a nonsmoker does cost more than a smoker. Table 1 summarizes the parameters and endogenous variables that characterize the general equilibrium, for the baseline distribution of smokers with fn = 0:8 and fs = 0:2:

14

~ Expected Health Expenditures per capita (Qi(t)hi(t))

0.35

0.3

0.25

0.2 NONSMOKER SMOKER 0.15

0.1

0.05

0 25

40

55

70

85

100

Age (t + 25)

Figure 3: Expected health expenditures Qi (t)e hi (t) as a function of age for both smokers and nonsmokers.

Regime Annuities, Exogenous Labor Bequests, Exogenous Labor

0.0360 0.0265

1 1

nh 0.000175 0.000175

r 0.0754 0.0765

b 0.445 0.443

B 0.0 0.0535

Table 1: Parameters and endogenous variables that characterize the baseline general equilibrium for each of the four regimes of the model.

15

4

Results

In the regime with annuities and exogenous labor, if we remove all smokers from the economy, the labor and capital tax rate needed to balance the government’s budget is y = 0:169. The tax rate increases by 0.96% because while H increases by 2.74%, a 3.34% increase in the capital stock and a 1.21% increase in the labor supply only leads to a 1.93% increase in output, or a 1.76% increase in net domestic product. Thus H increases faster than net domestic product, and taxes have to increase to accomodate the larger health expenditures.

A

Income-Expenditure Identity

To ensure the model is de…ned consistently, let us check that the incomeexpenditure identity, C(t) + H(t) + ( + g)K(t) = Y (t);

(56)

is satis…ed in both versions of the model.

A.1

Bequest Regime

Using the budget constraint (13), we get Z T C(t) = [fn Qn (s)cn (t; t s) + fs Qs (s)cs (t; t 0

=

fn 1+

c

fs + 1+ +

1 1+

Z

s)]ds

T

Qn (s) (1

y )w(t)(1

SS

ln (t; t

s))e(s)

0

c

c

Z

T

Qs (s) (1

y )w(1

SS

ls (t; t

@ks (t; t @t

s))e(s)

0

Z

@kn (t; t @t

s) s)

+ (1

+ (1

k )rkn (t; t

k )rks (t; t

s) ds s) ds

T

(fn Qn (s) + fs Qs (s)) [(1

B )B(t)

+

(t

s

Tr )b(t)] ds:

0

Using (??) and breaking apart the integrals, we get " Z T (1 + c )C(t) = fn (1 )w(t) Qn (s)(1 y SS

ln (t; t

s))en (s)ds + b(t)

0

Z

T

0

"

@kn (t; t Qn (s) @t

+fs (1 Z

0

T

y

s)

ds + (1

Qn (s)ds

Tr

k )r

Z

T

Qn (s)kn (t; t

s)ds + (1

B )B(t)

Z

T

Qs (s)(1

s)

ds + (1

Z

T

Qn (s)ds

0

ls (t; t

s))es (s)dt + b(t)

0

Z

T

Qs (s)ds

Tr

k )r

Z

0

16

T

0

SS )w(t)

@ks (t; t Qs (s) @t

Z

T

Qs (s)ks (t; t

s)ds + (1

B )B(t)

Z

0

T

Qs (s)ds

#

Substituting in (8) and (7) (1 +

c )C(t)

=

(1

SS )w(t)N + b(t)

y

Z

Z

T

[fn Qn (s) + fs Qs (s)]ds

Tr

T

fn Qn (s)

0

@kn (t; t @t

s)

+ fs Qs (s)

@ks (t; t @t

s)

ds + (1

k )rK(t)

+ (1

B )B(t)P:

Eqs. (18), (11), and (3) imply (1 +

c )C(t)

=

(1

Z

y )w(t)N

T

fn Qn (s)

0

+(1

k )rK(t)

Z

T

fn

0

@kn (t; t @t

s)

dQn (s) kn (t; t ds

+ fs Qs (s)

s) + fs

@ks (t; t @t

dQs (s) ks (t; t ds

s)

ds

s) ds

B B(t)P:

The constraint (17) implies C(t) + H(t)

= w(t)N + rK(t) Z

Z

T

fn Qn (s)

0

T

fn

0

dQn (s) kn (t; t ds

@kn (t; t @t

s) + fs

s)

+ fs Qs (s)

dQs (s) ks (t; t ds

@ks (t; t @t

s)

ds

s) ds:

Integrating by parts, we get C(t) + H(t)

= w(t)N + rK(t)

Z

T

fn Qn (s)

0

fn +

Z

T

0

dQn (s) kn (t; t ds

s) + fs

dkn (t; t fn Qn (s) ds

s)

@kn (t; t @t

s)

dQs (s) ks (t; t ds

+ fs Qs (s)

@ks (t; t @t

s)

ds

T

s) 0

dks (t; t + fs Qs (s) ds

s)

ds:

Using (14), this simpli…es to C(t) + H(t)

= w(t)N + rK(t) Z

0

Z

T

fn Qn (s)

0

T

fn Qn (s)

@kn (t; t @

s)

@kn (t; t @t

+ fs Qs (s)

s)

+ fs Qs (s)

@ks (t; t @

s)

@ks (t; t @t

s)

ds

ds:

Meanwhile, Z T dkn (t; t s) dks (t; t s) dK(t) = fn Qn (s) + fs Qs (s) ds dt dt dt 0 Z T dK(t) @kn (t; t s) @kn (t; t s) @ks (t; t s) @ks (t; t = fn Qn (s) + + fs Qs (s) + dt @t @ @t @ 0 (57) 17

s)

ds;

so

dK(t) = w(t)N + rK(t): dt

C(t) + H(t) + Finally, using (5) and (6), w(t)N + rK(t)

K(0) N

"

exp(gt)N +

=

(1

)

=

(1

)K(t) (exp(gt)N )1

#

1

K(0) N

K(t)

+ K(t) (exp(gt)N )1

K(t);

so we have the income-product identity w(t)N + rK(t) = Y (t) Thus C(t) + H(t) +

K(t):

(58)

dK(t) + K(t) = Y (t); dt

(59)

so C(t) + H(t) + ( + g)K(t) = Y (t); and we have de…ned the model with bequests consistently.

A.2

Annuities Regime

Now let us verify that the income-expenditure identity holds in the annuities regime. Z T C(t) = [fn Qn (s)cn (t; t s) + fs Qs (s)cs (t; t s)]ds 0

C(t)

=

fn 1+ + +

(1 +

c )C(t)

c

fs 1+ 1 1+ =

Z

T

Qn (s) (1

y )w(t)(1

ss

ln (t; t

s))e(s) + [(1

k )r

+ (1

a )zn (s)]kn (t; t

s)

0

c

c

Z

T

Qn (s) (1

y )w(t)(1

ss

ls (t; t

s))e(s) + [(1

k )r

+ (1

a )zn (s)]ks (t; t

0

Z

T

[fn Qn (s) + fs Qs (s)] (s

Tr )b(t)ds

0

(1

y )w(t)

ss

+(1

k )r

+(1

a)

+b(t) Z

0

Z

Z

Z

Z

T

[fn Qn (s)(1

0

T

[fn Qn (s)kn (t; t

l/n (t; t

s)) + fs Qs (s)(1

s) + fs Qs (s)ks (t; t

s)]ds

0 T

[fn Qn (s)zn (s)kn (t; t

s) + fs Qs (s)zs (s)ks (t; t

0

T

[fn Qn (s) + fs Qs (s)]ds

Tr

T

ls (t; t

fn Qn (s)

@kn (t; t @t 18

s)

+ fs Qs (s)

@ks (t; t @t

s)

ds

s)]ds

s))]e(s)ds

s)

Using (8), (7) and integrating the third term by parts, this simpli…es to (1 +

c )C(t)

=

(1

ss

+(1

a)

Z

y )w(t)N T

+ (1

fn Qn (s)

0

k )rK(t)

dkn (t; t ds

s)

+ fs Qs (s)

dks (t; t ds

s)

dt + b(t)

ss

y )w(t)N +(1

k )rK+(1

a)

S(t)

dK(t) + dt

SS w(t)N:

Using (23), we get C(t) + H(t) +

dK(t) = w(t)N + rK(t) = Y (t) dt

K(t):

Thus we do, indeed, get the income-expenditure identity.

B

The Intratemporal Problem

To determine the c and l that maximize (25), we can do a monotonic transformation to rewrite the problem as max ln c + (1

) ln l

c;l

subject to pc c + pl l = E 0

l(t)

1 8t:

This has the Lagrangian L = ln c + (1 @L @c @L @l

) ln l + [E

= =

pl

l

l):

= 0:

= 0 and l < 1. Then we have

pc c = pl l

pl l] + (1

pc = 0

c 1

Let us …rst consider the case where

pc c

=

19

1

:

T

Tr

Using (21), (11), and (22), this further simpli…es to (1+ c )C(t) = (1

Z

[fn Qn (s) + fs Qs (s

Thus

1

= E: E pc (1

c = l

=

pl

This is valid for

pl

E< In the other case,

)E

:

:

1

(60)

0 and l = 1. Then we have E

c= = =1

pl = 1

Since we must have

0, we get

pl pc

=

pc c

pl pl =1 E pl

1

E pl

E pl E pl

E

1

1+

E pl

1

1

1 =

1

1

;

1

which is the opposite of (60). Thus we have c(E; pc ; pl ) = and l(E; pc ; pl ) =

(

(

E pc E pl pc

(1

)E pl

1

E< E

pl 1 pl 1

E< E

pl 1 pl 1

Note that lim c(E; pc ; pl ) = p

E" 1

l

c(E; pc ; pl ) = lim p

E# 1

pl 1

l

20

:

pl pc

1 pl

pc

(61)

=

1

pl pc

(62)

E" 1

pl = 1 = lim l(E; pc ; pl ): p pl E# 1 l

1 1

lim l(E; pc ; pl ) = p l

Thus, both c(E; pc ; pl ) and l(E; pc ; pl ) are continuous with respect to E. Since u(c; l) is a smooth function of c l1 , V (E; pc ; pl ) is also continuous with respect to E. Finally, let = c l1 : Expressed as a function of (E; pc ; pl ), this is 8 1 < pc pl (E; pc ; pl ) = E pl :

1

E

pc

For E <

pl 1

pl 1

E

pl 1

pl 1

1

1 pc

> 0:

pl

; (E

E (E; pc ; pl ) =

1

pl ) pc

> 0:

1

lim p

E# 1

:

;

E (E; pc ; pl ) =

For E >

E<

l

E (E; pc ; pl )

=

pc

pl

1

1

1

=

pc

;

pl

so E is a continuous function of E. Therefore, is C 1 with respect to E and so also is V (E; pc ; pl ). Furthermore, since u(c; ) is strictly increasing in c, V is also strictly increasing in E. V itself is 8 1 > 1 < u E; E < 1pl pc pl V (E; pc ; pl ) = : (63) > pl : u Epcpl ; E 1 Since u0 (c; ) = c

VE (E; pc ; pl ) = This simpli…es to

for all c > 0 and

8 > <

pc

> :

VEE (E; pc ; pl ) =

1

E

8 > < > :

8 > <

pc

> :

1 pl

pc

+ 21

E<

pl 1

E

pl 1

:

1

1

E

(E pl ) (1 (1 pc

(1

1

(E pl ) pc

1 pl

1

1 pl

pc

E pl pc

VE (E; pc ; pl ) = Note further that

1 pl

> 0,

+

)

)

1

pl 1

E

pl 1

1

E<

pl 1

E>

pl 1

:

(64)

1

E ) (E

E<

pl ) (2 (1 pc

+ )

)

;

which is negative for E 6= 1pl . Thus VE is strictly decreasing and invertible. Let M be the inverse of VE given pc and pl . Let v (pc ; pl ) = VE

pl 1

; pc ; pl

=

so

"

pc

(1

v (pc ; pl ) = Then the inverse M is 8 > > > < M (x; pc ; pl ) = > > > :

pc

pc (1

pl

)

)

x

1

1

pc

pl 1

1

1

!

1

1

;

pl

:

(65)

1

x 1 +

1

+

pl

1

#1

+ pl

x > v (pc ; pl ) x

:

(66)

v (pc ; pl )

It is helpful to note that l(E; pc ; pl ) actually only depends on pc and E=pl . Let us de…ne (1 )x x < 1 1 l(x; pc ) = ; (67) 1 x 11

so we have l(E; pc ; pl ) = l

E ; pc : pl

Note that these results carry over to the special case when case, we always have pl = 1, E< 1

= 1. In that

and

C

c(E; pc ; pl )

=

l(E; pc ; pl )

=

E pc 0:

(68) (69)

Scaling with Respect to Wage

For an extensive macroeconomic variable x(t) that scales with w(t), we de…ne x e=

x(t) ; w(t)

22

(70)

where x e is independent of w. scales with w(t), we de…ne

For an extensive lifecycle variable x(t; ) that x( + s; ) ; w( + s)

x e(s) =

where the righthand side is independent of Then we can rewrite (41) as Z T Z 1 a e exp( (1 k )rs)Qi (s) Ei (s)dt = 0

(71)

for a balanced-growth equilibrium. T 1 k )rs)Qi (s)

exp( (1

a

[(1

y )e(s)+

ss

0

(72)

Meanwhile, let us suppose that a(

+ s; ) = ea (s)w( + s)

(1

+

where ea (s) is independent of w(0). Thus we have Ei ( + s; )

ea (0; )w( + s) Qi (0)

ei (s)w( + s) = M = E 1+

c ; (1

ss

y )w(

)

(s Tr )eb]ds:

;

(1

+

)

a

Qi (s) Qi (0)

exp([

(1

k )r]s)

;

+ s)e(s)):

There are two cases to consider. Suppose that ea (0; )w( + s) Qi (0)

(1

)

Qi (s) Qi (0)

(1

a

exp([

(1

k )r]s)

>

1+

)

(1

c

1

1 y )w(

ss

+ s)e(s)

That is

ea (0; ) Qi (0)

(1

a

Qi (s) Qi (0)

exp([

(1

k )r]s)

>

1+

)

1

1 (1

c

+

:

y )e(s)

ss

Then

ei (s)w( + s) E

0" 1+ = @

(1

c

y )w(s)e(

+ s)

Qi (s) Qi (0)

a

1

1

ea (0)w( + s) Qi (0)

0" 1+ = @

ss

(1

+

(1

c

)

y )e(s)

ss

exp([ 1

1 [1

w( + s)

+

1+

]

#1

1

#1

1

A (1

ea (0)

Qi (0)

!

1

k )r]s)

1

a

Qi (s) Qi (0)

exp([

A

(1

k )r]s)

so 0" 1+ e E(s) =@

c

(1

ss

y )e(s)

1 23

1

#1

ea (0)

Qi (0)

Qi (s) Qi (0)

a

exp([

(1

1 A

k )r]s)

1

:

In the other case, ea (0)

Qi (0)

(1

a

Qi (s) Qi (0)

exp([

(1

k )r]s)

1+

)

(1

c

1

1

+

;

y )e(s)

ss

and

ei (s)w( + s) E

(1 +

=

+(1

c)

(1

)

ea (0)w( + s) Qi (0)

y )w(

ss

(1

+

)

!

a

Qi (s) Qi (0)

exp([

(1

k )r]s

+ s)e(s);

so ei (s) = E

(1 +

c)

(1

)

ea (0)

Qi (s) Qi (0)

Qi (0)

!

a

exp([

(1

1

1 +

+(1

k )r]s

ss

All of which can be summarized by the result ei (s) = M E

ea (0)

Qi (0)

Qi (s) Qi (0)

!

a

exp([

(1

k )r]s); 1

+

c ; (1

ss

y )e(s)

(73)

This implies that li ( + s; ) is independent of w( + s) since li ( + s; )

= l = l

(1 (1

Ei ( + s; ) ;1 + y )w( + s)e(s) ! ei (s) E ;1 + c ; ss y )e(s)

c

ss

which implies that N is independent of w(t).

References [1] Barendregt, Jan J., Luc Bonneux, and Paul J. van der Maas, (1997), “The Health Care Costs of Smoking,” New England Journal of Medicine 337: 1052-1057. [2] Bullard, James and James A. Feigenbaum, (2007), “A Leisurely Reading of Lifecycle Consumption Data,” Journal of Monetary Economics 54: 23052320. [3] Feigenbaum, James A., (2008), “Can Mortality Risk Explain the Consumption Hump?” Journal of Macroeconomics 30: 844-872. [4] Hansen, Gary D. and Selahattin I·mrohoro¼ glu, (2008), “Consumption over the Life Cycle: The Role of Annuities,” Review of Economic Dynamics 11: 566-583. 24

:

y )e(s):

1

1 +

[5] Rogers, Richard G. and Eve Powell-Griner, (1991), “Life Expectancies of Cigarette Smokers and Nonsmokers in the United States,” Social Science and Medicine 32: 1151-1159.

25