III. Model
The model is neoclassical optimal consumption and portfolio choice, based on
Merton (1969, 1971), but augmented with Kreps-Portius type preferences and an
adjustment in the utility function for returns to scale in household size and for retirement
status.
The value function is expressed recursively to allow separation of risk aversion
and intertemporal substitution parameters.
V (t )  U (C, t )   1 E(V (t  1))
where
1
1
(1   )V (t )
 (V (t )) 
1 
.
With this formulation, γ is the coefficient of relative risk aversion, which governs the
certainty equivalence transformation, and θ is the elasticity of intertemporal substitution.
The budget constraint, or equation of motion for the state variable full wealth, is:
dM  rM  (  r)M  Cdt   dz
where M is full wealth, r is the risk-free rate of return, α is the share of full wealth
invested in risky assets, µ is the expected return on risky assets, and σ2 is the expected
variance of risky returns. Also, the budget constraint contains a Brownian motion term
for the diffusion process of the risky asset.
Finally, let the utility function be CRRA but with the additional term K which
encompasses both the returns to scale in household size and the retirement status
adjustments:
U (C , i, t )  K (i, t )
C (i, t )1
C (i, t )1
 ( g ( N (t ))  k ( H (i, t )))
1
1
where
g ( N (t ))  N (t )
representing a fixed cost of working; 0≥δ≥-1; N(t)=1 if working and N(t)=0 if retired.
Retired households don’t have the fixed costs of working and can substitute time for
expenditures, yielding more utility per dollar of expenditure. Retirement is assumed to
be deterministic and known, both in the model and in the construction of full wealth in
the data. Spouses cannot retire separately; a household is either working or retired.
The second element of K represents household size and makes the utility function
state-contingent on number of people in the household H(i,t). The household size for this
population is primarily the distinction between couple and single households.
Discussion of the model will be restricted to couples and singles although the model can
easily accommodate other household sizes.1 Change in household size is assumed to
happen only through death, which is stochastic. Expectation of divorce (or marriage for a
single household) is not currently allowed in the model. Returns to scale in household
size is expressed as follows:
k ( H (i, t )) 
H (i, t )
(a  bH (i, t ))
H(i,t) represents the initial state. So H(i,t)=1 for single households and H(i,t)=2 for
couples. The parameter η represents the degree of returns to scale in household size. A
couple gets more utility out of the same total dollar amount of consumption as a single
because each person in the couple gets utility out of common expenses such as housing.
Assigning values to a, b, and η allows calculation of the expression for K(i,t) for each
type of household. Let a=0.3, b=0.7, and η=.7.2
Retired Single Household:
1
K    1
1
Working Single Household: K  0 
Retired Couple Household:
1
K  
1
1
1
2
   1.38
1.7.7
The empirical analysis will show both the sample restricted to singles and couples as well as all
households.
2
These values are similar to those proposed by various authors in the context of poverty measurement as
well as the OECD equivalence scale. See a full discussion of such equivalence scales in Citro and Michael
(1995). In Citro and Michael’s proposed scale, they recommend the power element, representing the degree
of returns to scale, should be between 0.65 and 0.75. The other elements of their proposed scale focus on
the distinction between adults and children not on the distinction between the first and second adult as is
done here.
Working Couple Household: K  0 
2
 1.38
1.7.7
In addition to Merton, this model specification and the strategy for solving it owes
reference to Kimball (1993) for the treatment of Kreps-Portius type preferences; Kimball
& Mankiw (1989) for the state-contingent Bellman equation; Kimball & Shapiro (2003)
for the treatment of household size and cost of working in the utility function; and Hurd
(1999) for the treatment of couple mortality. The Bellman equation for this model is:
V ( M , i, t )  Vt ( M , i, t ) 
2
2
2


" (V (t ))
2  M
max U (C , i, t )  VM ( M , i, t ) X  VMM ( M , i, t ) 
VM ( M , i , t ) 

'
(
V
(
t
))
2



f ( N (t ),  (t ) j )
 (V ( M , j , t )   (V ( M , i, t ))dF ( N (t ), j )dt 


 ' (V ( M , i, t ) j

In the last term, j indicates a changed state relative to the initial state i. The term
f ( N (t ),  (t ) j ) indicates the probability of being in the changed state. If retirement
status is ignored for the moment, then f ( )   (t ) ; mortality hazards, λ, give the
probabilities of transitioning from the initial state to the changed state. The changed state
is death for a single household, H(j,t)=0; or widowing for a couple, H(j,t)=1. Since
transition occurs due to death, transition can only happen in one direction; households are
not allowed to increase in size. The mortality hazard  is time-varying based on age and
is gender specific.3 So the probability of a couple leaving the initial state in any period
(becoming widowed or a single household) is λ(t)husband + λ(t)wife.
Returning to the issue of retirement status, since retirement is deterministic and
known, the indicator variable N(t) is used rather than probabilities. The mortality hazards
are not affected by retirement status (death while working is allowed). So there are four
3
Households live at most until the terminal date, T (death is certain at T), and face an increasing mortality
risk up until T. For a single household, the definition of  is the probability of dying in period t conditional
on having survived to period t-1. Therefore, the unconditional probability of surviving to period t is
t

  dt
e
t0
. For a couple household, the unconditional probability of the household reaching period t intact is
t
e
 (  , Husband  ,Wife ) dt
t0
(see Hurd 1999).
possible changed states for a couple (two for a single): husband dies while household
working; wife dies while household working; husband dies while household retired; wife
dies while household retired. So the final term of the Bellman equation can be explicitly
expressed with four terms for a couple (or two for a single):
N (t ) (t ) Husband
 (V ( M , j  work & H 1, t )   (V ( M , i, t ))
 ' (V ( M , i, t )
N (t ) (t )Wife
 (V ( M , j  work & H 1, t )   (V ( M , i, t ))

 ' (V ( M , i, t )
(1  N (t )) (t ) Husband
 (V ( M , j  ret & H 1, t )   (V ( M , i, t ))

 ' (V ( M , i, t )
(1  N (t )) (t )Wife
 (V ( M , j  ret & H 1, t )   (V ( M , i, t ))

 ' (V ( M , i, t )
Since V(M,j,t) does not depend on which spouse dies, death of either spouse reduces
household size to one and results in the same value function, then the last term of the
Bellman equation for a household initially in the couple state is:
N (t ) (t ) husband   (t ) wife 
 (V ( M , j  work , t )   (V ( M , i, t ))
 ' (V ( M , i, t )
(1  N (t )) (t ) husband   (t ) wife 
 (V ( M , j  retired , t )   (V ( M , i, t ))

 ' (V ( M , i, t )
The value function in the retired state is different than the value function in the working
state (g(N) will always be zero in the retired state) so these terms cannot be combined in
the same way.
Abstracting from the states to begin with an intuitive solution, Merton’s original
version of the model can be solved for a steady state expression of the propensity to
consume, C/M, yielding a familiar expression under assumptions of infinite horizon (no
mortality hazards, therefore no state transitions) and constant expected returns (r, μ, and
σ) 4:
C 
1
(  r ) 2
  (1  )[r 
]
M 

2  2
4
The infinite horizon model assumes couples always remain as couples, so no divorce either.
This expression differs from that of Merton (1969) only in that risk aversion and
intertemporal substitution are represented separately, whereas in Merton θ=γ.
For the older households used in this analysis, the infinite horizon is likely to be a
poor approximation since encroaching mortality will increasingly impact their propensity
to consume out of remaining resources. With time-varying mortality there is no closed
form solution.5 However, since mortality hazards rates are well-known, the following
expression for C/M can be accurately estimated:
C

M
1

T
t
 
 (  r )2  
  ( t '  t ) 1 
(1/  )    (1 )  r 
2 
2    
 
 1 

e
where
and
t'
t

 1  t '
 Z ( ,retired ) d 
Z ( ,work ) d 


 1  t

dt'


 K ( j work , ) 
Z ( , work )   
 1  N ( )  ( ) 
  K ( i , )




 K ( j retired, ) 
Z ( , retired)   
 1  (1 N ( ))  ( ) 
 

K ( i , )

The first term in the brackets of the exponential function is the same expression as is in
the infinite horizon model. The second and third terms, involving the ratio K(j,t)/K(i,t),
represent the effect of mortality and retirement status on the consumption propensity,
incorporating the effect of returns to household scale. Looking at the form these latter
terms take for each type of household helps to understand this effect; these are shown in
Appendix A.
Appendix A: Expressions of the Model Solution by Household Type
5
Expected returns are still assumed to be constant. The model contains a fixed terminal date T (set equal to
age 100) to enable computation, but the mortality hazards get so high at late ages that the exact terminal
date within a reasonable range makes little difference in the calculation of C/M.
Expression for the propensity to consume out of full wealth:
C

M
1

T
t
 
 (  r )2  
  ( t '  t ) 1 
(1/  )    (1 )  r 
2 
2    
 
 1 

e
where
and
t'
t

 1  t '
 Z ( ,retired ) d 
Z ( ,work ) d 


 1  t



 K ( j work , ) 
Z ( , work )   
 1  N ( )  ( ) 
  K ( i , )




 K ( j retired, ) 
Z ( , retired)   
 1  (1 N ( ))  ( ) 
 

K ( i , )

First, in general:
N (t )  H ( j, t ) /(.3  .7 H ( j, t )) .7
K ( j  work , t ) / K (i, t ) 
N (t )  H (i, t ) /(.3  .7 H (i, t )) .7
H ( j, t ) /(.3  .7 H ( j, t )) .7
K ( j  ret , t ) / K (i, t ) 
H (i, t ) /(.3  .7 H (i, t )) .7
Retired Single Households:
Z ( , work )  0 for all τ
Z ( , retired )   ( ) for all τ {because K ( j, t )  0 when dead}
Retired Couple Households:
Z ( , work )  0 for all τ
 1

Z ( , retired )  
 1( ( ) husb   ( ) wife ) for all τ
.7
 2 /(1.7)

Working Single Households:
dt'
 

Z ( , work )  
 1  ( ) for τ before retirement
  1 
0
Z ( , retired )  0
for τ after retirement
for τ before retirement
  ( ) for τ after retirement
Working Couple Households:
  1

Z ( , work )  

1
( ( ) husb   ( ) wife ) for τ before retirement
.7


2
/(
1
.
7
)


0
Z ( , retired )  0
for τ after retirement
for τ before retirement
 1


 1( ( ) husb   ( ) wife ) for τ after retirement
.7
 2 /(1.7)
