of motion expressing next period’s capital as a sum of current capital and the difference between current production and consumption, the latter of which in-
cludes retirees’ consumption out of wages from the previous period. The accumula- tion process in thus described by a second order relation. Of course, in the standard
model, retirees consume all of their previous savings and all of capital’s share of the national product, which they receive as interest payments. The result is that capital
formation is due entirely to the savings of young workers, who must replace the entire past capital stock and then some, if they are to see growth in capital
intensity.
There are behavioral assumptions that could lead to accumulation from one period to the next, even if retirees hold all assets. One possibility is that retirees
refrain from spending all of savings and interest, as a consequence of either altruistic bequests or uncertainty regarding lifespan, then at least some of capital
could remain in the economy as unconsumed capital e.g. as savings of the young out of bequests. OLG models with altruism have received much attention in the
literature. For an overview see Smetters 1999. Other recent models include Michel and Pestieau 1998, Pecchenino and Pollard 1997, Hori 1997. A recent study of
the effects of uncertain lifespan are in Fuster 1999. All of these references assume, however, that next period’s capital stock depends entirely on current savings an
exception is Lines, 1999.
A second assumption, that retirees are not endowed with the entire capital stock, could also lead to capital accumulation. The following is a study of the dynamics
of capital accumulation under the latter hypothesis. In Section 2 an OLG model is developed. Existence and stability of steady states, and the dynamics in general, are
studied in Section 3. Concluding remarks are provided in Section 4.
2. Basic model
2
.
1
. Worker-capitalists The problem facing the representative agent is:
max
c
1t
U =
uc
1t
+ 1 + u
− 1
uc
2t + 1
subject to c
1t
+ s
t
5 w
t
c
2t + 1
5 1 + E
t
r
t + 1
s
t
c·, w·, u·, r· R
+
, s· R
+
0 5 u 5 E
t
r
t + 1
= r
t + 1
1 and variables are defined as follows: c
1t
is consumption in the working period at time t, c
2t + 1
is consumption in the retired period at time t + 1, w
t
is the wage in
period t, s
t
is savings in period t, r
t + 1
is the interest rate on one-period loans at time t + 1. The last line in Eq. 1 represents the assumption that individuals have perfect
foresight. The single parameter is u where 1 + u
− 1
is the subjective discount factor for future utility.
The instantaneous utility function assumed for consumption is the strictly concave, isoelastic function u· = ln·, which meets the usual requirements of a
utility function and guarantees that forward dynamics can be determined because substitution and income effects cancel each other exactly.
The first order condition determines the optimal amount of consumption and savings in the first period asc
1t
= 1 + u
2 + u w
t
s
t
w
t
= w
t
− c
1t
= w
t
2 + u ,
2 which is simply Keynes’ hypothesis that savings is a constant proportion of the
individual’s income. Note that if the individual values utility in the retirement period as much as current utility u = 0, he saves half of his wages for the second
period. If the future has no weight at all, u and s
t
0. Each member of the labor force N is characterized by the same utility function
and subjective discount factor so that total savings in any period is simply the product S
t
= N
t
s
t
. It is assumed that the labor force grows according to N
t
= N
1 + n
t
− 1 B n B 1
3 where n is given exogeneously.
2
.
2
. Technology Total output Y
t
is produced by capital K
t
and labor N
t
with function F that is homogeneous to the first degree, permitting output to be expressed in per capita
terms lower case variables Y
t
= FK
t
, N
t
= K
t a
N
t 1 − a
0 B a B 1 N
t
R
+
K
t
R
+
Y
t
N
t
= y
t
= k
t a
. 4
The per capita production function is well-behaved and satisfies the Inada conditions.
2
.
3
. Firms While production technology is well-defined in the standard model, the organiza-
tion of production is typically somewhat ambiguous. Diamond’s entrepreneurs in the competitive setup
1
become, in later analyses, profit-maximizing firms. The
1
‘Capital demanders are entrepreneurs who wish to employ capital for production in period t + 1.’ Diamond, 1965, p. 1130
former approach seems to indicate a second type of agent, although no other characteristics are offered. In the latter case alternative interpretations can be
considered. If firms are simply set up by retirees there are, at time t, N
t − 1
of them, and their growth rate is the same as that of the work force. If, instead,
firms are merely managed by retirees see, e.g. Boldrin, 1992 their number and growth rate are irrelevant. In either case, the profit-maximizing behavior of
retirees may need to be cast in the utility maximization problem. Would these owners or managers seek to maximize profits, or maximize the return to their
own capital?
Essentially, in the standard OLG model, production is operated by a profit- maximizing algorithm which solves the problem:
max
N
t
, K
t
V =
Y
t
− w
t
N
t
− r
t
K
t
subject to Y
t
5 FK
t
, N
t
. 5
First order conditions for problem 5 give optimal levels of labor and capital as those for which respective prices are equivalent to respective marginal prod-
ucts. Given the production function in Eq. 4 these are: MP
N
= w
t
= 1 − ak
t a
MP
K
= r
t
= a k
t a −
1
. 6
The algorithm employs the optimal factor levels and distributes wages and interest. Under the hypothesis that capital is entirely owned by retirees, the firm
algorithm is the mechanism by which output is distributed between the young worker and the retired capitalist generations. Thus is the classical antagonistic
framework of heterogeneous cohorts workers and capitalists transformed into that of homogeneous cohorts, heterogeneous generations. This formal description
is particularly useful for the study of transactions between generations for it focuses exactly on their opposing interests. It is less useful for studying the
timepath of genuine capital accumulation from one period to the next, and is certainly in great difficulty to explain the emergence of capitalism in a context of
few retirees, high discount rates andor subsistence wages.
Suppose, instead, that in the initial periods a portion of the capital stock is not assigned to retirees. Potential recipients of the unassigned part can be grouped
as: i pure capitalists, entrepreneurs, speculators, or any other agents whose consumption patterns can be ignored in a first approximation the extreme of the
classical savings hypothesis; or ii the firm algorithm which, in addition to its other tasks, reinvests any of capital’s share remaining after distribution to re-
tirees. The homogeneous cohorts vision is maintained by the latter assumption, but violated by the former. In either case, no further assumptions are necessary,
as long as the other owners of capital refrain from consuming.
2
.
4
. Equilibrium conditions The equilibrium condition in the single good market, investment equals savings,
is K
t + 1
− K
t
= Y
t
− C
1t
− C
2t
= r
t
K
t
+ w
t
N
t
− N
t
s
t
− 1 + r
t
N
t − 1
s
t − 1
= N
t
s
t
+ r
t
K
t
− 1 + r
t
N
t − 1
s
t − 1
7 where the second line makes use of Euler’s theorem for a first-degree homogeneous
production function. Notice that the dissavings of the retired generation is given by the last term on the RHS. If all of capital’s share of production goes to retirees
N
t − 1
s
t − 1
K
t
S
t − 1
and the model reduces to the standard version. The initial conditions permitting the relaxation of this hypothesis are studied in Section 3.4.
The labor market follows the standard model. Labor supply is inelastic, and the market is in equilibrium when w
t
in Eq. 6 induces firms to hire N
t
. The demand for capital is given by the marginal product function
KD
t
= N
t
a r
t 11 − a
. Under the hypotheses the supply of capital available at t, rearranging Eq. 7 is:
KS
t
= S
t − 1
+ 1 + r
t − 1
K
t − 1
− S
t − 2
. The equilibrium rate of rental for capital services is that which satisfies Eq. 6
and induces firms to hire the capital services available and fixed at time t.
3. The dynamics of accumulation
