February 7, 2021, Christopher D. Carroll SocSecAndKAccum
Consider a household with a 2-period lifetime, whose optimization problem is
|
Under logarithmic utility, handout 2PeriodLCModel shows that the solution to this problem is
| (1) |
The only role of government in this economy is to run a Social Security
program. Suppose that initially this economy had no Social Security system and
we are interested in the effects of introducing a Pay-As-You-Go Social Security
system that is expected to remain a constant size from generation to generation
from now on: while
, so that taxes are greater
than transfers when young and transfers are greater than taxes when
old.
The effects of Social Security on first period consumption can be seen by
writing out explicitly the value for from equation (1),
| (2) |
where the expression with the underbrace comes from the effect of introducing a
constant-sized PAYG Social Security system in handout GenAcctsAndGov. If
taxes paid when young are positive (as they are after the introduction of
the Social Security system) and the interest rate is positive, the expression with
the underbrace is a positive number, and since it is being subtracted from
it is clear that consumption in the first period of life will decline with the
introduction of the Social Security system.
Does the decline in consumption mean the saving rate rises? No - because saving is after-tax income minus consumption, and net taxes on the young have risen. For saving we have
| (3) |
So if then saving is less than before the introduction of Social
Security.
Now consider the implications in a Diamond (1965) OLG model where saving is the source of capital accumulation. Suppose there is no population growth so that
| (4) |
where as before in the OLGModel handout. Thus the
capital accumulation curve is shifted down. The dynamics of the introduction of
Social Security are captured in the figure, under the assumption that the
economy was at its steady-state equilibrium level
before the Social Security
system was introduced. The effect of introduction is an immediate increase in
consumption, as the old generation spends everything it gets and the
young generation doesn’t need to do as much retirement saving as before.
Over time the economy will converge to its new, lower level of capital
.
Diamond, Peter A. (1965): “National Debt in a Neoclassical Growth Model,” American Economic Review, 55, 1126–1150, http://www.jstor.org/stable/1809231.