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
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),
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
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
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.