February 7, 2021, Christopher D. Carroll GenAcctsAndGov
Consider a government that raises taxes , makes expenditures , and has an outstanding stock of debt at the beginning of period , on which it must pay interest at rate . The government can run a deﬁcit only by raising funds via the issuing of new bonds.
The government’s Dynamic Budget Constraint (DBC) is given by
 (1) 
But we can obtain a similar formula for in terms of , and substitute it into (1). Continued substitution gives
 (2) 
where denotes the present discounted value; this can be rewritten
 (3) 
Equation (3) should look familiar: recall that in the consumption problem we had an Intertemporal Budget Constraint that said
 (4) 
where is the beginningofperiod level of capital wealth (before interest has been earned).
In each case, the PDV of expenditures must be equal to the PDV of income plus current wealth. Thus, equations (2) and (3) are diﬀerent ways to express the Government Intertemporal Budget Constraint (GIBC).^{1}
Now let’s suppose that the only kind of expenditures the government engages in are transfers, so that simply reﬂects money handed out to some members of the population in period . Then will be equal to total net transfers among the members of the population at period . Note that there is nothing that says that must be positive or negative in any particular period. The GIBC only places restrictions on the present discounted value of net transfers.
The fact that government only has to satisfy the GIBC means that the government can potentially treat diﬀerent generations very diﬀerently from each other. It is therefore useful to have a mechanism to keep track of how diﬀerent generations are treated. The standard way of doing this is to construct a set of ‘generational accounts,’ as initially proposed by Auerbach, Kotlikoﬀ, and Gokhale (1991).
If we assume that consumers live twoperiod lives, the generational account for the generation born at time is:
 (5) 
In the US and most other countries, workingage people pay more in taxes than they receive in transfers, so is positive, while old people receive more in transfers than they pay in taxes, so is negative.
Note now that the aggregate total of net transfers can be subdivided into the net transfers of the two age groups in the population,
 (6) 
Now write out the GIBC (2) explicitly:

which again shows that the GIBC is consistent with any treatment of any particular generation; any pattern of generational accounts that satisﬁes the GIBC is feasible.
Consider an economy that initially has no government so that . Now consider introducing a Pay As You Go (PAYG) Social Security system at date , which is to remain of constant size forever after introduction,
 (7) 
Consider the generation born at time . It paid nothing into the Social Security system when young, yet gets out when old. Its generational account is therefore
 (8) 
so this generation beneﬁts from the introduction of SS because it paid no taxes yet receives beneﬁts.
The GA’s for succeeding generations are
 (9) 
so future generations are worse oﬀ by this amount.
The reason the introduction of Social Security makes future generations worse oﬀ is that without SS they could have invested the amount and earned interest on it of in period . Now the money is taken away from them when young and returned without interest when old. Thus, the loss is precisely the loss in interest income on in period , discounted back to the present.
Note that if there is zero population growth, the foregoing analysis all holds in percapita terms as well, so that the percapita change in generational accounts from introducing Social Security is
 (10) 
If there is perpetual population growth, it is possible to ﬁnance a positive rate of return on Social Security contributions. Deﬁne
 (11) 
and assume there is constant population growth, . If we assume that Social Security taxes per capita are constant, then we can achieve a positive rate of return on Social Security contributions equal to the growth rate of population:
 (12) 
Not only does this prove that it is possible for the Social Security system to pay a rate of return equal to the rate of population growth  it proves that the only rate of return that is consistent with constant percapita taxes on the young is a rate of return of .
Suppose there is wage growth betwen and , and suppose that workers contribute a constant percentage of their incomes to the Social Security system, . In this case it is possible to earn a rate of return on SS contributions equal to the product of the growth factor for wages and the growth factor for population:
 (13) 
so viewed from the perspective of the young generation in period , their Social Security contributions are returned to them larger by a factor of than what they paid in; the eﬀective rate of return is therefore .
Now consider the percapita generational accounts in an economy with constant population growth and constant wage growth and a Social Security system that imposes a constant tax of on the wages of the young:
 (14) 
Note that this expression will be negative if , meaning that the introduction of a Social Security system with a positive tax rate actually improves the lifetime budget constraint! This is another way of seeing that an economy is dynamically ineﬃcient if the return factor for capital is less than the product of the population growth and productivity growth factors. (Or, using approximations, the rate of return is less than the sum of the population growth rate and the productivity growth rate).