Generational Accounts and the Government

1The Government Budget Constraint¶

Consider a government that raises taxes $\TaxLev_{t}$, makes expenditures $\GovSpend_{t}$, and has an outstanding stock of debt $\Debt_{t}$ at the beginning of period $t$, on which it must pay interest at rate $\rfree_{t}$. The government can run a deficit only by raising funds via the issuing of new bonds (the alternative, printing money, leads to inflation and is the subject of monetary economics).

The government’s Dynamic Budget Constraint (DBC) is given by

But we can obtain a similar formula for $\Debt_{t+1}$ in terms of $\Debt_{t+2}$, and substitute it into (1). Continued substitution gives

where $\mathbb{P}$ denotes the present discounted value. This equation says the government must plan to repay its debts, but places no constraint on what the government can do in any single period. Unlike households, the government has an infinite horizon and can obligate future generations to pay for the spending of current generations (Japan once offered 100-year mortgages, illustrating how intergenerational obligations can extend across multiple lifetimes). This can be rewritten

Equation (3) should look familiar: recall that in the consumption problem we had an Intertemporal Budget Constraint that said

where $\Kap_{t}$ is the beginning-of-period level of capital wealth (before interest has been earned), and $\mathbb{P}_{t}(Y) = H_{t}$ is human wealth.

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). The $\Rfree_{t}$ multiplying debt reflects the timing convention that interest payments occur at the beginning of the period; for the consumer we were thinking about the situation after any interest income was received. The sign difference reflects the fact that $D$ is debt while $B$ represents asset balances.

Now let’s suppose that the only kind of expenditures the government engages in are transfers, so that $\GovSpend_{t}$ simply reflects money handed out to some members of the population in period $t$. Then $\Surplus_{t}$ will be equal to total net transfers among the members of the population at period $t$ (this is the primary surplus if we think of government activities more generally). Note that there is nothing that says that $\Surplus_{t}$ 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 et al. (1991).

If we assume that consumers live two-period lives, the generational account for the generation born at time $t$ is:

Note that either of these $Z$ terms can be negative, which would signify net transfers to households rather than taxes collected.

In the US and most other countries, working-age people pay more in taxes than they receive in transfers, so $\Surplus_{1,t}$ is positive, while old people receive more in transfers than they pay in taxes, so $\Surplus_{2,t}$ is negative. The largest such transfers in the US are Social Security and Medicare.

Note now that the aggregate total of net transfers can be subdivided into the net transfers of the two age groups in the population,

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.

2Social Security and Generational Accounts¶

Consider an economy that initially has no government so that $\Surplus_{1,t}=\Surplus_{2,t}=\Surplus_{2,t-1} = 0$. Now consider introducing a Pay As You Go (PAYG) Social Security system at date $s$, which is to remain of constant size forever after introduction,

Consider the generation born at time $s-1$. It paid nothing into the Social Security system when young, yet gets $\Surplus_{2,s}$ out when old. Its generational account is therefore

so this generation benefits from the introduction of SS because it paid no taxes yet receives benefits. Since $\Surplus_{2,s}$ is negative (a transfer to the old), this generation is better off than without Social Security.

The generational accounts for succeeding generations are

so future generations are worse off by this amount. Since taxes are positive, the effect on the lifetime budget constraint is the negative of the expression on the RHS of (10).

The reason the introduction of Social Security makes future generations worse off is that without SS they could have invested the amount $\Surplus_{1,t}$ and earned interest on it of $\rfree_{t+1}\Surplus_{1,t}$ in period 2. 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 $\Surplus_{1,t}$ in period $t+1$, discounted back to the present.

Note that if there is zero population growth, the foregoing analysis all holds in per-capita terms as well, so that the per-capita change in generational accounts from introducing Social Security is

2.1Effects of Population Growth¶

If there is perpetual population growth, it is possible to finance a positive rate of return on Social Security contributions. It will often be convenient to write generational accounts in per-capita terms rather than in aggregate terms, using lower-case letters for per-capita quantities. Define per capita contributions as (note that $\surplus_{2,t+1}$ is divided by $L_{t}$ rather than $L_{t+1}$ to express the return from the perspective of the contributing generation):

and assume there is constant population growth, $\PopGro=L_{t+1}/L_{t}$. 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:

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 per-capita taxes on the young is a rate of return of $\PopGro$. In an economy with perpetual population growth, it is not only possible but necessary to pay a positive return.

2.2Effects of Productivity Growth and Population Growth¶

Suppose there is wage growth ${\WGro}$ betwen $t$ and $t+1$, and suppose that workers contribute a constant percentage of their incomes to the Social Security system, $\surplus_{1,t} = \zeta \Wage_{1,t}$. 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:

so viewed from the perspective of the young generation in period $t$, their Social Security contributions are returned to them larger by a factor of ${\WGro}\PopGro$ than what they paid in; the effective rate of return is therefore ${\WGro}\PopGro$.

2.3Generational Accounts in a Growing Economy¶

Now consider the per-capita generational accounts in an economy with constant population growth and constant wage growth and a Social Security system that imposes a constant tax of $\zeta$ on the wages of the young:

Note that this expression will be negative if ${\WGro}\PopGro>\Rfree_{t+1}$, meaning that the introduction of a Social Security system with a positive tax rate $\zeta$ actually improves the lifetime budget constraint! This is another way of seeing that an economy is dynamically inefficient if the return factor for capital $\Rfree$ 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).