Productivity Growth In a Pure Pay as You Go System

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In a pure pay as you go system, Social Security taxes equal Social Security benefits in every time period. There is no Trust Fund.

Social Security Taxes = Benefits

SS Taxes (T) = Total Wages (W) * The tax rate (t)

Benefits = Average benefit (b) * Retirees (R)

Therefore:

W*t = b*R

The rate of tax that workers would have to pay would be:

t = (b*R) / W

Total wages (W) is equal to the average wage (w) times the number of workers (N).

W = w * N

Thus the social security tax rate can be written as follows:

t = (b*R) / (w * N)

If we rearrange terms, that becomes:

t = (b/w) * (R/N)

The first term (b/w) is called the Replacement Rate (RR).

The second term (R/N) is called the Dependency Rate (DR)

Thus the social security tax rate can be written as follows:

t = RR * DR

The growth rate of the tax rate can be expressed as follows:

gt = gRR + gDR

If the replacement rate is constant (gRR = 0), the growth rate of the tax rate will be the same as the growth rate of the dependenccy rate. Demography is all important.

gt = gDR

If productivity grows, the wage rate will increase and the replacement rate will fall. Economics takes over.

gRR = gb - gw

The growth rate of the tax rate can be expressed as follows:

gt = gb - gw + gDR

If productivity growth is large enough, it can offset the growth in the benefit rate and the growth in the dependency rate and there need be no increase in the Social Security tax rate.