Deficit = rBt-1 + Gt – Tt (22.1)
where all variables are expressed in real terms, B is real debt, and r is the real interest rate. The text notes that
official deficit measures typically substitute nominal interest payments (iBt-1) for real interest payments (rBt-1) in
equation (26.1). The deficit is linked to the increase in debt by the government budget constraint:
Bt–Bt-1 = rBt-1 + Gt – Tt (22.2)
Defining the primary deficit as (Gt – Tt), government debt can be expressed as
Bt = (1 + r)Bt-1 + Primary Deficit. (22.3)
This relationship implies that, starting from zero debt and a zero primary deficit, a one-unit increase in the
primary deficit for one period will generate a debt of Bt = (1 + r)t after t periods. To repay this debt after t periods,
the government must run a primary surplus of (1 + r)t. If spending is unchanged, this means that a reduction in
taxes today implies an increase in future taxes of equal present value. If the government seeks to stabilize the
debt instead of repaying it, then the government must run a primary surplus equal to the real interest payment on
the debt in every future period. In other words, the government must eliminate the inflation-adjusted deficit,
defined as the primary deficit plus real interest payments on the debt (or equivalently, as the official deficit minus
inflation times real debt.)
To examine the evolution of the debt-to-GDP ratio, rewrite the government budget constraint as
Bt/Yt = (1 + r)(Yt-1/Yt)(Bt-1/Yt-1) + (Gt – Tt)/Yt,
which can be approximated as
(Bt/Yt) – (Bt-1/Yt-1) = (r – g)(Bt-1/Yt-1) + (Gt – Tt)/Yt, (22.5)
where g is the growth rate of output. Given some initial debt, equation (22.5) implies that the debt-to-GDP ratio
will grow when there is a primary deficit and when the real interest rate exceeds the growth rate of output. To
understand the latter effect, suppose the primary deficit is zero. Then, debt (the numerator) will grow at rate r and
output (the denominator) will grow at rate g. The difference in growth rates is approximately the change in the
debt-to-GDP ratio.
3. Ricardian Equivalence, Cyclical Adjusted Deficits, and War Finance
i. Ricardian equivalence. Ricardian equivalence is the proposition that neither deficits nor debt affect
economic activity. For example, given unchanged government spending, a tax cut today implies a tax increase of
equal present value in the future. Therefore, consumer wealth is unchanged, and private consumption is
unaffected. An increase in today’s government deficit will be matched with an equal increase in private saving.
In practice, however, tax increases that are distant and uncertain are likely to be ignored by consumers, because
they may not live to see them or because they do not think that far into the future. As a result, although
expectations certainly affect economic behavior, it is unlikely that Ricardian equivalence holds in strict form.
ii. Deficits, output stabilization, and the cyclically-adjusted deficit. The fact that deficits reduce
investment does not mean that they should be avoided at all times, but rather that deficits during recessions should
be offset by surpluses during booms. In this way, fiscal policy will not lead to a steady increase in debt. The
cyclically-adjusted deficit removes the effect of the business cycle from the deficit. Thus, it can be used to assess
whether fiscal policy is consistent with no systematic increase in debt over time. Estimating the
cyclically-adjusted deficit requires knowledge of two facts: the reduction in the deficit that would occur if output
were to increase by 1% and the difference between current output and its natural level. The first fact is relatively
easy to determine. As a rule of thumb, a 1 percent decrease in output increases the deficit by 0.5% of GDP. The
deficit increases because most taxes are proportional to output, but most spending does not depend on output.
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22-103