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Long-Run Implications of Alternative Fiscal Policies 105
28. Even with an omc of less than unity it is likely that the impact of the tax on bequests handed
down from one generation to the next would gradually disappear so that W and K would eventually
be unaffected by the tax-financed expenditure. But this is in the very long run indeed.
29. See, e.g., J. Tobin and H. W. Guthrie, “Intergenerations Transfers of Wealth and The Theory of
Saving,” Cowles Foundation Discussion Paper No. 98, November 1960.
30. By so doing we are also deliberately by-passing the other major issue of fiscal policy, that of the
distribution of the burden between income classes.
31. In order to avoid side issues, we will assume that the coupon rate on these bonds in such as to
make their market value also equal to dD, and that no change occurs in the government purchase of
goods and services, G.
32. Just for reference, we may note that according to the Modigliani-Brumberg model, if income
were growing at approximately exponential rate, W would be growing at the same rate.
33. This conclusion is strictly valid only in so far as the fall in disposable income brought about by
the fall in K is matched by an equal fall in consumption. To the extent, however, that consumption
falls somewhat less, cumulated saving may fall somewhat more, pushing W and K to a lower position
than in our figure: but this extra adjustment will in any event tend to be of a second order of
magnitude. The nature and size of this adjustment can be exhibited explicitly with reference to the
Modigliani-Brumberg model of consumption behaviour. As indicated earlier, this model implies that,
in the long run, the aggregate net worth of consumers tends to be proportional to their (disposable)
income, or (1) W = gY, where the proportionality constant is a decreasing function of the rate of
growth of income. Suppose initially income is stationary as population and technology are both stationary.
We also have the identity (2) W = K + D, where D denotes the national debt. With population
and technology given, the effect of capital on income can be stated by a “production function”
(3) Y = f(K). We have stated in the text that a gratuitous increase in D, or more generally an increase
in D which does not result in government capital formation or otherwise change the production function,
will tend to reduce K by dD and Y by r*dD: i.e., we have asserted
dK dY
- 1 and ª -r*,
dD dD
where
df
r* = = f¢
r.
dK
By means of equations (1)–(3) we can now evaluate these derivatives exactly. Solving (2) for K and
using (1) and (3) we have K = gf(K) - D. Hence
dK
dD
Similarly,
dY
dD
gf dK dK -1
= ¢ - =
dD dD - gf ¢ = -1
1 or
1 1-
gr* .
r dW gr
= - *
= - *
and
1-
gr * dD 1-
gr * .
Thus, if r* r = 0.05 and g is in the order of 4, then
dK
dY
is -1.25 instead of -1 and is -0.625 instead of -0.05.
dD
dD
I am indebted to Ralph Beals, presently a graduate student at Massachusetts Institute of Technology,
for pointing out that these formulæ are not entirely general, for, within the Modigliani-Brumberg
model, the second-order effect is not independent of the nature of taxes employed to defray the interest
bill. In fact, the formulæ derived above are strictly valid only if the revenue is raised entirely
through an income tax on non-property income. With other kinds of taxes, one obtains somewhat
more complicated formulæ. For instance, if the taxes are levied on property income this will depress