Preston J. Miller

Thomas J. Sargent

English

Research Papers in Economics

Should Currency Be Priced Like Cars?  Thomas M. Supel 
Richard M. Todd (p. 3) 
Some Pleasant Monetarist Arithmetic  Michael R. Darby (p. 15) 
A Reply to Darby  Preston J. Miller 
Thomas J. Sargent (p. 21) 
District Conditions 
Urban Strength, Rural Weakness  (p. 27) Federal Reserve Bank of Minneapolis 
Quarterly Review vol. 8, no 2 issn 0271-5287 
This publication primarily presents economic research aimed at improving policymaking 
by the Federal Reserve System and other governmental authorities. 
Produced in the Research Department. Edited by Preston J. Miller and Inga Velde. 
Graphic design by Phil Swenson and typesetting by Barbara Cahlander and Terri Desormey, Graphic Services Department. 
Address requests for additional copies to the Research Department, 
Federal Reserve Bank, Minneapolis, Minnesota 55480. 
Articles may be reprinted if the source is credited and the Research 
Department is provided with copies of reprints. 
The views expressed herein are those of the authors and not necessarily those 
of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. Federal Reserve Bank of Minneapolis Quarterly Review/Spring 1984 
A Reply to Darby 
Preston J. Miller 
Monetary Adviser 
Research Department 
Federal Reserve Bank of Minneapolis 
Thomas J. Sargent 
Adviser 
Research Department 
Federal Reserve Bank of Minneapolis 
and Professor of Economics 
University of Minnesota 
A key result of Sargent and Wallace's " Some Unpleasant 
Monetarist Arithmetic" (1981) is that a permanently 
higher government deficit must eventually be accommo-
dated by increases in the monetary base. In "Some 
Pleasant Monetarist Arithmetic" (in this issue), Darby 
argues that this result does not currently apply to the U.S. 
economy because it depends on an assumption which is 
not supported by the data. In this reply to Darby we 
explain why we find his argument unconvincing and why 
we remain concerned about the longer-term monetary 
implications of high prospective federal budget deficits. 
Background 
Sargent and Wallace describe an economic model in 
which the real growth rate y and the real interest rate r are 
assumed to be constants for all time. It is also assumed that 
monetary and budget policies initially imply a steady-state 
equilibrium where the real interest rate exceeds the real 
growth rate (r > y). Given these assumptions, Sargent and 
Wallace show that any attempt to run a permanently 
higher deficit net-of-interest is simply not feasible unless 
the supply of base money is eventually increased. Without 
an eventual increase in the base-money supply, a perma-
nent increase in the deficit would cause the ratio of interest-
bearing government bonds to national income to diverge to 
infinity (see Figure 1), so at some point that ratio would 
outstrip the ratio of total wealth to income. That is, the 
government would eventually be unable to command the 
resources needed to pay its debt. 
Darby's model retains Sargent and Wallace's assump-
tions that the real interest rate and real growth rate are 
constants, but it departs from their model by assuming that 
the real growth rate is greater than the real interest rate 
(y>r). As a result, the consequences of a permanent 
increase in the deficit are very different from those claimed 
by Sargent and Wallace. Under Darby's assumption that 
y > rt a permanent increase in the deficit is feasible even 
with no change in the supply of base money from its initial 
path. Such a policy change simply leads to a higher 
eventual level for the ratio of interest-bearing government 
bonds to national income (see Figure 2). Although the 
level of the bond-to-income ratio may be higher, the 
government can always command the resources needed to 
pay its debt. 
Darby's Evidence 
Darby examines U.S. data to determine whether Sargent 
and Wallace's or his own model is more appropriate as a 
description of the current situation facing the United 
States. His evidence, simple and direct, rests on taking the 
formal Sargent-Wallace model and his own version of 
their model quite literally. Since both models assume a 
constant real interest rate and real growth rate, the sign of 
the difference between these constant rates is the critical 
distinction between the two models. Darby's procedure is 
simply to compute the historical average of the difference 
between the real growth rate and the real interest rate. He 
interprets the sign of this difference as settling the matter: 
if y > r so that y — r is positive, Darby's model applies; 
if r > y so that y — r is negative, Sargent-Wallace's applies. 
As long as the formal models explicitly analyzed by 
Sargent-Wallace and Darby are taken literally, Darby's 
test is the natural one to perform. 
21 Figures 1 and 2 
Financing a Higher Deficit Without 
Increasing the Supply of Base Money 
Effect on Federal Bonds Held by the Public 
(as a percentage of GNP) 
Figure 1 With Sargent-Wallace 
Assumption (r > y) 
Historical 
Bond/GNP Deficit Policy 
% cf ^ 0 
A 
Higher Deficit Policy 
d + Ad Beginning 
at Time tn 
Time t 
Figure2 With Darby Assumption (y>r) 
H istorical 
Bond/GNP Deficit Policy 
% d^O 
Higher Deficit Policy 
d + Ad Beginning 
at Time t0 
Time t 
Is Darby's Evidence Sufficient? 
Our own interpretation of Darby's evidence, however, is 
conditioned by our preference not to take literally Sargent 
and Wallace's assumption that the real interest rate is 
constant and, therefore, independent of budget policy. 
Instead, Sargent and Wallace can be regarded as using 
their model to approximate a more complicated one in 
which the real interest rate is itself a function of the budget 
and monetary policies in place.
1 Their intent was to point 
out the choices facing monetary policymakers whenever 
budget policies cause the real interest rate to exceed the 
real growth rate. Such was the situation when Sargent and 
Wallace wrote their paper in the first half of 1981, a time 
when the real interest rate exceeded the real growth rate 
and when monetary and budget policymakers were an-
nouncing moves that seemed to imply that the real interest 
rate would continue to exceed the real growth rate for a 
sustained period. 
In models more general than the one analyzed by 
Sargent and Wallace, the real interest rate on government 
bonds is not a constant but is partly determined as a rising 
function of the ratio of interest-bearing government bonds 
to base money. In such models, the difference between the 
real growth rate and real interest rate depends on the 
monetary and budget policies in place. From the perspec-
tive of these more general models, Darby's empirical 
evidence does not resolve the matter of the relevance of 
Sargent and Wallace's argument for the choices currently 
facing U.S. policymakers. 
Viewed through such models, evidence that the real 
interest rate has averaged less than the real growth rate 
over some historical period may be interpreted as partly 
reflecting the monetary and budget policies in place during 
that historical period. The average difference between the 
real growth rate and the real interest rate would not be 
expected to remain the same after a change in monetary 
and budget policies, such as a switch to a regime with 
higher average deficits.
2 Moreover, in the context of more 
general models in which the real interest rate depends on 
monetary and budget policies, much more is involved than 
simply comparing a real growth rate with a real interest 
rate to determine the feasibility of financing a permanent 
increase in the deficit without resorting to printing more 
base money. 
1 In their concluding remarks, Sargent and Wallace indicate that they had in 
mind a model where the real interest rate depends on policy. They state that their 
assumption of a constant real interest rate was made "because it seems to be 
maintained by many of those who argue for a low rate of growth of money no matter 
how big the current deficit is. If we were to replace that assumption, we would 
instead assume that the public's demand for government bonds is an increasing 
function of the real rate of return on bonds . . ." (p. 6). 
2 That the distribution of the sequence of equilibrium real interest rates de-
pends on budget and monetary policies is an application of the principles under-
lying the Lucas critique to the problem at hand. See Lucas 1976. 
22 Federal Reserve Bank of Minneapolis Quarterly Review/Spring 1984 
More Is Involved 
In order to show that more is involved, we make use of a 
simple form of the steady-state budget constraint:
3 
(1) d = (7r+y)m(r, if) + (y~r)b(rf 7r) 
where the following are ratios to gross national product 
(GNP), 
d = deficit net-of-interest 
m — demand for base money 
b — demand for government bonds 
and where the following are rates per unit of time, 
7r = inflation 
y = real growth 
r — real interest. 
We assume the markets for money and bonds clear (that 
is, the supply of money and bonds equals the demand), 
so that 
(2) m(r, tt) = MJYt and b(r, 7r) = BJYt 
where 
M= stock of base money 
B = stock of government bonds 
Y = nominal GNP 
and t subscripts denote a given time period. The steady-
state budget constraint states that the deficit net-of-interest 
d must be financed by seignorage on money (77+ y)m(r, TT) 
and by seignorage on bonds (y—r)b(r, TT). (Seignorage is 
the revenue the government earns from issuing money and 
bonds.) A deficit policy is feasible only when the govern-
ment can earn enough seignorage on its money and bond 
issue to finance its deficit net-of-interest. 
The steady-state budget constraint helps to make 
precise our analysis of the insufficiency of Darby's em-
pirical evidence and to suggest what empirical evidence 
is sufficient in the context of more general models. 
Suppose the economy is initially in equilibrium such that, 
given a deficit net-of-interest d and a path of money M over 
time, the money-to-GNP and bond-to-GNP ratios (MJYt 
and BJY) and the inflation rate, real growth rate, and real 
interest rate (TT, y, and r) are all constant over time. The 
issue is whether it is possible to raise the deficit net-of-
interest for all time to a new level d + Ad while keeping the 
path of money fixed at {M} and have the economy con-
verge to a new equilibrium where the ratios MJY[ and 
B't/Y't and the rates 7r', y', and r' once again are all constant 
over time. From the steady-state budget constraint this is 
possible only if 
(3) d + Ad = (n'+y'MrTT') + (y'-r')b(r
,t TT'). 
Equation (3) says that the deficit net-of-interest can be 
permanently raised to a higher level only if the total 
seignorage from money and bonds calculated at the new 
equilibrium rates (r\ y', and TT', consistent with a fixed 
path of money) is enough to cover the higher deficit. 
In general, to determine empirically if it is possible to 
finance a higher deficit net-of-interest d + Ad by bond 
issue alone, we would require estimates of the elasticities 
of money demand and bond demand (in order to determine 
how these demands change when the rates of real interest 
and inflation change to r and TT'). We would also need an 
estimate of how the economy's real growth rate varies with 
the rates of real interest and inflation. It would then be 
possible to determine quantitatively how the seignorage 
represented on the right side of the steady-state budget 
constraint changes when the deficit net-of-interest on the 
left side changes. 
Since the equilibrium rates depend on the policies in 
place, it is not sufficient to examine the average difference 
between the rates of real growth and real interest (y—r) 
over some extended historical period, as Darby does. If 
historically the government has been running a deficit d, 
then y and r will reflect the rates corresponding to that 
policy—the rates y and r in equation (1). In order to 
determine the feasibility of financing a higher deficit 
d + Ad by bond issue alone, however, estimates of y' and 
r' are required—as shown in equation (3). Given a fixed 
path for the monetary base, a proposed policy of running a 
persistently higher deficit net-of-interest than has been 
observed historically would be expected to raise the real 
interest rate relative to the real growth rate, thereby shifting 
their relationship toward that supposed by Sargent and 
Wallace. 
Another reason that simply looking at the difference 
between y and r does not suffice is because the change 
3The constraint is derived in Miller 1983, pp. 12-13. Its notation has been 
altered slightly here to conform with Darby's. 
23 Figures 3 and 4 
Two Cases Where Darby's Evidence is Insufficient 
When r Rises With the Ratio of Bonds to Money 
Figure 3 Case One: r>r*  Figure 4 Case Two: r <r* 
Bond 
Seignorage 
A 
Maximum Level 
Initial Level for d 
New Level for 
d+Ad 
Decrease 
in Bond Seignorage as d 
is Increased to d+Ad 
Bond 
- Seignorage 
(y~r)b(r,n) 
y Real Interest 
Rater 
Maximum Level 
Initial Level for d 
in seignorage caused by a change in deficit policy depends 
on other factors—such as how the inflation rate changes 
and how the new rates of interest and inflation affect the 
demands for money and bonds (see equation [3]). For 
example, an increase in the real interest rate could 
generally be expected to cause people to shift their asset 
holdings from money to bonds, thereby lowering the 
demand for money while raising the demand for bonds. 
A Model That Shows Darby's Evidence 
Isn't Sufficient 
In order to illustrate that Darby's evidence is not sufficent, 
we consider a simple monetarist model, which is taken 
from our more general class where the real interest rate 
rises with the ratio of bonds to money. This model (see 
Miller 1982) has the following features: 
a. The real growth rate y is fixed. 
b. The inflation rate tt is determined by the growth 
in the stock of base money.
4 
c. The real interest rate r is inversely related to 
the stock of private capital. 
From the steady-state budget constraint, the change in 
seignorage as the deficit d is increased to the new level 
d + Ad while the path of money is held constant is 
given by 
(4) A seignorage = tBAb + AtBb 
where 
tB — y — r, the rate of seignorage on bonds 
AtB = r — r\ the change in the rate of sei-
gnorage on bonds 
Ab = b(r7r) — b(r, 7r), the change in the 
demand for bonds. 
Such a change in policy is feasible only if the seignorage 
increases by the same amount as the deficit. 
In this model such a change in deficit policy has both a 
quantity and a price effect on bond seignorage (tBAb and 
AtBb, respectively). When y > r, the quantity effect is 
positive as the real demand for bonds increases at a 
positive seignorage rate (y—r). The price effect, however, 
4 We assume a simple quantity demand function for money so that m j
 =0. 
Bond 
<— Seignorage 
(y-r)b(r.n) 
Required Increase 
in Bond Seignorage as d 
f is Increased by Ad>Ad* 
Maximum Increase 
in Bond Seignorage as d 
is Increased by Ad* 
y Real Interest 
Rater 
Bond 
Seignorage 
A 
Required New Level 
for Ad > Ad* Federal Reserve Bank of Minneapolis Quarterly Review/Spring 1984 
is negative as additional government debt drives out 
private capital and raises the real interest rate at which the 
government must borrow. In this model there is a maxi-
mum amount of bond seignorage that can be generated. 
For the additional deficit to be financed by bond issue 
alone, it is necessary that the seignorage from bonds 
initially be less than that maximum. Because the quantity 
and price effects are of opposite sign, seignorage is 
maximized at some real interest rate r* < y, as shown in 
Figures 3 and 4. 
Two Cases 
This simple model indicates two cases where we could 
observe historically that y > r and where it still would not 
be possible to finance a permanent increase in the deficit 
net-of-interest by bond issue alone. The first case is when 
the real interest rate is greater than or equal to the real 
interest rate at which seignorage from bonds is maximized: 
r > r*. In this case, any further increase in bond issue will 
cause the seignorage from bonds to decrease (see Figure 
3). So even though y > rf it would not be possible to 
finance any increase in deficits by bond issue alone. The 
second case is when the real interest rate is less than the 
rate at which seignorage from bonds is maximized, r < r*, 
but the increase in the deficit is too large. Since there is a 
finite maximum to bond seignorage, it is always possible 
that the increase in the deficit will exceed the maximum 
increase in bond seignorage (see Figure 4). 
Which Arithmetic Best Applies Currently? 
Current observations of the relationship between the real 
interest rate and real growth rate seem consistent with the 
implications of our simple model and the more general 
models of which ours is an example. It appears that since 
1980, the real interest rate has tended to rise relative to the 
real growth rate as it became increasingly evident that the 
government had embarked on a historically unprece-
dented policy of large and persistent deficits net-of-
interest. With nominal interest rates currently far above 
the inflation rates predicted for and realized in 1984, the 
real interest rate seems to be high and the difference 
between the real growth rate and real interest rate (y—r) 
seems to be negative. These observations favor Sargent 
and Wallace's position. 
Perhaps more direct evidence favoring Sargent and 
Wallace's position is contained in recent Congressional 
Budget Office (CBO) projections of a steady rise in the 
government's bond-to-GNP ratio (see Figure 5 and 
compare with Figures 1 and 2). The CBO projects this 
Figure 5 
CBO Projections of the Bond-to-income Ratio 
Federal Bonds Held by the Public 
(as a percentage of G N P)* 
Bonds/GNP Bonds/GNP 
'Plotted for the end of fiscal years 
Source: CBO (U.S. Congress 1984, p.10) 
ratio assuming that the budget policies in place at the 
beginning of 1984 are maintained and that monetary 
policy does not accommodate the deficits (U.S. Congress 
1984). Even under its assumption of modestly declining 
real interest rates (which already seems overly optimistic), 
the CBO's projected bond-to-GNP ratio rises steadily. 
Because of the apparent error in the CBO's assumption 
for the real interest rate, the bond-to-GNP ratio would 
still be expected to rise for real growth rates higher than 
the roughly 3.5 percent assumed by the CBO (U.S. Con-
gress 1984, p. 12). Thus, even under more optimistic as-
sumptions about real growth, current deficit policy would 
still not be sustainable without future monetary accom-
modation. 
25 References 
Lucas, Robert E. 1976. Econometric policy evaluation: A critique. In The Phillips 
curve and labor markets, ed. K. Brunner and A. H. Melzer. Carnegie-
Rochester Conference Series on Public Policy 1: 19-46. Amsterdam: 
North-Holland. 
Miller, Preston J. 1982. A monetarist approach to federal budget control. 
Research Department Working Paper 210. Federal Reserve Bank of 
Minneapolis. 
. 1983. Budget deficit mythology. Federal Reserve Bank of 
Minneapolis Quarterly Review 7 (Fall): 1-13. 
Sargent, Thomas J., and Wallace, Neil. 1981. Some unpleasant monetarist 
arithmetic. Federal Reserve Bank of Minneapolis Quarterly Review 5 
(Fall): 1-17. 
U.S. Congress. 1984. Congressional Budget Office. Baseline budget projections 
for fiscal years 1985-1989. A report to the Senate and House committees 
on the budget—Part II. Washington, D.C.: U.S. Government Printing 
Office. 

A monetarist approach to federal budget control. Research Department Working Paper 210. Federal Reserve Bank of Minneapolis.

Budget deficit mythology. Federal Reserve Bank of

Congressional Budget Office. Baseline budget projections for fiscal years 1985-1989. A report to the Senate and House committees on the budget—Part II.

Econometric policy evaluation: A critique.

Some unpleasant monetarist arithmetic.

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