System dynamics of interest rate effects on aggregate demand

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SYSTEM DYNAMICS OF INTEREST RATE EFFECTS ON AGGREGATE DEMAND
By: Linwood Tauheed and L. Randall Wray
Introduction
‘Perverse’ Interest Rate Effects on Aggregate Demand?
Heterodox economics has always been skeptical of the Fed’s ability to “fine-tune” the
economy, in spite of the long-running Monetarist claims about the efficacy of monetary policy
(even if orthodox wisdom used to disdain discretion). The canonization of Chairman Greenspan
over the past decade and a half has eliminated most orthodox squeamishness about a
discretionary Fed, while currently fashionable theory based on the “new monetary consensus”
has pushed monetary policy front and center. As Galbraith argues, lack of empirical support for
such beliefs has not dampened enthusiasm. Like Galbraith, the followers of Keynes have always
insisted that “[b]usiness firms borrow when they can make money and not because interest rates
are low”. (Galbraith 2004, p. 45) Even orthodox estimates of the interest rate elasticity of
investment are so low that the typical rate adjustments used by the Fed cannot have much effect.
Conventional belief can still point to interest rate effects on consumption, with two main
channels. Consumer durables consumption, and increasingly even consumption of services and
nondurables rely on credit and, thus, might be interest-sensitive. Second, falling mortgage rates
lead to refinancing, freeing up disposable income for additional consumption. Ultimately,
however, whether falling interest rates might stimulate consumption must depend on different
marginal propensities to consume between creditors and debtors. In reality, many consumers are
simultaneously debtors and creditors, making analysis difficult because a reduction of rates
lowers both debt payments and interest income. If we can assume that these do not have
asymmetric effects (a highly implausible assumption), we can focus only on net debtors and

creditors. The conventional wisdom has always been that net creditors have lower MPCs than do
net debtors, so we can assume that lower rates stimulate consumption by redistributing after-
interest income to debtors. Still, the consumer lives in the same business climate as firms, and if
the central bank lowers rates in recession, the beneficial impacts can be overwhelmed by
employment and wage and profit income effects. Further, as society ages and net financial
wealth becomes increasingly concentrated in the hands of the elderly whose consumption is
largely financed out of interest income, it becomes less reasonable to assume a low MPC for net
creditors. Perhaps the MPC of creditors is not much different from that of debtors—which makes
the impact of rising rates on consumption all the more ambiguous. This does not mean that
lowering rates in recession (or raising them in expansion) is bad policy, but it could account for
the observation made by Galbraith and others before him that monetary policy is ineffectual.
Most conventional and even unconventional explorations of interest rate effects have
focused on demand-side effects. It is also possible that raising interest rates can have a perverse
impact on prices coming from the supply side. Interest is a business cost much like energy costs
that will be passed along if competitive pressures permit. The impact on aggregate demand
arising from this might be minimal or ambiguous, but it is conceivable that tight monetary policy
might add to cost-push inflationary pressures while easy policy would reduce them. For
monetary policy to work in the conventional manner on constraining inflation, interest rate
effects on aggregate demand would have to dominate such supply-side impacts. This seems
ambiguous at best, given the uncertain impacts of interest rates on demand.
What has been largely ignored is the impact that interest rate changes have on
government spending, and hence on aggregate demand through the “multiplier” channel. In a
somewhat different context to the issue to be pursued here, some Post Keynesian authors have

recognized that rising rates tend to increase the size of government budget deficits. Indeed, in the
late 1980s, a number of countries with large budget deficits would have had balanced budgets if
not for interest payments on outstanding debt—Italy was a prime example, with government
interest payments amounting to more than 10% of GDP (Brazil was another). It was recognized
at the time by a few analysts that lowering rates was probably the only way to reduce the
government’s deficit—a path later successfully followed. However, conventional wisdom holds
that high deficits cause high interest rates, hence, government can quickly find itself in an
“unsustainable” situation (rising rates increase deficits that cause markets to raise rates even
higher) that can be remedied only through “austerity”—raising taxes and cutting non-interest
spending. In truth, for countries on a floating exchange rate, the overnight interbank lending rate
(fed funds rate in the US) is set by the central bank, and government “borrowing” rates are
determined relative to this by arbitragers mostly in anticipation of future overnight rate targets.
The heterodox literature on rate-setting by the central bank is large and the arguments need not
be repeated here. What is important is to recognize the government’s ability to reduce its deficit
spending by lowering interest rates.
What we want to investigate is the nearly ignored possibility that lowering/raising interest
rates will lower/raise aggregate demand in a manner opposite to normal expectations. This would
occur if lowering rates lowered government deficits by reducing interest payments—which are
essentially the same as any other transfer payments from government to the private sector.
Assume an economy in which private debt is small relative to GDP, and in which the interest
elasticity of private investment (and other private spending) is small. By contrast, government
debt is assumed to be large with holdings distributed across “widows and orphans” with high
spending propensities. Raising interest rates will have little direct effect on the private sector,

which carries a low debt load and whose spending is not interest sensitive, in any case. However,
rising interest rates increase government interest payments; to the extent that the debt is short-
term or at variable interest rates, the simulative impact on private sector incomes and spending is
hastened.
This is, of course, the most favorable case. However, on not implausible assumptions
about private and public debt ratios and interest rate elasticities, it is possible for the government
interest payment channel to overcome the negative impact that rising rates have on private
demand. We will proceed as follows. First, we will briefly introduce the methodology to be used,
System Dynamics modeling. We then set out the model and explain the variables and
parameters. For the first part of our analysis, we will use historical US data to set most
parameters. We will determine given those parameters (debt ratios and interest elasticities), at
what level of interest rates can we begin to obtain “perverse” results—where further increases
actually stimulate demand. For the final part of the analysis, we will determine values for
government debt ratios at which “perverse” results can be obtained, for different levels of
interest rates. Throughout we will assume uniform MPCs, and so leave for further research the
questions about distribution effects. Further, we will not include interest rate effects on consumer
borrowing—this is equivalent to assuming that consumer borrowing is not interest sensitive, or
that any interest rate effects on net (private) debtors is exactly offset by effects on net (private)
creditors. Finally, as we will briefly mention below, further research will be needed to explore
implications arising from international debtor/creditor relations (Americans hold foreign
liabilities, while foreigners hold American liabilities—both private and public—and even if
domestic MPCs are the same across debtors and creditors, the MPCs of foreigner creditors and
debtors could be different).

System Dynamics Modeling
System Dynamics is a numerical method for creating dynamic models of systems. The
origin of System Dynamics (SD) modeling is to be found in its development by MIT’s Jay
Wright Forrester in the 1950’s initially as a method for designing and understanding electronic
feedback control systems known as servomechanisms. Its later expansion from engineering into
social applications came with Forrester’s writing of Industrial Dynamics (Forrester, 1961), and
later Urban Dynamics (Forrester, 1969) where SD was applied to city planning.1 As SD
modeling expanded its scope from engineering applications to business management to social
systems, its underlying method remained consistent - “the application of feedback control
systems principles and techniques to managerial, organizational, and socioeconomic problems.”
(Roberts, 1978, 3).2
The use of SD allows a form of experimentation with the theoretical model under
analysis, by allowing the observance of the effects of manipulation of relevant variables. This
facility becomes most useful when dealing with complex models of social systems with many
feedbacks loops that cannot be solved analytically by a closed system of equations. (see
Forrester, 1995).
In this study, interest rate effects on Aggregate Demand are observed in both ‘static’ and
dynamic modes. A detailed discussion of SD is beyond the scope of this paper. For additional
information on SD modeling, particularly in a Heterodox context, refer to (Radzicki, 1988, ,
1990).

Method
The Model
The model used in this study, diagrammed in Figure 1, is the familiar Aggregate
Demand/GDP model. The definition and description of model variables are in Table 1.
Variables are set to 2000 National Income and Product Account (NIPA) values as listed in the
Statistical Abstract of the United States (2001).
Figure 1 - Aggregate Demand/GDP Model
AD
NI
Ig
C
mpc G
rIg
DR
DS
Dp
Tx
DI
alpha
epsilon
Xn
GDP
XP
Sc_ect
Depr_etc
Czero
GDPinit
dGDP
XPg
rAdjust rControl
STOP
rDS
0.39
0.09
1.74
-0.37
1.83
9.96
6.76
6.99
3.92
9.96
0.26
8.00
1.29
0.00

Table 1 - Equations for Aggregate Demand/GDP Model (values in Trillions)
Variable Definition Description
AD C + G + Ig + Xn Aggregate Demand
alpha (

) 1.0 coefficient of Investment Function
C mpc * (DI) + Czero Consumption Function
Czero .473540 estimated Autonomous Consumption
given C and mpc
Depr_etc -.3706 + .3749 + 1.2571 +
.7696 - .0279 - 4.042
NI adjustments (Depr, Factor Income,
etc.
dGDP (AD - GDP) change in GDP per quarter (in static
Model dGDP = 0)
DI NI – Tx + XP – Sc_etc Disposable Income
Dp DR * GDP Public Debt
DR (3.4101 + .5114) / 9.9631 Debt Ratio (Public + FRB Debt) / GDP
DS MAX(Dp * rDS, .01 * Dp) Debt Service (max of Dp*rDS or l%*Dp)
epsilon (

) -.25 interest rate elasticity of investment
G 1.7437 Government Spending
GDP GDPinit (set to GDPinit at start of simulation)
GDPinit 9.9631 GDP initial value for simulation
Ig alpha * rIgepsilon Investment Function
mpc .90 assumed marginal propensity to consume
NI GDP - Depr_etc National Income
rDS rIg - rAdjust average Debt Service Rate (rounded)
rIg .09 average Prime Rate (rounded)
rAdjust .09 - (.2552 / 3.9215) =
0.024923
prime rate adjustment to Debt Service
rate
rControl IF((ABS(dGDP) < .01),
.01,0) / Timestep
increases rIg by l% when simulation
reaches temporary ‘‘equilibrium’’
Sc_etc .01990 Personal Income adjustment from NI
(includes corporate savings)
STOP STOPIF(r > .2) stops simulation when rIg is > 20%.
Tx .7056 + .286 + 1.2919 personal and corporate tax and Social
security contributions
Xn -.3707 Net Exports
XP XPg + DS Transfer Payments (includes Federal
Debt service)
XPg 1.036 Government Transfer Payments
Model Structure
In the model (see Figure 1), there are three feedback loops to note.

1. The positive loop from GDP to NI (national income) to DI (disposable income) to
C (consumption) to AD (aggregate demand) to dGDP (change of GDP) to GDP.
2. The positive loop from GDP to Dp (public debt) to DS (public debt service) to XP
(transfer payments) to DI…back to GDP.
3. The negative loop from dGDP to GDP back to dGDP.
The first two loops drive the increase/decrease in GDP, as producers, with some delay,
increase/decrease Aggregate Supply in response to the perceived increase/decrease in Aggregate
Demand (AD). The third loop prevents the positive loops from driving the system explosively
and brings GDP into ‘equilibrium’ with Aggregate Demand, again with some delay, by
decreasing the change in GDP (dGDP) as the AD-GDP gap decreases.
In addition, although not strictly loops, two feedback effects are the basic effects
analyzed in this study.
4. The positive Debt Service ‘loop’ from rDS (average interest rate on public debt)
to DS…interfacing with loop #2 back to GDP.
5. The negative Investment ‘loop’ from rIg (average prime rate) to Ig (investment) to
AD …back to GDP.
The Investment Function
A constant elasticity Investment Function is derived as follows:
1.



 )()( g
Irdr
g
dI
2. rdr
g
I
g
dI


3. cr
g
I )ln()ln(

4.

r
c
e
g
I
5. c
elet 

...

6.


r
g
I

A cursory review of the literature on interest rate elasticity found a fairly wide range of
estimates, with most below -0.25, although a few were substantially larger. We also found that
setting ε = -.25, the investment function approximates the 2000 value for Gross Private Domestic
Investment (Ig = 1,832.7) when α = 1 and rIg =.09 -- (Ig = 1.8257). Hence, we use -0.25 as the
interest rate elasticity of investment for this analysis. The graph of the function for interest rates
from 1% to 20% is shown in Figure 2.
Figure 2 – Constant Elasticity Investment Function
Investment Function
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
Interest Rate (rIg)
Ig
The Debt Service Function and the Debt Ratio
Government Debt Service (DS) is a linear function of the Prime Rate (rIg) and Public
Debt (Dp). In 2000 this debt was 3.9215 (3.4101 owed to the public and .5114 owed to the Fed).
Payment on this debt equaled .2552. This corresponds to an effective average interest rate of
6.5077%. This is, of course lower than the Prime Rate used in the Investment Function. The
difference is used as an adjustment (rAdjust) to the Prime Rate for computing rDS, the Debt

Service rate, and equals approximately 2.4923%. This value is used throughout as the spread
between rIg and the DS rate (rDS)3.
The Public Debt (Dp) to GDP Debt Ratio (DR) is calculated from 2000 values
(Dp=3.9215/GDP=9.9631) and equals .3936. This ratio is held constant for the first parts of this
study.
Static Analysis
The model is initialized in ‘equilibrium’ at 2000 values with GDP = 9.9631 and other
variables set accordingly. In order to obtain ‘static’ results from an essentially dynamic model,
the linkage between AD and GDP is broken. This linkage is modeled by the flow ‘dGDP’ that
functions to gradually increase or decrease the level of GDP if the gap between AD and GDP
(AD-GDP) is positive or negative respectively. To perform a ‘static’ analysis, dGDP is forced to
zero (0). Thus, GDP (as well as NI and Dp) will remain constant regardless of changes in AD
(see Figure 5).
Dynamic Analysis
For the dynamic analysis, the model is also initialized in ‘equilibrium’ at 2000 values
with GDP = 9.9631, other variables set accordingly. However, the linkage between AD and
GDP is fully active since dGDP is set to AD-GDP. The functioning of a ‘flow’ variable such as
dGDP in an SD model is that it does not respond immediately to its value, but over some
adjustment time-period. In this simulation, the dGDP adjustment time is one year, which is four
times the simulation time-step of one quarter. This causes a gradual increase or decrease in the
level of GDP as the gap between AD and GDP (AD-GDP) is positive or negative respectively.

Results
Static Analysis
Under the control of the variable rControl, the value of rIg is increased from 9% to 20%
and then stops. The results are shown in Table 2, Figure 3, and Figure 4.
As the value of rIg increases from 9% to 20%, Ig decreases while DS increases. Ig
decreases in accordance with it functional form. Since there is no change in Dp, DS increases
linearly as rDS (rIg – rAdjust) increases (see Figure 3 and Table 2).
AD changes value in response to the change in Ig (ΔIg) plus the change in DS (ΔDS),
both resulting from the change in rIg alone. The total effect on ΔAD, with mpc = .9, is ΔIg +
.9*ΔDS, but since ΔDS is constant all change in ΔAD is attributable to ΔIg. As can be seen in
Figure 4 and Table 2 the combined effect on ΔAD is initially negative but becomes positive
between an rIg of 12% and 13% (remember that this represents the Investment Function interest
rate, the Debt Service function interest rate is lower by approximately 2.4923 percent). The
effect on AD can be seen in Figure 5 as the slope of the AD curve become positive at this point.
Table 2 - Static Mode Interest Rate Effects on Key Variables4
AD (s) GDP rIg Ig DS (s) ΔIg ΔDS ΔAD=ΔIg+.9*ΔDS Dp
9.9631 9.9631 0.09 1.8257 0.2552 na na na 3.9215
9.9509 9.9631 0.10 1.7783 0.2944 -0.0475 0.0392 -0.0122 3.9215
9.9444 9.9631 0.11 1.7364 0.3336 -0.0419 0.0392 -0.0066 3.9215
9.9423 9.9631 0.12 1.6990 0.3728 -0.0374 0.0392 -0.0021 3.9215
9.9439 9.9631 0.13 1.6654 0.4121 -0.0337 0.0392 0.0016 3.9215
9.9486 9.9631 0.14 1.6348 0.4513 -0.0306 0.0392 0.0047 3.9215
9.9560 9.9631 0.15 1.6069 0.4905 -0.0280 0.0392 0.0073 3.9215
9.9656 9.9631 0. 16 1.5811 0.5297 -0.0257 0.0392 0.0096 3.9215
9.9771 9.9631 0.17 1.5574 0.5689 -0.0238 0.0392 0.0115 3.9215
9.9903 9.9631 0.18 1.5353 0.6081 -0.0221 0.0392 0.0132 3.9215
10.0049 9.9631 0.19 1.5146 0.6474 -0.0206 0.0392 0.0147 3.9215
10.0209 9.9631 0.20 1.4953 0.6866 -0.0193 0.0392 0.0160 3.9215

Figure 3 – Static vs. Dynamic Mode Debt Service Function
Static vs . Dynamic Debt Service by Interest Rate
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
Interest Rate (rIg)
DS ( d)
DS ( s )
Figure 4 - Static Mode Interest Rate Effects on Investment and Debt Service and their Combined
Effect on Aggregate Demand
Static Change in Aggregate Demand/Investment/Debt Service by Interest Rate
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
Interest Rate (rIg)
ΔIg
ΔDS
ΔAD=ΔIg+.9*ΔDS

Figure 5 - Static Mode Interest Rate Effects on Aggregate Demand and GDP
Static Aggr egate Demand/GDP by Interest Rate
9.75
9.88
10.00
10.13
10.25
0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
Interest Rate (rIg)
AD (s)
GDP
Dynamic Analysis
As in the ‘static’ case, under the control of the variable rControl the value of rIg is
increased from 9% to 20% and then stops. The results are shown in Table 3, Figure 6, and
Figure 7. The results are those of the relevant variables at the time-step in which the simulation
reaches a temporary ‘equilibrium’ and before rIg is increased by 1%.5
As the value of rIg increases from 9% to 20%, Ig decreases while DS increases. Again,
this results in a decrease in Ig in accordance with its functional form, and an increase in DS.
However, since Dp increases/decreases as GDP increases/decreases the DS function no longer
yields linear results with respect to rIg (see Figure 3 and Table 3).
The AD to GDP interface is fully active through dGDP, and AD now changes value not
only in response to the change in Ig (ΔIg), which as in the ‘static’ mode depends only on the
change in rIg, but also in response to the change in DS (ΔDS) which varies as both Dp and rIg
vary6. The total effect on AD from ΔIg and ΔDS is as before, ΔIg + .9*ΔDS. However, there
is also an effect on DI (and therefore C and therefore AD) through NI that is an effect of the

change in GDP. GDP changes nonlinearly in the direction of AD as seen in Figure 7. Thus, the
total effect on AD is the sum of all changes in GDP, Ig and DS as the simulation moves from
‘equilibrium’ to ‘equilibrium’ and equals ∑(ΔIg + .9*(Δ_GDP + ΔDS)) (see the note for Table
3).
As can be seen in Figure 6 (right axis) the combined effect on ΔAD is initially negative
but becomes positive between an rIg of 13% and 14%. This is later than in the ‘static’ case and
will be explained below. The effect on AD can be seen in Figure 7 as the slope of the AD curve
become positive at this point with AD overtaking GDP between an rIg of 14% and 15%.
The slope of GDP increases dramatically around an rIg of 16% (Δ_GDP becomes
positive between an rIg of 15% and 16%, Table 3) causing accelerated feedback effects through
DS by way of Dp, and AD by way of DI - (Figure 6).
Table 3 - Dynamic Mode Interest Rate Effects on Key Variable7
Qtrs AD (d) GDP rIg ∑Δ_GDP DS (d) ∑ΔIg ∑ΔDS
ΔAD=∑(ΔIg+
.9*(Δ_GDP+ΔDS)
)
0 9.9631 9.9631 0.09 0.00000 0.2552 0.0000 0.0000 0.0000
12 9.9226 9.9325 0.10 -0.03055 0.2935 -0.0475 0.0383 -0.0405
42 9.8261 9.8359 0.11 -0.09661 0.3294 -0.0419 0.0359 -0.0965
55 9.7900 9.8000 0.12 -0.03596 0.3667 -0.0374 0.0374 -0.0361
56 9.7887 9.7975 0.13 -0.00250 0.4052 -0.0337 0.0385 -0.0013
57 9.7908 9.7953 0.14 -0.00220 0.4437 -0.0306 0.0385 0.0021
58 9.7964 9.7942 0.15 -0.00113 0.4822 -0.0280 0.0385 0.0057
68 9.8287 9.8187 0.16 0.02457 0.5220 -0.0257 0.0399 0.0323
130 10.0567 10.0468 0.17 0.22808 0.5737 -0.0238 0.0517 0.2280
206 10.3549 10.3450 0.18 0.29818 0.6314 -0.0221 0.0577 0.2982
299 10.7420 10.7321 0.19 0.38715 0.6973 -0.0206 0.0659 0.3871
409 11.2329 11.2229 0.20 0.49073 0.7734 -0.0193 0.0761 0.4908
note: Δ_GDP is the actual change in GDP per time-step rather than the AD/GDP
gap (dGDP=AD-GDP) which is roughly four times Δ_GDP per time-step.

Figure 6 - Dynamic Mode Interest Rate Effects on Investment, Debt Service, and GDP and their
Combined Effect on Aggregate Demand
Dynamic Change in Aggregate Demand/Investment/Debt Service by Interest Rate
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
Interest Rate (rIg)
-0.20
-0.10
0.00
0.10
0.20
0.30
0.40
0.50
0.60
∑ΔIg (left axis)
∑Δ DS (left axis)
∑Δ _GDP
ΔAD=∑(ΔIg+
.9*(Δ_GDP+ΔDS))
Figure 7 - Dynamic Mode Interest Rate Effects on Aggregate Demand and GDP
Dynamic Aggregate Demand/GDP by Interest Rate
9.50
10.00
10.50
11.00
11.50
0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
Interest Rate (rIg)
AD (d)
GDP

Static vs. Dynamic Comparison
As seen in Figure 8 the feedback effects of GDP on AD causes dynamic AD to initially
drop below static AD, but eventually to overcome static AD at an rIg between 16% and 17%.
The rates of change are dramatically different before an rIg of 12% and after an rIg of 16%.
This outcome depends on the functional form of the Investment Function (see Figure 2).
Before an rIg of 12% the negative change in Ig (ΔIg) dominates the positive change in DS (ΔDS)
becoming approximately equal at an rIg of 12%. After an rIg of 16% the combined positive
effects of ΔDS and Δ_GDP dominates the increasing smaller negative ΔIg effect8.
The shift to positive ΔAD occurs later for the dynamic case than for the static case
(between 13%-14% rather than between 12%-13% see Figure 9) because of the ‘inertia’ caused
by the lower GDP level which takes longer for dGDP (AD-GDP) to overcome.
Figure 8- Comparison of Static vs. Dynamic Aggregate Demand.
Static vs . Dynamic Aggregate Demand by Interest Rate
9.75
9.85
9.95
10.05
10.15
10.25
0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
Interest Rate (rIg)
AD (d)
AD (s)

Figure 9 - Static vs. Dynamic Change in Aggregate Demand by Interest Rate
Static vs. Dynamic Change in A ggregate Demand by Interest Rate
-0.04
-0.04
-0.03
-0.03
-0.02
-0.02
-0.01
-0.01
0.00
0.01
0.01
0.12 0.13 0.14
Interest Rate (rIg)
ΔAD=ΔIg+.9*ΔDS
ΔAD=∑(ΔIg+
.9*(Δ_GDP+ΔDS) )
Static Model with Variable Debt Ratio
In this section we make the government’s debt ratio variable to find the “knife edge” or
“tipping point” at which an increase of the interest rate will actually stimulate the economy. We
derive the Debt Ratio (DR) where interest rate change effect on AD from change in Ig and
change in DS equals zero, i.e. ΔIg (r) + ΔDS(r) = ΔAD(r) = 0.
Let:
1.


r
g
I
2. 1... 

let
3.


rrr
g
I )(
4. rDRGDPmpcDS ***
5. rDRGDPmpcrrrDRGDPmpcDS 




 ***])[(***
6. 0

 ADDS
g
Ilet
7. rDRGDPmpcrrr  ***])[(


8. rGDPmpc rrr
DR 



**
])[(


We obtain the following results for DR with mpc = .9, ε = -.25, Δr = .01 and various
values for GDP, for Investment interest rates from 3% to 20%. (For the change in Prime Rate the
change in the Debt Service Interest Rate is the same, i.e.
)(])[(])[( rAdjustrrrAdjustrrrrr



 ).
Table 4 - ‘Equilibrium” Debt Ratios for various GDP/Interest Rate Combinations
Investment
Interest
Rate Investment
Change In
Investment
'Equilibrium'
Debt Ratio at
GDP1
'Equilibrium'
Debt Ratio at
GDP2
'Equilibrium'
Debt Ratio at
GDP3
rIg=r
ε
ΔI
g
=
(r+Δr)ε - rε
DR1= - ΔIg/
(mpc*GDP1*Δr)
DR2= - ΔIg/
(mpc*GDP2*Δr)
DR3= - ΔIg/
(mpc*GDP3*Δr)
0.03 2.4028 0.1667 3.7191 1.8596 0.9298
0.04 2.2361 0.1213 2.7061 1.3531 0.6765
0.05 2.1147 0.0942 2.1017 1.0508 0.5254
0.06 2.0205 0.0764 1.7037 0.8519 0.4259
0.07 1.9441 0.0638 1.4237 0.7118 0.3559
0.08 1.8803 0.0546 1.2169 0.6085 0.3042
0.09 1.8257 0.0475 1.0586 0.5293 0.2647
0.10 1.7783 0.0419 0.9339 0.4670 0.2335
0.11 1.7364 0.0374 0.8334 0.4167 0.2083
0.12 1.6990 0.0337 0.7508 0.3754 0.1877
0.13 1.6654 0.0306 0.6819 0.3409 0.1705
0.14 1.6348 0.0280 0.6235 0.3118 0.1559
0.15 1.6069 0.0257 0.5736 0.2868 0.1434
0.16 1.5811 0.0238 0.5305 0.2652 0.1326
0.17 1.5574 0.0221 0.4928 0.2464 0.1232
0.18 1.5353 0.0206 0.4597 0.2299 0.1149
0.19 1.5146 0.0193 0.4305 0.2152 0.1076
0.20 1.4953 0.0181 0.4044 0.2022 0.1011
As can be seen in Table 4 and Figure 10 below, for plausible “real world” prime rates in
the range of 10% to 12%, and for a “real world” GDP of just under $10 trillion, government debt
ratios well under 50% of GDP are sufficient to obtain perverse monetary policy results.

Figure 10 - ‘Equilibrium’ Debt Ratios for various GDP/Interest Rate Combinations
'Equilibrium' Debt Ratio at GDP1-2-3
by Interest Rate
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
Investment Interest Rate
DR1=-ΔIg/
(mpc*GDP1*Δr)
DR2=-ΔIg/
(mpc*GDP2*Δr)
DR3=-ΔIg/
(mpc*GDP3*Δr)
Static Model with Variable Level of Public Debt
It is also useful to find the level of outstanding government debt at which the debt service
impact of raising interest rates overwhelms the negative impact on aggregate demand resulting
from effects of interest rates on investment. We might call this the “equilibrium” public debt.
Here we derive Public Debt where interest rate change effect on AD from change in Ig and
change in DS equals zero, i.e. dIg(r) + dDS(r) = dAD(r) = 0
1.


r
g
I
2. 1... 

let
3. drr
g
dI 1
*



4. )(** rAdjustrDpmpcDS 
5. drDpmpcdDS **
6. 0

 dADdDS
g
dIlet
7. Dpmpcr *]
1
*[ 





8. mpc
r
Dp ]
1
*[ 





Results for ‘Equilibrium’ Public Debt (Dp) for mpc = .9, ε = -.25, and Investment
interest rates of 3% to 20% are shown in Table 5 and Figure 11. With an interest rate of 10%, if
the public debt is $4.94 trillion—or about 50% of GDP, interest rate hikes can stimulate
aggregate demand.
Table 5 - ‘Equilibrium’ Public Debt
Investment
Interest Rate
Change In
Investment
'Equilibrium'
Public Debt
r
dIg=
ε*r(ε - 1)
Dp= -
dIg/mpc
0.03 20.0234 22.2483
0.04 13.9754 15.5282
0.05 10.5737 11.7486
0.06 8.4188 9.3542
0.07 6.9433 7.7148
0.08 5.8759 6.5288
0.09 5.0715 5.6350
0.10 4.4457 4.9397
0.11 3.9464 4.3849
0.12 3.5397 3.9330
0.13 3.2027 3.5585
0.14 2.9193 3.2437
0.15 2.6781 2.9757
0.16 2.4705 2.7450
0.17 2.2902 2.5447
0.18 2.1323 2.3692
0.19 1.9930 2.2144
0.20 1.8692 2.0769
Figure 11 - ‘Equilibrium’ Public Debt by Interest
Rate
'Equilibrium' Public Debt by Interest Rate
0.00
5.00
10.00
15.00
20.00
25.00
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
0.19
Investm ent Interest Rate
Dp=-dIg/mpc
Admittedly, this is a simple model that includes only one negative effect—the interest
rate elasticity of investment. If other private spending is also interest-rate sensitive, and if the
usual assumption that net debtors have higher spending propensities holds, then higher debt
ratios will be required to obtain perverse results. However, note that Italy had a government debt
ratio above 100% and interest rates on government debt above 10% in the late 1980s. Turkey

flirts with interest rates of 28% and government deficits above 25% of GDP—perhaps
sufficiently high that lowering rates would actually cool the economy. Finally, Japan has the
highest deficits and debt ratio in the developed world, with high net private saving ratios and
substantial private sector financial wealth, all in the context of zero interest rates. It is entirely
possible that raising rates in Japan would actually stimulate the economy by increasing private
sector interest income.
Conclusion
This exercise has demonstrated that under not-too-implausible conditions, raising interest
rates could actually stimulate aggregate demand through debt service payments made by
government on its outstanding debt. This is more likely if private sector indebtedness is small, if
private spending is not interest-rate elastic, if interest rates are high, and if government debt is
large (above 50%) relative to GDP. In addition, the reset period of the government’s debt affects
the rapidity with which interest rate changes are transmitted to spending. Our analysis used fairly
simple models and hence represents a first attempt at modeling these impacts.
In future work we need to take account of the distribution of ownership of the public
debt—domestically and internationally. Foreign holdings of sovereign debt presumably would
diminish the debt service effects we have explored; further, institutional ownership of the debt
probably reduces the spending propensities of received government interest payments. Most
government debt is not directly held by “widows and orphans”, but rather is held indirectly
through financial institutions, pension funds, and so on. The “pass through” effects of increased
government interest payments on household income and thus on spending are not immediately
clear in the case of institutionalized holdings. Still, it is plausible that part of the reason that

empirical evidence does not support the conventional wisdom about monetary policy probably
can be attributed to the debt service effects analyzed here.
References
Forrester, J. W. (1961). Industrial dynamics. Cambridge, Mass.: M.I.T. Press.
Forrester, J. W. (1969). Urban dynamics. Cambridge, Mass.: M.I.T. Press.
Forrester, J. W. (1995). Counterintuitive behavior of social systems. Retrieved May 27, 2003,
from http://sysdyn.clexchange.org/sdep/Roadmaps/RM1/D-4468-2.pdf
Galbraith, J. K. (2004). The economics of innocent fraud: truth for our time. Boston: Houghton
Mifflin.
Radzicki, M. J. (1988). Institutional Dynamics: An extension of the Institutionalist approach to
socioeconomic analysis. Journal of Economic Issues, 22(3), 633.
Radzicki, M. J. (1990). Institutional Dynamics, deterministic chaos, and self-organizing systems.
Journal of Economic Issues, 24(1), 57.
Richardson, G. P. (1991). Feedback thought in social science and systems theory. Philadelphia:
University of Pennsylvania Press.
Roberts, E. B. (Ed.). (1978). Managerial applications of system dynamics. Cambridge: MIT
Press.
Simon, H. A. (1957). Models of man: social and rational; mathematical essays on rational
human behavior in a social setting. New York: Wiley.
United States. Bureau of the Census. (2001). Statistical abstract of the United States (121st ed.).
Washington, DC.
Wiener, N. (1948). Cybernetics. New York: J. Wiley.

Endnotes
1 The term System Dynamics pertains to the practice of Industrial Dynamics applied to social
systems modeling.
2 It should be noted that the application of feedback theory to problems of understanding
organizational behavior had been previously explored by Simon (1957) and to cybernetic models
of human behavior by Wiener (1948). See Richardson (1991) for additional examples.
3 To avoid negative DS rates, in the model the DS rate floor is set at 1%. In this study since rIg
never goes below 9%, rDS never goes below 6.5077%.
4 Bold type in columns AD (s) and GDP identifies the point of return to (near) initial values.
Bold type in the ΔAD column identifies the point where ΔAD changes from negative (AD
declining) to positive.
5 The data points generated from the simulation are variably spaced in time (see Figure iv1 and
Table 3 variable ‘Qtrs’). Each data point represents the ‘equilibrium’ level of AD/GDP
immediately before the next change in rIg. As can be seen this occurs at irregular intervals
becoming smaller from Quarter #0 (start of simulated time) to Quarter #55, and longer after
Quarter #58. This has the effect of distorting the slope of the variables relative to the changes in
interest rate. This should be kept in mind when interpreting the results temporally.
Figure iv1 - Dynamic Aggregate Demand/GDP by Time in Quarters
Dynamic Aggregate Demand/GDP by Time in Quarters
9.50
10.00
10.50
11.00
11.50
0 50 100 150 200 250 300 350 400 450
Time in Quarter s
AD (d)
GDP
6 The total change on ΔDS as Dp and rIg vary is the sum of the partial changes in both Dp and
rDS, i.e. DprDSDprDSdDS **  .
7 Bold type in columns ∑ΔIg and ∑ΔDS identifies the point where the cumulative negative effect
on ΔIg (because of increases in rIg) equals the cumulative positive effect on ΔDS. Bold type in
the ΔAD column identifies the point where ΔAD changes from negative (AD declining) to

positive. Bold type in the ∑Δ_GDP column identifies the point where ∑Δ_GDP changes from
negative (GDP declining) to positive.
8 In this model Ig(r) is independent of GDP and AD. A refinement might be to make Ig
dependent on expected or perceived AD, shifting Ig(r) as a function of dGDP.