Uncertainty and Economic growth session at 12th International Conference

1. Macrodynamics of Debt-financed Investment-led
Growth with Interest Rate Rules
Soumya Datta
Faculty of Economics,
South Asian University,
New Delhi, INDIA
The 12th International Post-Keynesian Conference
Kansas City, Missouri
September 24-28, 2014

2. Introduction: Objectives of Study
This study attempts to answer the following primary questions:
Can financial considerations provide endogenous bounds to an
otherwise unstable demand-constrained closed economic systems?
In other words, does the financial sector play a stabilizing role?
Can these considerations give rise to persistent growth cycles, or
cyclical patterns in the growth rates of macroeconomic variables?
Can these cycles break down to more complex dynamical
possibilities?
How effective is a monetary policy, in the form of interest rate rules,
in achieving its desired objectives?

3. A Preview of the Model
We explicitly model the possibilities of borrowers defaulting on
payment commitments. This encourages lenders to discriminate
between borrowers, leading to credit rationing and red-lining.
During the upward phase of business cycle, financial variables
deteriorate due to credit expansion (Fisher, Minsky). Two kinds of
credit expansion: credit deepening and credit widening.
We provide an alternative macroeconomic story to Minsky’s
arguments.
To keep dimension low, at the moment we do not include income
distribution considerations. Prices remain constant. Hence,
monetary policy (Taylor Rule) is suitably modified – with capacity
utilization as a proxy for inflation.

4. Basic Model
A simple continuous time model of closed economy with no
government.
Two social classes: workers earning wages (W) and capitalists
earning profits (P).
National income by income method: Y (t) = W (t) + P (t).
Workers do not save. Capitalists save a fraction sp of profits. Hence
consumption, C (t) = W (t) + (1 − sp) P (t).
Price is a fixed-up markup over wage costs of production. Hence
P (t) = Y (t), where is the share of profits in national income.
Aggregate demand consists of consumption and investment:
AD (t) = C (t) + I (t).
Investment is financed either internally out of retained earnings, or
externally out of debt or equity.

5. Basic Model (Cont’d)
Potential output, Y ⋆ (t) =

6. K (t) where

7. : fixed output capital
ratio given by existing technology. Availability of the capital is the
binding constraint on production.
Actual level of output: Y (t) = min [AD (t) , Y⋆ (t)]. For all
AD Y ⋆, aggregate demand acts as the main constraint on
production. In this case output is determined by aggregate demand.
Rate of capacity utilization, u (t) =
Y (t)
Y ⋆ (t)
2 ]0, 1[.
Rate of investment, g (t)
I (t)
K (t)

8. Goods Market Equilibrium Investment Function
Goods market equilibrium: Level of output measured by income
method equals aggregate demand, i.e.
W (t) + P (t) = C (t) + I (t)
) Y (t) =
1
sp
I (t)
and g (t) = sp

9. u (t)
Post-Keynesian investment function:
g⋆ (t) = ¯
+
(t) u (t)
) g⋆ (t) = ¯
+
(t) g (t)
sp

10. where
is the sensitivity of desired rate of investment to capacity
utilization and is endogenously determined by financial factors. ¯
is
the exogenous component of investment (Dum´enil L´evy 1999).

11. Financial sector
In our model, we primarily examine debt as the main financial variable.
Debt dynamics affect the real sector via investment through two possible
routes:
By directly affecting the cost of financing investment.
Through various forms of risks associated with debt, for instance,
the possibility of the borrower defaulting on its payment
commitments.

12. Dynamics of Debt
The total outstanding debt commitment in period, t given by a
history of borrowing, B, at a rate of interest, r , and repayment, R:
D (t) =
t
Z
τ=0
(B () − R ()) er(t)(t−τ)d
) ˙D(t) = B (t) − R (t) + r (t)D (t)
Define macroeconomic index of financial fragility:
(t) =
(q + r (t))D (t)
P (t)
=
k (q + r (t)) spd (t)
g (t)
where d (t)
D (t)
K (t)
and g (t)
I (t)
K (t)

13. Dynamics of Debt (Cont’d)
Repayment of debt
Let the actual repayment in period t be a fraction (t) of the
outstanding debt commitment, i.e. R (t) = (t)D (t).
(t) depends on
1 Ability of firms to repay, given by the level of retained profits,
P. Higher retained profits would enable borrowers to repay
larger fraction of outstanding debt commitments without
altering its capital structure.
2 Index of financial fragility, . Higher would be associated
with a borrower profile where the firms have higher gearing
ratios. Hence, they would be forced to repay back a higher
fraction of outstanding debt commitments.
We adopt above in a simple multiplicative form:
(t) = mP (t) (t)
Substituting from the values of P and :
(t) = m(q + r (t)) d (t)

14. Dynamics of Debt (Cont’d)
Borrowing Financial Structure
In any period, t, let a fraction a (t) of the total investment I (t)
made by the firm sector be financed by fresh borrowing, i.e.
B (t) = a (t) I (t), where the fraction a (t) will be determined by the
financial structure of the firm.
For a given level of profits, we expect a higher rate of
investment to result in a higher proportion of investment
financed by outside sources.
Between two sources of external finance, there might be an
increasing preference for debt as the rate of investment
increases.
An increase in the level of financial fragility, , might
necessitate financing a higher proportion of the cost of
investment through debt.
) a (t) = a (g (t) , (t)) ; ag 0, aλ 0. With a simple
multiplicative form, we have
a (t) =
k (q + r (t)) s
d (t)

15. Creditworthiness and Borrower Profile
Consider the process of loan application by lenders. Broadly, the
quantitative factors determining the creditworthiness of a loan
application might be categorized into two classes:
1 Idiosyncratic factors: A preliminary assessment consisting of
factors which remain unchanged across various stages of a
business cycle, e.g. credit history, long-term repayment records,
reputation etc. Based on these factors, the lending institutions
might assign a credit rating or score to each loan applicant,
classifying them as prime or sub-prime.
2 Systemic factors: For a final decision, the lending institutions
take into account additional criteria, including the current
income of the loan applicants, evaluation of their proposed
projects in terms of their expected future income and risk
associated. These factors would depend on the specific stage
of business cycle one is in.

16. Creditworthiness and Borrower Profile (Cont’d)
We formalize the first by introducing 2 [0, 1], an indicator of the
proportion of borrowers with a high perceived risk of default (i.e.
the sub-prime borrowers) in the macroeconomic distribution of debt.
Periods of prosperity accompanied with a gradual worsening of the
profile of borrowers, leading to inclusion of borrowers with higher
perceived risk of default (sub-prime borrowers). This might happen
because:
During periods of prosperity, greater number of loan applicants
will qualify a given set of prudential norms.
In addition, typically prosperity leads to a relaxation of
prudential norms, both directly as well as indirectly from
financial innovation and predatory lending practices of
organized lenders, leading to emergence of new financial
instruments.
Formalizing this: (t) = gg (t) ; g 2 ]0, 1/gmax]

17. Creditworthiness and Borrower Profile (Cont’d)
Cumulative Index of Risk of Default
Construct a cumulative index of risk of default:
(t) = η (t) + λ (t)
where η and λ represent the sensitivity of to and .
The cumulative index of risk of default, , consist of two separate
risk components, and , emerging from two different kinds of risk
involved in credit expansion:
1 Credit widening, or inclusion of new borrowers with lower
credit rating, captured by .
2 Credit deepening, or an increase in the gearing ratio of existing
borrowers. captures a combination of both credit widening
and credit deepening.
This makes a more comprehensive macroeconomic indicator of
the risk of default than some of the more conventional indicators.

18. Financial Determinants of Investment
Risk of default negatively affects the sensitivity of the rate of
investment to capacity utilization,
.
Managers are concerned with risk of default, since in case of a
default, a firm might face a change in ownership through a
hostile takeover, threatening the job of managers. Thus, an
increase in would make them cut back on investment.
Lenders are concerned with risk of default, and might resort to
rationing and red-lining of credit if increases to unacceptable
levels. While this will affect only a section of borrowers, all
borrowers will cut back on investment in order to avoid getting
credit rationed or red-lined.

19. Financial Determinants of Investment (Cont’d)
The rate of interest negatively affects the sensitivity of the rate of
investment to capacity utilization,
.
Rate of interest directly affects the cost of servicing debt for
both past and new loans. This increases the cost of financing
investment.
An increase in the rate of interest increases the possibility of
adverse selection of risky projects. This might prompt lending
institutions to increase credit rationing and red-lining.
Formalizing:
(t) = ¯μ − ˆμ (t) − r (t)
where is the sensitivity of the accelerator to the rate of interest,
and ˆμ is the sensitivity of the accelerator to the cumulative risk of
default, .

20. Monetary Policy
Modified version of Taylor-type interest rate rule, which, instead of
targeting the inflation or the output gap, targets the rate of
capacity utilization as a proxy for the level of economic activity.
The Central Bank adjusts the rate of interest as a response to the
gap between the desired and the actual rate of capacity utilization,
i.e.
˙ r (t)
r (t)
= l [u (t) − u⋆]
where u⋆ 2 ]0, 1[ is the rate of capacity utilization desired by the
Central Bank.

21. Dynamics of Investment
Let the rate of investment be continuously adjusted so as to meet a
fraction, h, of the gap between the actual and the desired rate of
investment, i.e.
g˙ (t)
g (t)
= h (g⋆ (t) − g (t))
where h represents the speed of adjustment of the actual investment
to the desired level by the investors.
With suitable substitutions:
g˙ (t) =
¯μ
sp

22. !
− 1
g (t) −
ˆμηg
sp

23. {g (t)}2 −
ˆμλkq

24. d (t) −
ˆμλk

25. r (t) d (t)
−
sp

26. g (t) r (t) +
¯
sp

27. #
hg (t)

28. Complete Model
g˙ (t) =
¯μ
s

29. − 1
g (t) −
ˆμηg
s

30. {g (t)}2 −
ˆμλkq

31. d (t) −
ˆμλk

32. r (t) d (t) −
s

33. g (t) r (t) +
¯
s

34. hg (t)
˙ r (t) = l
g (t)
s

35. − u
⋆
r (t)
˙d
(t) =
kqs
− 1
g (t) +
ks
d (t)
g (t) r (t) − mqd (t) − mr (t) d (t) + r (t)
These dynamics resemble the generalized predator-prey class of
models with two predators and one prey. Both r and d are
analogous to the predators, whereas g is analogous to prey.
Underlying such an analogy with ecological models, however, there
is a complex interaction of several macroeconomic feedback effects.

36. Macroeconomic Feedback Effects
Multiplier-Accelerator Relationship:
g
multiplier
−−−−−−! Y −! u
accelerator
−−−−−−! g
⋆ −! g
Financial Feedback I:
g
multiplier
−−−−−−! Y −! u
Taylor rule
−−−−−−! r
investment function
−−−−−−−−−−−! g
⋆ #−! g #
Financial Feedback II:
g −! −!
investment function
−−−−−−−−−−−! g
⋆ #−! g #
Financial Feedback III:
g −! B −! d −! −!
investment function
−−−−−−−−−−−! g
⋆ #−! g #
Secondary Financial Feedback:
(a) g
multiplier
−−−−−−! Y −! u
Taylor rule
−−−−−−! r −! −!
investment function
−−−−−−−−−−−! g
⋆ #−! g #
(b) g
multiplier
−−−−−−! Y −! u
Taylor rule
−−−−−−! r −! B −! d −!
−!
investment function
−−−−−−−−−−−! g
⋆ #−! g #

37. Summary of Results
The dynamical system has only one economically meaningful steady
state. The steady state rate of investment:
¯g = sp

38. u⋆
Note that the steady state rate of investment is completely
determined by the monetary policy of the Central Bank.
Steady state is stable provided l ˆl
, i.e. monetary policy is
sufficiently passive.
For a wide range of numerical values
@ˆl
/@u⋆ 0 8 u⋆ 2 ]0, 1[ : Targeting a higher rate of capacity
utilization will affect the effectiveness of monetary policy.
@ˆl
/@h 2 ]0,1[ 8 h 2 ]−1,1[: Faster adjustment by private
investors will leave more room for central bank to conduct
monetary policy.

39. Summary of Results (Cont’d)
Comparative Dynamics
We note that the steady state rate of growth depends directly on
the propensity to save out of profits. In other words paradox of
thrift does not operate in long run.
Given that we begun with a post-Keynesian investment function,
this result might seem to be a departure from standard
post-Keynesian literature and more in line with Harrodian literature.
In fact, higher the target rate of capacity utilization by Central
Bank, closer is the steady state rate of growth to the classic
Harrod’s result.
However, unlike the Harrodian literature, the steady state of growth
does not stabilize at an exogenously given natural rate, but at the
rate targeted by the Central Bank.

40. Summary of Results (Cont’d)
Away from the steady state, depending on the values of the
parameters, dynamical possibilities include
convergence to steady state, or
divergence away from the steady state, or
emergence of stable/unstable limit cycles around the steady
state (from non-degenerate Hopf Bifurcation, using h as the
control parameter), or/and
emergence of invariant torus around Hopf bifurcation limit
cycles and its eventual breakdown, bifurcation of homoclinic
and heteroclinic Shil’nikov orbits etc.

41. Summary of Results (Cont’d)
Bifurcation
A variety of bifurcations are shown to be possible:
1 Codim 1 bifurcation: Non-degenerate Hopf-bifurcation, using h as
the control parameter, leading to emergence of stable/unstable limit
cycles.
2 Codim 2 bifurcation: Using h and l as the control parameters, it is
possible to derive:
Neimark-Sacker bifurcation leading to emergence of invariant
torus.
Saddle-node bifurcation, and disappearance of saddle-nodes
through Shil’nilov bifurcation, emergence of infinite number of
periodic orbits.
Fold-Hopf (Gavrilov-Guckenheimer) bifurcation, triggering off
appearance and bifurcation of Shil’nikov homoclinic and
hetroclinic orbits, appearance of invariant torus and its
breakdown leading to chaos.
Double-zero (Bogdanov-Takens) bifurcation.

42. Conclusions
Even a simple model of real-financial interaction in a
demand-constrained economy leads to a complex interaction of
macroeconomic feedback effects.
Depending on the strengths of these effects, and the lags in them, a
wide variety of complex dynamical possibilities exist.
Under certain conditions, financial factors can endogenously bound
a demand-constrained economic system.
Even a purely deterministic system can give rise to complex
dynamics, and be sensitive to initial conditions. This can have
computational implications.
Monetary policy in the form of interest rate rules can determine the
steady state in our model. This conclusion, however, comes with
several riders.

43. Limitations
Areas for future research
By holding prices and share of profits fixed, we do not explicitly
model income distribution considerations in this model. An
immediate extension of this model, therefore, could be to look into
the effect of the macroeconomic feedback effects discussed here on
the distribution of income between various social classes.
We do not include complications arising out of changes in asset
prices in our model. Hence, we miss an important area which has
received a considerable attention in the literature, involving asset
price dynamics leading to boom-bust cycles.
We note that large number of dynamical possibilities exist in this
model. It is difficult to symbolically impose restrictions on
parameters to restrict the set of outcomes. One possible extension,
therefore, might be to suitably calibrate the model with the help of
real world data.

44. Thank you!
Feedbacks welcome.
soumya@econ.sau.ac.in