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 ConferenceKansas City, MissouriSeptember 24-28, 2014

Introduction: Objectives of Study

This study attempts to answer the following primary questions:

Can financial considerations provide endogenous bounds to anotherwise 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, orcyclical patterns in the growth rates of macroeconomic variables?Can these cycles break down to more complex dynamicalpossibilities?

How effective is a monetary policy, in the form of interest rate rules,in achieving its desired objectives?

A Preview of the Model

We explicitly model the possibilities of borrowers defaulting onpayment commitments. This encourages lenders to discriminatebetween borrowers, leading to credit rationing and red-lining.

During the upward phase of business cycle, financial variablesdeteriorate due to credit expansion (Fisher, Minsky). Two kinds ofcredit expansion: credit deepening and credit widening.

We provide an alternative macroeconomic story to Minsky’sarguments.

To keep dimension low, at the moment we do not include incomedistribution considerations. Prices remain constant. Hence,monetary policy (Taylor Rule) is suitably modified – with capacityutilization as a proxy for inflation.

Basic Model

A simple continuous time model of closed economy with nogovernment.

Two social classes: workers earning wages (W ) and capitalistsearning profits (P).

National income by income method: Y (t) = W (t) + P (t).

Workers do not save. Capitalists save a fraction sp of profits. Henceconsumption, C (t) = W (t) + (1− sp)P (t).

Price is a fixed-up markup over wage costs of production. HenceP (t) = ψY (t), where ψ 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, orexternally out of debt or equity.

Basic Model (Cont’d)

Potential output, Y ⋆ (t) = βK (t) where β: fixed output capitalratio given by existing technology. Availability of the capital is thebinding constraint on production.

Actual level of output: Y (t) = min [AD (t) ,Y ⋆ (t)]. For allAD ≤ Y ⋆, aggregate demand acts as the main constraint onproduction. In this case output is determined by aggregate demand.

Rate of capacity utilization, u (t) =Y (t)

Y ⋆ (t)∈ ]0, 1[.

Rate of investment, g (t) ≡I (t)

K (t)

Goods Market Equilibrium & Investment Function

Goods market equilibrium: Level of output measured by incomemethod equals aggregate demand, i.e.

W (t) + P (t) = C (t) + I (t)

∴ Y (t) =1

spψI (t)

and g (t) = spψβu (t)

Post-Keynesian investment function:

g⋆ (t) = γ + γ (t) u (t)

⇒ g⋆ (t) = γ +γ (t) g (t)

spψβ

where γ is the sensitivity of desired rate of investment to capacityutilization and is endogenously determined by financial factors. γ isthe exogenous component of investment (Dumenil & Levy 1999).

Financial sector

In our model, we primarily examine debt as the main financial variable.Debt dynamics affect the real sector via investment through two possibleroutes:

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 paymentcommitments.

Dynamics of Debt

The total outstanding debt commitment in period, t given by ahistory of borrowing, B, at a rate of interest, r , and repayment, R :

D (t) =

t∫

τ=0

(B (τ) − R (τ)) er(t)(t−τ )dτ

⇒ D (t) = B (t)− R (t) + r (t)D (t)

Define macroeconomic index of financial fragility:

λ (t) =(q + r (t))D (t)

σP (t)=

k (q + r (t)) spd (t)

σg (t)

where d (t) ≡D (t)

K (t)and g (t) ≡

I (t)

K (t)

Dynamics of Debt (Cont’d)Repayment of debt

Let the actual repayment in period t be a fraction φ (t) of theoutstanding debt commitment, i.e. R (t) = φ (t)D (t).

φ (t) depends on

1 Ability of firms to repay, given by the level of retained profits,σP . Higher retained profits would enable borrowers to repaylarger fraction of outstanding debt commitments withoutaltering its capital structure.

2 Index of financial fragility, λ. Higher λ would be associatedwith a borrower profile where the firms have higher gearingratios. Hence, they would be forced to repay back a higherfraction of outstanding debt commitments.

We adopt above in a simple multiplicative form:

φ (t) = mσP (t) λ (t)

Substituting from the values of P and λ:

φ (t) = m (q + r (t)) d (t)

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 thefinancial structure of the firm.

For a given level of profits, we expect a higher rate ofinvestment to result in a higher proportion of investmentfinanced by outside sources.Between two sources of external finance, there might be anincreasing preference for debt as the rate of investmentincreases.An increase in the level of financial fragility, λ, mightnecessitate financing a higher proportion of the cost ofinvestment through debt.

∴ a (t) = a (g (t) , λ (t)) ; ag > 0, aλ > 0. With a simplemultiplicative form, we have

a (t) =k (q + r (t)) s

σψd (t)

Creditworthiness and Borrower Profile

Consider the process of loan application by lenders. Broadly, thequantitative factors determining the creditworthiness of a loanapplication might be categorized into two classes:

1 Idiosyncratic factors: A preliminary assessment consisting offactors which remain unchanged across various stages of abusiness cycle, e.g. credit history, long-term repayment records,reputation etc. Based on these factors, the lending institutionsmight 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 institutionstake into account additional criteria, including the currentincome of the loan applicants, evaluation of their proposedprojects in terms of their expected future income and riskassociated. These factors would depend on the specific stageof business cycle one is in.

Creditworthiness and Borrower Profile (Cont’d)

We formalize the first by introducing η ∈ [0, 1], an indicator of theproportion 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 theprofile of borrowers, leading to inclusion of borrowers with higherperceived risk of default (sub-prime borrowers). This might happenbecause:

During periods of prosperity, greater number of loan applicantswill qualify a given set of prudential norms.In addition, typically prosperity leads to a relaxation ofprudential norms, both directly as well as indirectly fromfinancial innovation and predatory lending practices oforganized lenders, leading to emergence of new financialinstruments.

Formalizing this: η (t) = ηgg (t) ; ηg ∈ ]0, 1/gmax]

Creditworthiness and Borrower Profile (Cont’d)Cumulative Index of Risk of Default

Construct a cumulative index of risk of default:

Λ (t) = Ληη (t) + Λλλ (t)

where Λη and Λλ represent the sensitivity of Λ to η and λ.

The cumulative index of risk of default, Λ, consist of two separaterisk components, η and λ, emerging from two different kinds of riskinvolved in credit expansion:

1 Credit widening, or inclusion of new borrowers with lowercredit rating, captured by η.

2 Credit deepening, or an increase in the gearing ratio of existingborrowers. λ captures a combination of both credit wideningand credit deepening.

This makes Λ a more comprehensive macroeconomic indicator ofthe risk of default than some of the more conventional indicators.

Financial Determinants of Investment

Risk of default negatively affects the sensitivity of the rate ofinvestment to capacity utilization, γ.

Managers are concerned with risk of default, since in case of adefault, a firm might face a change in ownership through ahostile takeover, threatening the job of managers. Thus, anincrease in Λ would make them cut back on investment.Lenders are concerned with risk of default, and might resort torationing and red-lining of credit if Λ increases to unacceptablelevels. While this will affect only a section of borrowers, allborrowers will cut back on investment in order to avoid gettingcredit rationed or red-lined.

Financial Determinants of Investment (Cont’d)

The rate of interest negatively affects the sensitivity of the rate ofinvestment to capacity utilization, γ.

Rate of interest directly affects the cost of servicing debt forboth past and new loans. This increases the cost of financinginvestment.An increase in the rate of interest increases the possibility ofadverse selection of risky projects. This might prompt lendinginstitutions to increase credit rationing and red-lining.

Formalizing:γ (t) = µ− µΛ (t)− αr (t)

where α is the sensitivity of the accelerator to the rate of interest,and µ is the sensitivity of the accelerator to the cumulative risk ofdefault, Λ.

Monetary Policy

Modified version of Taylor-type interest rate rule, which, instead oftargeting the inflation or the output gap, targets the rate ofcapacity utilization as a proxy for the level of economic activity.

The Central Bank adjusts the rate of interest as a response to thegap between the desired and the actual rate of capacity utilization,i.e.

r (t)

r (t)= l [u (t)− u⋆]

where u⋆ ∈ ]0, 1[ is the rate of capacity utilization desired by theCentral Bank.

Dynamics of Investment

Let the rate of investment be continuously adjusted so as to meet afraction, h, of the gap between the actual and the desired rate ofinvestment, i.e.

g (t)

g (t)= h (g⋆ (t)− g (t))

where h represents the speed of adjustment of the actual investmentto the desired level by the investors.

With suitable substitutions:

g (t) =

[(

µ

spψβ− 1

)

g (t) −µΛηηg

spψβ{g (t)}

2−µΛλkq

σψβd (t) −

µΛλk

σψβr (t) d (t)

−α

spψβg (t) r (t) +

γ

spψβ

]

hg (t)

Complete Model

g (t) =

[(

µ

sβ− 1

)

g (t) −µΛηηg

sβ{g (t)}

2−µΛλkq

σψβd (t) −

µΛλk

σψβr (t) d (t) −

α

sβg (t) r (t) +

γ

sβ

]

hg (t)

r (t) = l

{

g (t)

sβ− u

⋆}

r (t)

d (t) =

[(

kqs

σψ− 1

)

g (t) +ks

σψg (t) r (t) − mqd (t) − mr (t) d (t) + r (t)

]

d (t)

These dynamics resemble the generalized predator-prey class ofmodels with two predators and one prey. Both r and d areanalogous to the predators, whereas g is analogous to prey.

Underlying such an analogy with ecological models, however, thereis a complex interaction of several macroeconomic feedback effects.

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 ↑−→ η ↑−→ Λ ↑investment function

−−−−−−−−−−−→ g⋆

↓−→ g ↓

Financial Feedback III:

g ↑−→ B ↑−→ d ↑−→ λ ↑−→ Λ ↑investment function

−−−−−−−−−−−→ g⋆

↓−→ g ↓

Secondary Financial Feedback:

(a) g ↑multiplier

−−−−−−→ Y ↑−→ u ↑Taylor rule

−−−−−−→ r ↑−→ λ ↑−→ Λ ↑

investment function−−−−−−−−−−−→ g

⋆↓−→ g ↓

(b) g ↑multiplier

−−−−−−→ Y ↑−→ u ↑Taylor rule

−−−−−−→ r ↑−→ B ↑−→ d ↑−→ λ ↑

−→ Λ ↑investment function

−−−−−−−−−−−→ g⋆

↓−→ g ↓

Summary of Results

The dynamical system has only one economically meaningful steadystate. The steady state rate of investment:

g = spψβu⋆

Note that the steady state rate of investment is completelydetermined by the monetary policy of the Central Bank.

Steady state is stable provided l < l , i.e. monetary policy issufficiently passive.

For a wide range of numerical values

∂ l/∂u⋆ < 0 ∀ u⋆ ∈ ]0, 1[ : Targeting a higher rate of capacityutilization will affect the effectiveness of monetary policy.∂ l/∂h ∈ ]0,∞[ ∀ h ∈ ]−∞,∞[: Faster adjustment by privateinvestors will leave more room for central bank to conductmonetary policy.

Summary of Results (Cont’d)Comparative Dynamics

We note that the steady state rate of growth depends directly onthe propensity to save out of profits. In other words paradox ofthrift 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 standardpost-Keynesian literature and more in line with Harrodian literature.In fact, higher the target rate of capacity utilization by CentralBank, closer is the steady state rate of growth to the classicHarrod’s result.

However, unlike the Harrodian literature, the steady state of growthdoes not stabilize at an exogenously given natural rate, but at therate targeted by the Central Bank.

Summary of Results (Cont’d)

Away from the steady state, depending on the values of theparameters, dynamical possibilities include

convergence to steady state, ordivergence away from the steady state, oremergence of stable/unstable limit cycles around the steadystate (from non-degenerate Hopf Bifurcation, using h as thecontrol parameter), or/andemergence of invariant torus around Hopf bifurcation limitcycles and its eventual breakdown, bifurcation of homoclinicand heteroclinic Shil’nikov orbits etc.

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 asthe control parameter, leading to emergence of stable/unstable limitcycles.

2 Codim 2 bifurcation: Using h and l as the control parameters, it ispossible to derive:

Neimark-Sacker bifurcation leading to emergence of invarianttorus.Saddle-node bifurcation, and disappearance of saddle-nodesthrough Shil’nilov bifurcation, emergence of infinite number ofperiodic orbits.Fold-Hopf (Gavrilov-Guckenheimer) bifurcation, triggering offappearance and bifurcation of Shil’nikov homoclinic andhetroclinic orbits, appearance of invariant torus and itsbreakdown leading to chaos.Double-zero (Bogdanov-Takens) bifurcation.

Conclusions

Even a simple model of real-financial interaction in ademand-constrained economy leads to a complex interaction ofmacroeconomic feedback effects.

Depending on the strengths of these effects, and the lags in them, awide variety of complex dynamical possibilities exist.

Under certain conditions, financial factors can endogenously bounda demand-constrained economic system.

Even a purely deterministic system can give rise to complexdynamics, and be sensitive to initial conditions. This can havecomputational implications.

Monetary policy in the form of interest rate rules can determine thesteady state in our model. This conclusion, however, comes withseveral riders.

LimitationsAreas for future research

By holding prices and share of profits fixed, we do not explicitlymodel income distribution considerations in this model. Animmediate extension of this model, therefore, could be to look intothe effect of the macroeconomic feedback effects discussed here onthe distribution of income between various social classes.

We do not include complications arising out of changes in assetprices in our model. Hence, we miss an important area which hasreceived a considerable attention in the literature, involving assetprice dynamics leading to boom-bust cycles.

We note that large number of dynamical possibilities exist in thismodel. It is difficult to symbolically impose restrictions onparameters to restrict the set of outcomes. One possible extension,therefore, might be to suitably calibrate the model with the help ofreal world data.

Thank you!Feedbacks welcome.

soumya@econ.sau.ac.in