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Macrodynamics of Debt-Financed Investment-Led Growth with Interest Rate Rules


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Uncertainty and Economic growth session at 12th International Conference

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Macrodynamics of Debt-Financed Investment-Led Growth with Interest Rate Rules

  1. 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. 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. 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. 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. 5. Basic Model (Cont’d) Potential output, Y ⋆ (t) =
  6. 6. K (t) where
  7. 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. 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. 9. u (t) Post-Keynesian investment function: g⋆ (t) = ¯ + (t) u (t) ) g⋆ (t) = ¯ + (t) g (t) sp
  10. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 22. ! − 1 g (t) − ˆμηg sp
  23. 23. {g (t)}2 − ˆμλkq
  24. 24. d (t) − ˆμλk
  25. 25. r (t) d (t) − sp
  26. 26. g (t) r (t) + ¯ sp
  27. 27. # hg (t)
  28. 28. Complete Model g˙ (t) = ¯μ s
  29. 29. − 1 g (t) − ˆμηg s
  30. 30. {g (t)}2 − ˆμλkq
  31. 31. d (t) − ˆμλk
  32. 32. r (t) d (t) − s
  33. 33. g (t) r (t) + ¯ s
  34. 34. hg (t) ˙ r (t) = l g (t) s
  35. 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. 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. 37. Summary of Results The dynamical system has only one economically meaningful steady state. The steady state rate of investment: ¯g = sp
  38. 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. 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. 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. 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. 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. 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.
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