Dynamical Systems and Financial Instability

M. R. Grasselli Mainstream Dynamical Systems and Financial Instability Alternative approaches

SFC models

Conclusions M. R. Grasselli

Mathematics and Statistics - McMaster University and Fields Institute for Research in Mathematical Sciences

ICTP-SAIFR Seminar, May 30, 2014 A brief history of Macroeconomics

Dynamical Systems and Classics (Smith, Ricardo, Marx): no distinction between Financial Instability micro and macro, Say’s law, emphasis on long run. M. R. Grasselli Beginning of the 20th century (Wicksell, Fisher): natural Mainstream rate of interest, quantity theory of money. Alternative approaches Keynesian revolution (1936): shift to demand, fallacies of

SFC models composition, role of expectations, and much more! Conclusions Neoclassical synthesis - 1945 to 1970 (Hicks, Samuelson, Solow): Keynesian consensus. Rational Expectations Revolution - 1972 (Lucas, Prescott, Sargent): internal consistency, microfoundations. Start of Macro Wars: Real Business Cycles versus New Keynesian. 1990’s: impression of consensus around DSGE models, but with diﬀerent ﬂavours. Dynamic Stochastic General Equilibrium

Dynamical Systems and Financial Instability Seeks to explain the aggregate economy using theories

M. R. Grasselli based on strong microeconomic foundations.

Mainstream Collective decisions of rational individuals over a range of

Alternative variables for both present and future. approaches

SFC models All variables are assumed to be simultaneously in

Conclusions equilibrium. The only way the economy can be in disequilibrium at any point in time is through decisions based on wrong information. Money is neutral in its eﬀect on real variables. Largely ignores uncertainty by simply subtracting risk premia from all risky returns and treat them as risk-free. Really bad economics: hardcore (freshwater) DSGE

Dynamical Systems and Financial Instability

M. R. Grasselli The strand of DSGE economists aﬃliated with RBC Mainstream theory made the following predictions after 2008: Alternative approaches 1 Increases government borrowing would lead to higher SFC models interest rates on government debt because of “crowding

Conclusions out”. 2 Increases in the money supply would lead to inflation. 3 Fiscal stimulus has zero effect in an ideal world and negative effect in practice (because of decreased confidence). Wrong prediction number 1

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models

Conclusions

Figure: Government borrowing and interest rates. Wrong prediction number 2

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models

Conclusions

Figure: Monetary base and inﬂation. Wrong prediction number 3

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models

Conclusions

Figure: Fiscal tightening and GDP. Better (but still bad) economics: soft core (saltwater) DSGE

Dynamical Systems and Financial Instability The strand of DSGE economists aﬃliated with New M. R. Grasselli Keynesian theory got all these predictions right.

Mainstream They did so by augmented DSGE with ‘imperfections’ Alternative (wage stickiness, asymmetric information, imperfect approaches competition, etc). SFC models

Conclusions Still DSGE at core - analogous to adding epicycles to Ptolemaic planetary system. For example: “Ignoring the foreign component, or looking at the world as a whole, the overall level of debt makes no diﬀerence to aggregate net worth – one person’s liability is another person’s asset.” (Paul Krugman and Gauti B. Eggertsson, 2010, pp. 2-3) Finance in DSGE models

Dynamical Systems and The ﬁnancial sector merely serve as intermediaries Financial Instability channeling savings from households to business.

M. R. Grasselli Banks provide indirect finance by borrowing short and lending long (business loans), thereby solving the problem Mainstream of liquidity preferences (Diamond and Dybvig (1986) Alternative approaches model). SFC models Financial market provide direct finance through shares, Conclusions thereby introducing market prices and discipline. Financial Frictions (e.g borrowing constraints, market liquidity) create persistence and amplification of real shocks (Bernanke and Gertler (1989), Kiyotaki and Moore (1997) models) See Brunnermeier and Sannikov (2013) for a recent contribution to this strand of literature in light of the financial crisis, in particular in the context of macro-prudential regulation. Frictions literature still missing the point

Dynamical Turner 2013 observes that: Systems and Financial Instability “Quantitative impacts suggested by the models were far

M. R. Grasselli smaller than those empirically observed in real world episodes such as the Great Depression or the 2008 crisis” Mainstream

Alternative “Most of the literature omits consideration of approaches behaviourally driven ‘irrational’ cycles in asset prices”. SFC models

Conclusions “the vast majority of the literature ignores the possibilities of credit extension to ﬁnance the purchase of already existing assets”. “the dominant model remains one in which household savers make deposits in banks, which lend money to entrepreneurs/businesses to pursue ‘investment projects’. The reality of a world in which only a small proportion (e.g. 15%) of bank credit funds ‘new investment projects’ has therefore been left largely unexplored.” Turner (2013) slide

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models

Conclusions A parallel history of Macroeconomics

Dynamical Systems and Classical 19th century monetarism (Bagehot, Allan Financial Instability Young): role of banks in trade (Britain) and development

M. R. Grasselli (U.S.), central banking.

Mainstream Several prominent disciples of Keynes (Kaldor, Robinson,

Alternative Davidson) immediately rejected the Neoclassical synthesis approaches as “bastardized Keynesianism”. SFC models Flow of Funds accounting - 1952 (Copeland): alternative Conclusions to both Y = C + I + G + X − M (finals sales) and MV = PT (money transactions) by tracking exchanges of both goods and financial assets. Gurley, Shaw, Tobin, Minsky: financial intermediation at centre stage. Kindleberger (1978): detailed history of financier crises. Stock-flow consistent models (Godley, Lavoie) Revival of interest after the 2008 crisis. Key insight 1: money is not neutral

Dynamical Systems and Money is hierarchical: currency is a promise to pay gold Financial Instability (or taxes); deposits are promises to pay currency;

M. R. Grasselli securities are promises to pay deposits. Financial institutions are market-makers straddling two Mainstream

Alternative levels in the hierarchy: central banks, banks, security approaches dealers. SFC models The hierarchy is dynamic: discipline and elasticity change Conclusions in time. Key insight 2: money is endogenous

Dynamical Systems and Financial Instability M. R. Grasselli Banks create money and purchasing power. Mainstream Reserve requirements are never binding. Alternative approaches

SFC models

Conclusions Key insight 3: private debt matters

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models

Conclusions

Figure: Change in debt and unemployment. Key insight 4: ﬁnance is not just intermediation

Dynamical Systems and Market never clear in all states: set of events is larger than Financial Instability what can be contracted. M. R. Grasselli The financial sector absorbs the risk of unfulfilled promises. Mainstream The cone of acceptable losses defines the size of the real Alternative economy. approaches

SFC models

Conclusions

Figure: Cherny and Madan (2009) Much better economics: SFC models

Dynamical Systems and Financial Instability Stock-ﬂow consistent models emerged in the last decade

M. R. Grasselli as a common language for many heterodox schools of thought in economics. Mainstream

Alternative Consider both real and monetary factors from the start approaches

SFC models Specify the balance sheet and transactions between sectors Goodwin model Keen model Accommodate a number of behavioural assumptions in a Ponzi ﬁnancing Noise and Stock way that is consistent with the underlying accounting Prices Stabilizing structure. government Great Moderation Reject silly (and mathematically unsound!) hypotheses The Ultimate Model such as the RARE individual (representative agent with Conclusions rational expectations). See Godley and Lavoie (2007) for the full framework. Balance Sheets

Dynamical Systems and Financial Instability

M. R. Grasselli Firms Balance Sheet Households Banks Central Bank Government Sum current capital Mainstream Cash +Hh +Hb −H 0 Alternative Deposits +M +M −M 0 approaches h f Loans −L +L 0 SFC models Bills +Bh +Bb +Bc −B 0 Goodwin model Keen model Equities +pf Ef + pbEb −pf Ef −pbEb 0 Ponzi ﬁnancing Advances −A +A 0 Noise and Stock Prices Capital +pK pK Stabilizing government Sum (net worth) Vh 0 Vf Vb 0 −B pK Great Moderation The Ultimate Model Table: Balance sheet in an example of a general SFC model. Conclusions Transactions

Dynamical Systems and Financial Instability Transactions Firms Households Banks Central Bank Government Sum M. R. Grasselli current capital

Consumption −pCh +pC −pCb 0 Mainstream Investment +pI −pI 0 Alternative Gov spending +pG −pG 0 approaches Acct memo [GDP] [pY ] SFC models Wages +W −W 0 Goodwin model Keen model Taxes −Th −Tf +T 0 Ponzi ﬁnancing Interest on deposits +rM .Mh +rM .Mf −rM .M 0 Noise and Stock Prices Interest on loans −rL.L +rL.L 0 Stabilizing government Interest on bills +rB .Bh +rB .Bb +rB .Bc −rB .B 0 Great Moderation Proﬁts +Πd + Πb −Π +Πu −Πb −Πc +Πc 0 The Ultimate Sum S 0 S − pI S 0 S 0 Model h f b g Conclusions Table: Transactions in an example of a general SFC model. Flow of Funds

Dynamical Systems and Financial Instability

M. R. Grasselli Flow of Funds Firms Households Banks Central Bank Government Sum current capital Mainstream Cash +H˙ h +H˙ b −H˙ 0 ˙ ˙ ˙ Alternative Deposits +Mh +Mf −M 0 approaches Loans −L˙ +L˙ 0 ˙ ˙ ˙ ˙ SFC models Bills +Bh +Bb +Bc −B 0

Goodwin model Equities +pf E˙ f + pbE˙ b −pf E˙ f −pbE˙ b 0 Keen model Advances −A˙ +A˙ 0 Ponzi ﬁnancing Noise and Stock Capital +pI pI Prices Stabilizing Sum Sh 0 Sf Sb 0 Sg pI government Change in Net Worth (Sh +p ˙ f Ef +p ˙ bEb)(Sf − p˙ f Ef +pK ˙ − pδK)(Sb − p˙ bEb) Sg pK˙ + pK˙ Great Moderation The Ultimate Model Table: Flow of funds in an example of a general SFC model. Conclusions General Notation

Dynamical Systems and Financial Instability

M. R. Grasselli Employed labor force: ` Mainstream Production function: Y = f (K, `) Alternative approaches Y Labour productivity: a = ` SFC models K Goodwin model Capital-to-output ratio: ν = Keen model Y Ponzi ﬁnancing ` Noise and Stock Employment rate: λ = Prices N Stabilizing government Change in capital: K˙ = I − δK Great Moderation p˙ The Ultimate Inﬂation rate: i = Model p Conclusions Goodwin Model (1967) - Assumptions

Dynamical Systems and Assume that Financial Instability βt N = N0e (total labour force) M. R. Grasselli αt a = a0e (productivity per worker) Mainstream K Alternative Y = min , a` (Leontief production) approaches ν SFC models Goodwin model Assume further that Keen model Ponzi ﬁnancing Noise and Stock K Prices Y = = a` (full capital utilization) Stabilizing government ν Great e Moderation w˙ = Φ(λ, i, i )w (Phillips curve) The Ultimate Model pI = pY − w` (Say’s Law) Conclusions NOTE: In the original paper, Goodwin assumed that w above was the real wage rate, so all quantities were normalized by p. Goodwin Model - SFC matrix

Dynamical Systems and Financial Firms Balance Sheet Households Sum Instability current capital M. R. Grasselli Capital +pK pK

Sum (net worth) 0 0 Vf pK Mainstream Transactions Alternative approaches Consumption −pC +pC 0 Investment +pI −pI 0 SFC models Acct memo [GDP] [pY ] Goodwin model Keen model Wages +W −W 0 Ponzi financing Noise and Stock Profits −Π +Πu 0 Prices Sum 0 0 0 0 Stabilizing government Great Flow of Funds Moderation Capital +pI pI The Ultimate Model Sum 0 0 Πu pI Conclusions Change in Net Worth 0 pI +pK ˙ − pδK pK˙ + pK˙ Table: SFC table for the Goodwin model. Goodwin Model - Differential equations

Dynamical Systems and Financial Deﬁne Instability wL w M. R. Grasselli ω = = (wage share) pY pa Mainstream L Y Alternative λ = = (employment rate) approaches N aN SFC models Goodwin model Keen model It then follows that Ponzi ﬁnancing Noise and Stock Prices ω˙ w p˙ a˙ e Stabilizing = − − = Φ(λ, i, i ) − i − α government ω w p a Great Moderation ˙ The Ultimate λ 1 − ω Model = − α − β − δ Conclusions λ ν In the original model, all quantities were real (i.e divided by p), which is equivalent to setting i = i e = 0. Example 1: Goodwin model

Dynamical Systems and Financial Instability w = 0.8, λ = 0.9 0 0 1 M. R. Grasselli

Mainstream 0.98 Alternative approaches 0.96 SFC models Goodwin model Keen model

Ponzi ﬁnancing λ 0.94 Noise and Stock Prices Stabilizing government 0.92 Great Moderation The Ultimate Model 0.9 Conclusions

0.88 0.7 0.75 0.8 0.85 0.9 0.95 1 ω Example 1 (continued): Goodwin model

Dynamical Systems and Financial ω Instability w = 0.8, λ = 0.9, Y = 100 λ 0 0 0 1 6000 M. R. Grasselli Y

Mainstream 0.95 5000 Alternative approaches 0.9 4000 SFC models Goodwin model Keen model λ , Ponzi ﬁnancing 0.85 3000 Y ω Noise and Stock Prices Stabilizing government 0.8 2000 Great Moderation The Ultimate Model 0.75 1000 Conclusions

0.7 0 0 10 20 30 40 50 60 70 80 90 t Goodwin Model - Extensions, structural instability, and empirical tests

Dynamical Systems and Financial Instability Desai 1972: Inflation leads to a stable equilibrium. M. R. Grasselli Ploeg 1985: CES production function leads to stable Mainstream equilibrium. Alternative approaches Goodwin 1991: Pro-cyclical productivity growth leads to SFC models explosive oscillations. Goodwin model Keen model Ponzi financing Solow 1990: US post-war data shows three sub-cycles with Noise and Stock Prices a “bare hint of a single large clockwise sweep” in the Stabilizing government (ω, λ) plot. Great Moderation The Ultimate Harvie 2000: Data from other OECD confirms the same Model Conclusions qualitative features and shows unsatisfactory quantitative estimations. Testing Goodwin on OECD countries

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models Goodwin model Keen model Ponzi ﬁnancing Noise and Stock Prices Stabilizing government Great Moderation The Ultimate Model Conclusions

Figure: Harvie (2000) Correcting Harvie

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models Goodwin model Keen model Ponzi ﬁnancing Noise and Stock Prices Stabilizing government Great Moderation The Ultimate Model Conclusions

Figure: Grasselli and Maheshwari (2012) SFC table for Keen (1995) model

Dynamical Firms Systems and Balance Sheet Households Banks Sum Financial current capital Instability Deposits +D −D 0 Loans −L +L 0 M. R. Grasselli Capital +pK pK

Sum (net worth) Vh 0 Vf 0 pK Mainstream Transactions Alternative Consumption −pC +pC 0 approaches Investment +pI −pI 0 SFC models Acct memo [GDP] [pY ] Goodwin model Wages +W −W 0 Keen model Interest on deposits +rD −rD 0 Ponzi ﬁnancing Interest on loans −rL +rL 0 Noise and Stock Prices Proﬁts −Π +Πu 0

Stabilizing Sum Sh 0 Sf − pI 0 0 government Great Flow of Funds Moderation Deposits +D˙ −D˙ 0 The Ultimate Model Loans −L˙ +L˙ 0 Capital +pI pI Conclusions Sum Sh 0 Πu 0 pI

Change in Net Worth Sh (Sf +pK ˙ − pδK)pK ˙ + pK˙ Table: SFC table for the Keen model. Keen model - Investment function

Dynamical Systems and Financial Instability Assume now that new investment is given by M. R. Grasselli

Mainstream K˙ = κ(1 − ω − rd)Y − δK Alternative approaches where κ(·) is a nonlinear increasing function of profits SFC models Goodwin model π = 1 − ω − rd. Keen model Ponzi financing Noise and Stock This leads to external financing through debt evolving Prices Stabilizing according to government Great Moderation The Ultimate D˙ = κ(1 − ω − rd)Y − (1 − ω − rd)Y Model Conclusions Keen model - Differential Equations

Dynamical Systems and Financial Instability M. R. Grasselli Denote the debt ratio in the economy by d = D/Y , the model Mainstream can now be described by the following system Alternative approaches ω˙ = ω [Φ(λ) − α] SFC models Goodwin model κ(1 − ω − rd) Keen model ˙ Ponzi ﬁnancing λ = λ − α − β − δ (1) Noise and Stock ν Prices Stabilizing κ(1 − ω − rd) government ˙ Great d = d r − + δ + κ(1 − ω − rd) − (1 − ω) Moderation ν The Ultimate Model Conclusions Keen model - equilibria

Dynamical Systems and The system (1) has a good equilibrium at Financial Instability ν(α + β + δ) − π M. R. Grasselli ω = 1 − π − r α + β Mainstream λ = Φ−1(α) Alternative approaches ν(α + β + δ) − π SFC models d = Goodwin model α + β Keen model Ponzi ﬁnancing Noise and Stock with Prices −1 Stabilizing π = κ (ν(α + β + δ)), government Great Moderation which is stable for a large range of parameters The Ultimate Model It also has a bad equilibrium at (0, 0, +∞), which is stable Conclusions if κ(−∞) − δ < r (2) ν Example 2: convergence to the good equilibrium in a Keen model

Dynamical Systems and Financial ω 7 ω = 0.75, λ = 0.75, d = 0.1, Y = 100 Instability λ x 10 0 0 0 0 8 1.3 1 2 Y M. R. Grasselli d 1.8 7 1.2 Mainstream 0.95 1.6 Alternative 6 λ approaches 1.1 1.4 0.9 SFC models 5 1.2 Goodwin model 1 Keen model d λ d Y Ponzi ﬁnancing 4 ω 0.85 1 Noise and Stock ω Prices 0.9 0.8 Stabilizing 3 government 0.8 Great 0.6 Moderation 0.8 The Ultimate 2 Model 0.4 Y 0.75 0.7 Conclusions 1 0.2

0 0.7 0 0 50 100 150 200 250 300 time Example 3: explosive debt in a Keen model

Dynamical Systems and Financial ω = 0.75, λ = 0.7, d = 0.1, Y = 100 6 Instability 0 0 0 0 x 10 6000 35 1 2.5 M. R. Grasselli ω λ 0.9 Y 30 Mainstream 5000 d 0.8 2 Alternative approaches 25 0.7 4000 SFC models λ 0.6 1.5 Goodwin model 20 Keen model Y d λ d Y Ponzi ﬁnancing 3000 ω 0.5 Noise and Stock Prices 15 0.4 1 Stabilizing government 2000 Great 0.3 Moderation 10 The Ultimate Model 0.2 0.5 1000 Conclusions 5 ω 0.1

0 0 0 0 0 50 100 150 200 250 300 time Basin of convergence for Keen model

Dynamical Systems and Financial Instability

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Mainstream

Alternative approaches 10

SFC models 8 Goodwin model Keen model 6 Ponzi ﬁnancing d Noise and Stock 4 Prices Stabilizing 2 government Great 0 Moderation 0.4 The Ultimate 0.5 Model 0.5 0.6 Conclusions 0.7 0.8 1 0.9 1 1.1 1.5 ω λ Ponzi ﬁnancing

Dynamical Systems and Financial Instability To introduce the destabilizing eﬀect of purely speculative M. R. Grasselli investment, we consider a modiﬁed version of the previous

Mainstream model with

Alternative approaches D˙ = κ(1 − ω − rd)Y − (1 − ω − rd)Y + P SFC models Goodwin model P˙ = Ψ(g(ω, d)P Keen model Ponzi financing Noise and Stock Prices where Ψ(·) is an increasing function of the growth rate of Stabilizing government economic output Great Moderation The Ultimate Model κ(1 − ω − rd) g = − δ. Conclusions ν Example 4: effect of Ponzi financing

Dynamical Systems and Financial ω = 0.95, λ = 0.9, d = 0, p = 0.1, Y = 100 Instability 0 0 0 0 0 1 M. R. Grasselli 0.9 Mainstream No Speculation 0.8 Ponzi Financing Alternative approaches 0.7 SFC models Goodwin model 0.6 Keen model Ponzi ﬁnancing ω 0.5 Noise and Stock Prices Stabilizing 0.4 government Great Moderation 0.3 The Ultimate Model 0.2 Conclusions 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λ Stock prices

Dynamical Systems and Financial Instability Consider a stock price process of the form M. R. Grasselli

Mainstream dSt (µt ) = rbdt + σdWt + γµt dt − γdN Alternative St approaches SFC models where Nt is a Cox process with stochastic intensity Goodwin model Keen model µt = M(p(t)). Ponzi ﬁnancing Noise and Stock Prices The interest rate for private debt is modelled as Stabilizing government rt = rb + rp(t) where Great Moderation The Ultimate ρ3 Model rp(t) = ρ1(St + ρ2) Conclusions Example 5: stock prices, explosive debt, zero speculation

Dynamical Systems and Financial Instability 1 M. R. Grasselli ω λ 0.5 Mainstream

Alternative 0 approaches 0 10 20 30 40 50 60 70 80 90 100

SFC models 2 1000 Goodwin model p Keen model d Ponzi ﬁnancing 1 500 Noise and Stock Prices Stabilizing 0 0 government 0 10 20 30 40 50 60 70 80 90 100 Great Moderation 200 The Ultimate Model S 150 t Conclusions 100

0 0 10 20 30 40 50 60 70 80 90 100 Example 6: stock prices, explosive debt, explosive speculation

Dynamical Systems and Financial 3 Instability ω 2 M. R. Grasselli λ 1 Mainstream 0 Alternative 0 10 20 30 40 50 60 70 80 90 100 approaches 10 1000 SFC models 8 p 800 Goodwin model 6 d 600 Keen model 4 400 Ponzi ﬁnancing Noise and Stock 2 200 Prices 0 0 Stabilizing 0 10 20 30 40 50 60 70 80 90 100 government Great 10000 Moderation S The Ultimate t Model 5000 Conclusions

0 0 10 20 30 40 50 60 70 80 90 100 Example 7: stock prices, ﬁnite debt, ﬁnite speculation

Dynamical Systems and Financial Instability 1 M. R. Grasselli ω 0.9 λ

Mainstream 0.8 Alternative 0.7 approaches 0 10 20 30 40 50 60 70 80 90 100

SFC models 0.011 0.5 Goodwin model Keen model Ponzi ﬁnancing 0.01 p 0 Noise and Stock d Prices Stabilizing 0.009 −0.5 government 0 10 20 30 40 50 60 70 80 90 100 Great Moderation 400 The Ultimate Model 300 S t Conclusions 200

100

0 0 10 20 30 40 50 60 70 80 90 100 Stability map

Dynamical Systems and Financial Stability map for ω = 0.8, p = 0.01, S = 100, T = 500, dt = 0.005, # of simulations = 100 Instability 0 0 0 1 0.9

0.6 M. R. Grasselli 0.85 0.7

0.9 0.7 0.8 0.6 0.65 0.9 0.75 0.65 0.85 0.6 0.8 0.7 Mainstream 0.65 0.6 0.8 0.6 0.8 0.7 Alternative 0.850.8 0.65 0.6 0.55 0.6 0.6 approaches 0.7 0.8 0.75 0.7 0.550.6 0.55 0.55 0.7 0.75 0.7 0.65 SFC models 0.6 0.6 0.7 0.6 Goodwin model 0.6 0.65 0.7 Keen model 0.6 0.65 0.6 0.55 0.65 0.550.65

d 0.6 Ponzi ﬁnancing 0.5 0.55 0.65 0.65 0.6 Noise and Stock 0.7 0.55 Prices 0.6 0.4 0.65 Stabilizing 0.55 0.5 0.6 government 0.7 0.65 0.65 0.6 0.6 0.55 0.6 Great 0.3 0.5 Moderation 0.55 0.55 0.55 0.6 The Ultimate 0.5 0.55 0.5 Model 0.2 0.6 0.5 0.5 0.5 0.5 0.55 0.45 Conclusions 0.55 0.55 0.5 0.1 0.6 0.6 0.6 0.55 0.45 0.5 0.55 0.5 0.45 0.55 0.5 0.6 0.6 0.55 0 0.7 0.75 0.8 0.85 0.9 0.95 λ Introducing a government sector

Dynamical Systems and Following Keen (and echoing Minsky) we add discretionary Financial Instability government subsidied and taxation into the original system

M. R. Grasselli in the form

Mainstream G = G1 + G2 Alternative T = T + T approaches 1 2 SFC models where Goodwin model Keen model ˙ ˙ Ponzi financing G1 = η1(λ)Y G2 = η2(λ)G2 Noise and Stock Prices ˙ ˙ Stabilizing T1 = Θ1(π)Y T2 = Θ2(π)T2 government Great Moderation Defining g = G/Y and τ = T /Y , the net profit share is The Ultimate Model now Conclusions π = 1 − ω − rd + g − τ, and government debt evolves according to B˙ = rB + G − T . Differential equations - reduced system

Dynamical Systems and Notice that π does not depend on b, so that the last Financial Instability equation can be solved separately. M. R. Grasselli Observe further that we can write Mainstream π˙ = −ω˙ − rd˙ +g ˙ − τ˙ (3) Alternative approaches leading to the ﬁve-dimensional system SFC models Goodwin model Keen model ω˙ =ω [Φ(λ) − α] , Ponzi ﬁnancing Noise and Stock Prices λ˙ =λ [γ(π) − α − β] Stabilizing government Great g˙2 =g2 [η2(λ) − γ(π)] (4) Moderation The Ultimate Model τ˙2 =τ2 [Θ2(π) − γ(π)] Conclusions π˙ = − ω(Φ(λ) − α) − r(κ(π) − π) + (1 − ω − π)γ(π)

+ η1(λ) + g2η2(λ) − Θ2(π) − τ2Θ2(π) Good equilibrium

Dynamical Systems and The system (4) has a good equilibrium at Financial Instability ν(α + β + δ) − π η1(λ) − Θ1(π) M. R. Grasselli ω = 1 − π − r + α + β α + β Mainstream λ = Φ−1(α) Alternative approaches π = κ−1(ν(α + β + δ)) SFC models Goodwin model g 2 = τ 2 = 0 Keen model Ponzi ﬁnancing Noise and Stock and this is locally stable for a large range of parameters. Prices Stabilizing government The other variables then converge exponentially fast to Great Moderation The Ultimate ν(α + β + δ) − π η (λ) Model 1 d = , g 1 = Conclusions α + β α + β Θ (π) τ = 1 1 α + β Bad equilibria - destabilizing a stable crisis

Dynamical Systems and Financial Instability M. R. Grasselli Recall that π = 1 − ω − rd + g − τ.

Mainstream The system (4) has bad equilibria of the form Alternative approaches (ω, λ, g2, τ2, π) = (0, 0, 0, 0, −∞) SFC models Goodwin model (ω, λ, g2, τ2, π) = (0, 0, ±∞, 0, −∞) Keen model Ponzi ﬁnancing Noise and Stock Prices If g2(0) > 0, then any equilibria with π → −∞ is locally Stabilizing government unstable provided η (0) > r. Great 2 Moderation The Ultimate On the other hand, if g (0) < 0 (austerity), then these Model 2 Conclusions equilibria are all locally stable. Persistence results

Dynamical Systems and Financial Proposition 1: Assume g (0) > 0, then the system (4) is Instability 2 eπ-UWP if either M. R. Grasselli 1 λη1(λ) is bounded below as λ → 0, or Mainstream 2 Alternative η2(0) > r. approaches Proposition 2: Assume g (0) > 0 and τ (0) = 0, then the SFC models 2 2 Goodwin model system (4) is λ-UWP if either of the following three conditions Keen model Ponzi ﬁnancing is satisﬁed: Noise and Stock Prices Stabilizing 1 λη1(λ) is bounded below as λ → 0, or government Great Moderation 2 η2(0) > max{r, α + β}, or The Ultimate Model 3 r < η2(0) ≤ α + β and Conclusions −r(κ(x) − x) + (1 − x)γ(x) + η1(0) − Θ1(x) > 0 for γ(x) ∈ [η2(0), α + β]. Hopft bifurcation with respect to government spending.

Dynamical Systems and Financial Instability OMEGA M. R. Grasselli

Mainstream 0.692

Alternative approaches 0.69

SFC models Goodwin model 0.688 Keen model Ponzi ﬁnancing Noise and Stock 0.686 Prices Stabilizing government 0.684 Great Moderation The Ultimate Model 0.682 Conclusions 0.68 0.28 0.285 0.29 0.295 0.3 0.305 0.31 0.315 0.32 0.325 eta_max The Great Moderation in the U.S. - 1984 to 2007

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models Goodwin model Keen model Ponzi ﬁnancing Noise and Stock Prices Stabilizing government Great Moderation The Ultimate Model Conclusions

Figure: Grydaki and Bezemer (2013) Possible explanations

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream Real-sector causes: inventory management, labour market Alternative approaches changes, responses to oil shocks, external balances , etc. SFC models Financial-sector causes: credit accelerator models, ﬁnancial Goodwin model Keen model innovation, deregulation, better monetary policy, etc. Ponzi ﬁnancing Noise and Stock Prices Grydaki and Bezemer (2013): growth of debt in the real Stabilizing government sector. Great Moderation The Ultimate Model Conclusions Bank credit-to-GDP ratio in the U.S

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models Goodwin model Keen model Ponzi ﬁnancing Noise and Stock Prices Stabilizing government Great Moderation The Ultimate Model Conclusions

Figure: Grydaki and Bezemer (2013) Cumulative percentage point growth of excess credit growth, 1952-2008

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models Goodwin model Keen model Ponzi ﬁnancing Noise and Stock Prices Stabilizing government Great Moderation The Ultimate Model Conclusions

Figure: Grydaki and Bezemer (2013) Excess credit growth moderated output volatility during, but not before the Great Moderation

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models Goodwin model Keen model Ponzi ﬁnancing Noise and Stock Prices Stabilizing government Great Moderation The Ultimate Model Conclusions Figure: Grydaki and Bezemer (2013) Example 8: strongly moderated oscillations

Dynamical Systems and Financial

Instability ω = 0.9, λ = 0.91, d = 0.1, p = 0.01, Y = 100, κ'(π ) = 20 0 0 0 0 0 eq 12 3500 0.9 1 180 M. R. Grasselli

160 0.8 0.9 Mainstream 3000 10 Alternative 140 0.7 0.8 approaches 2500 8 120 SFC models 0.6 ω 0.7 Goodwin model λ 2000 Keen model Y 100 d p λ d Y Ponzi ﬁnancing 6 ω 0.5 0.6 p Noise and Stock 80 Prices 1500 Stabilizing 0.4 0.5 government 4 60 Great 1000 Moderation 0.3 0.4 The Ultimate 40 Model 2 500 0.2 0.3 Conclusions 20

0 0 0.1 0.2 0 0 10 20 30 40 50 60 70 80 90 100 time Example 9 (cont): Shilnikov bifurcation

Dynamical Systems and Financial

Instability ω = 0.9, λ = 0.91, d = 0.1, p = 0.01, Y = 100, κ'(π ) = 20 0 0 0 0 0 eq M. R. Grasselli

Mainstream

Alternative approaches 12 SFC models 10 Goodwin model Keen model 8 Ponzi ﬁnancing

Noise and Stock d 6 Prices Stabilizing 4 government

Great 2 Moderation 1 The Ultimate 0.95 Model 0 0.45 0.9 0.5 0.55 0.85 Conclusions 0.6 0.65 0.8 0.7 0.75 0.75 0.8 0.85 0.9 0.7 λ

ω Shortcomings of Goodwin and Keen models

Dynamical Systems and No independent speciﬁcation of consumption (and Financial Instability therefore savings) for households:

M. R. Grasselli C = W , Sh = 0 (Goodwin) Mainstream ˙ Alternative C = (1 − κ(π))Y , Sh = D = Πu − I (Keen) approaches SFC models Full capacity utilization. Goodwin model Keen model Everything that is produced is sold. Ponzi ﬁnancing Noise and Stock Prices No active market for equities. Stabilizing government Skott (1989) uses prices as an accommodating variable in Great Moderation The Ultimate the short run. Model Conclusions Chiarella, Flaschel and Franke (2005) propose a dynamics for inventory and expected sales. Grasselli and Nguyen (2013) provide a synthesis, including equities and Tobin’s portfolio choices. Concluding remarks

Dynamical Systems and Macroeconomics is too important to be left to Financial Instability macroeconomists. M. R. Grasselli Since Keynes’s death it has developed in two radically

Mainstream different approaches: 1 Alternative The dominant one has the appearance of mathematical approaches rigour (the SMD theorems notwithstanding), but is based SFC models on implausible assumptions, has poor fit to data in general, Conclusions and is disastrously wrong during crises. Finance plays a negligible role 2 The heterodox approach is grounded in history and institutional understanding, takes empirical work much more seriously, but is generally averse to mathematics. Finance plays a major role. It’s clear which approach should be embraced by mathematical finance “to boldly go where no man has gone before” ··· Qatlho!

Dynamical Systems and Financial Instability

M. R. Grasselli

Mainstream

Alternative approaches

SFC models

Conclusions