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
M. R. Grasselli
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
50
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