Lecture 5: Modelling the Dual Price Hypothesis Honours Finance (Advanced Topics in Finance: Nonlinear Analysis) Lecture 5: Modelling the Dual Price Hypothesis

Recap Last lecture we considered Goodwin’s predator-prey model of cyclical growth as a foundation for the “Dual Price Level” hypothesis This lecture we’ll Analyse the model in more detail Add private debt, Prices and a Government sector to develop a series of models of the Dual Price Level hypothesis Simulate the models and discuss the results © Steve Keen 2005, www.debunking-economics.com

Foundations The basic Goodwin model is Properties of this simple model illustrate why nonlinear systems are so different to linear ones Like predator-prey system, equilibrium is neutral: model neither converges to nor diverges from equilibrium; Deviations above & below equilibrium don’t “cancel each other out”: equilibrium is NOT the average Property not a result simply of “quirky” functions (like Phillips curve) but nature of nonlinear systems E.g., simple predator-prey system has just 4 constants and 2 variables: no nonlinear functions… © Steve Keen 2005, www.debunking-economics.com

Foundations Yet equilibrium of system is not average of system: Divergence gets much more extreme with more complex models So time & history matter: can’t just treat ups & downs of trade cycle as on average equal to equilibrium! Reason: asymmetries can apply because of nonlinear forces System can go much further in one direction than other © Steve Keen 2005, www.debunking-economics.com

Foundations Asymmetry increases as more realism brought into model Basic model Only nonlinearity is Phillips curve Capitalists assumed to invest all profits But unrealistic: implies capitalists destroy capital if profit falls below zero Investment a function of (expectations of) profit Keynes: investors extrapolate existing conditions forward Expectations low during times of low profit, high during times of high Nonlinear investment function advisable © Steve Keen 2005, www.debunking-economics.com

Nonlinear Investment Function Replacing linear with nonlinear investment function yields Many possible forms, but basic property that d(k[p])/dt an increasing function of p. We’ll use © Steve Keen 2005, www.debunking-economics.com

Nonlinear investment function Nonlinear investment function means desired (and executed) investment during boom exceeds profits desired (and executed) investment during slump less than profits © Steve Keen 2005, www.debunking-economics.com

Nonlinear Investment Function Nonlinear investment function makes little change to nature of basic model: Still closed cycle But asymmetry much more obvious: © Steve Keen 2005, www.debunking-economics.com

(1) Fisher = Goodwin with debt Basic Goodwin model has no debt finance Debt introduced as stock buffer to firms investment plans Aggregate position supported by, of all people, Fama & French: “Debt seems to be the residual variable in financing decisions. Investment increases debt, and higher earnings tend to reduce debt.” (1997) “The source of financing most correlated with investment is long-term debt… These correlations confirm the impression that debt plays a key role in accommodating year-by-year variation in investment.” (1998) © Steve Keen 2005, www.debunking-economics.com

A Banking Sector Banking sector added by assuming all debt finances investment: Work out new dynamics of debt/output ratio: Product rule: Substitute: Debt ratio grows if investment exceeds profits… Falls as growth rises Complex interaction because growth encourages investment… © Steve Keen 2005, www.debunking-economics.com

A Banking Sector This results in the following three-dimensional system: Wages share of output Employment ratio Debt to output ratio What are systemic dynamics now? 3 dimensional system… Basic insight of Poincare; modern version from Robert May: “And Three Means Chaos” Continuous time dynamic system with only 2 variables must either… © Steve Keen 2005, www.debunking-economics.com

A bit of chaos Converge to equilibrium; or Explode to infinite cycles; or Follow a fixed orbit; or Converge to a limit cycle Reason: basic property of ODE models Solution gives a time path and initial conditions Each solution & set of initial conditions unique No two time paths can have the same initial conditions No two time paths can intersect 2 dimensional system exists in a plane Can’t have one time path cross another; so system either spirals in to equilibrium, blasts out indefinitely, or converges to closed path © Steve Keen 2005, www.debunking-economics.com

A bit of chaos Good illustration of this is Van Der Pol equation for sustained signal in an electronic circuit Also nice example of eigenvalue analysis of stability… Currents with passive component (resistor) have damped oscillations: cycles die out when external current removed: Currents with active component (transistor) replacing resistor have sustained oscillations: cycles continue even when external current removed: © Steve Keen 2005, www.debunking-economics.com

A bit of chaos Can also be written as 2nd order ODE by using I=dV/dt Exercise: convert to pair of coupled 1st order equations… OK, a hint: define Y0=V, Y1=dV/dt, Y2=d2Vdt2… Coupled 1st order equations are Work out equilibrium & stability… Substitute terms in Y0 & Y1 only: © Steve Keen 2005, www.debunking-economics.com

A bit of chaos From 1st equation, Y1=0 Put into second; only valid for all t if Y0=0 So equilibrium is (0,0). Is it stable? Work it out! Express model as matrix equation Work out Jacobian Calculate value at equilibrium If dominant eigenvalue < 0, it is stable… © Steve Keen 2005, www.debunking-economics.com

A bit of chaos Next, create matrix of partial derivatives if f1 & f2 with respect to Y0 and Y1: Now evaluate this at the equilibrium (0,0): © Steve Keen 2005, www.debunking-economics.com

A bit of chaos We get a simple matrix: Does it describe a stable or unstable system? It’s now the matrix expression of a simple second order linear ODE. Describes whether small divergence from equilibrium is damped or amplified Work out its eigenvalues: Express as matrix equation: Work with these two bits… Assume solution of form Then © Steve Keen 2005, www.debunking-economics.com

A bit of chaos Rearrange using matrix rules: Only possible for non-trivial v if determinant is zero… © Steve Keen 2005, www.debunking-economics.com

A bit of chaos Determinant is: Its roots are: Equilibrium unstable for all values of m Near equilibrium initial values deviate from equilibrium Far from equilibrium ones converge to the limit cycle © Steve Keen 2005, www.debunking-economics.com

A bit of chaos Phase plot shows instability and convergence to limit cycle 3rd order version of model (and experiments with vacuum tube circuits) gave initial instance of chaos in 1920s Basic point here: 2 dimensional continuous systems cannot display chaos Topography limits dynamic paths to convergence, divergence, or limit cycle Can’t “cut through” space defined by limit cycle… © Steve Keen 2005, www.debunking-economics.com

“And three means chaos” Third dimension means one time path can move anywhere in 3D volume without ever intersecting a different time path Hence Lorenz’s complex dynamics So basic Goodwin model can’t display chaos But introduction of debt gives 3rd dimension… Basics of chaos can (but not necessarily must) apply: Sensitive dependence on initial conditions Large scale divergence of time paths with very similar initial conditions Complex (non limit cycle or aperiodic) cycles © Steve Keen 2005, www.debunking-economics.com

A Banking Sector Without debt, the model gave us the following patterns: © Steve Keen 2005, www.debunking-economics.com

Analysing Goodwin’s model of cyclical growth Eigenvalue analysis of Goodwin model: Function: Rate of growth equals labour productivity plus population growth Non-trivial equilbrium: Wage demands equal rate of growth of productivity © Steve Keen 2005, www.debunking-economics.com

Analysing Goodwin’s model of cyclical growth Calculating the equilibrium (given parameter values): Next, what are the stability properties of this non-trivial equilibrium? Calculate Jacobian of model at equilibrium: © Steve Keen 2005, www.debunking-economics.com

These generate complex numbers Analysing Goodwin’s model of cyclical growth Jacobian at equilibrium: These generate complex numbers These two obviously zero Eigenvalues have zero real part Neutral equilibrium, closed limit cycle Any initial value generates fixed orbit around equilibrium © Steve Keen 2005, www.debunking-economics.com

Properties of debt model Equilibrium employment rate exists Coefficients in Phillips curve function Equilibrium profit share: Coefficients in investment function Investment/output ratio Equilibrium debt level: Profit share Growth rate © Steve Keen 2005, www.debunking-economics.com

A Banking Sector Simulating the model near its equilibrium, we get: © Steve Keen 2005, www.debunking-economics.com

A Banking Sector For a more dynamic view… All variables spiral towards equilibrium But away from equilibrium… © Steve Keen 2005, www.debunking-economics.com

“And three means chaos” Stability properties of Symbolic answer would be roots of 3rd order polynomial… Messy! For the record, roots of ax3+bx2+cx+d=0 are: 4th order (when government introduced) far worse; and impossible for 5th & above anyway: instead calculate numerical eigenvalues, Lyapunov exponents, etc.… © Steve Keen 2005, www.debunking-economics.com

Finance & Economic Breakdown Numerical calculations give equilibrium of system as: System stable around this equilibrium, but Debt value implausible in real economy: equilibrium value equates to capitalist savings being 3.6 times GDP Echoes of Fisher: away from this equilibrium, “instability ensues”… © Steve Keen 2005, www.debunking-economics.com

A Banking Sector With different initial conditions: © Steve Keen 2005, www.debunking-economics.com

A Banking Sector Debt cycles now rise over time: Model replicates Fisher’s insight: Stable near equilibrium Unstable further away… © Steve Keen 2005, www.debunking-economics.com

Stable equilibrium... © Steve Keen 2005, www.debunking-economics.com

Unstable “far” from equilibrium © Steve Keen 2005, www.debunking-economics.com

Interesting properties of model Equilibrium wage share a negative linear function of interest rate: Slope depends on interest rate; narrow feasible range, but result differs from neoclassical income distribution theory consistent with Kaleckian effect far more pronounced out of equilibrium when d can be very large © Steve Keen 2005, www.debunking-economics.com

Interesting properties of model Equilibrium: Now includes equilibrium debt to output ratio Investment - profit ratio Rate of growth © Steve Keen 2005, www.debunking-economics.com

A Stable Equilibrium… within limits Jacobian is: With equilibrium conditions: Where we is a function of the rate of interest © Steve Keen 2005, www.debunking-economics.com

A Stable Equilibrium… within limits As a function of r, the Jacobian is... A mess! But the essential point is The real part of the dominant eigenvalue is negative for almost all values of r Therefore the equilibrium of the model is stable, and The linearised version is stable However outside the equilibrium, the nonlinear forces dominate… © Steve Keen 2005, www.debunking-economics.com

Of models and theories To relate Dual Price Level Hypothesis model to the theory: Fisher/Keynes/Minsky analysis argues economy fundamentally cyclical corporate debt crucial to understanding capitalism deflation exacerbates impact of debt government counter-cyclical spending can counteract tendency to corporate bankruptcy during slump after debt-financed boom All these features emerge in these models © Steve Keen 2005, www.debunking-economics.com

Cyclical Economy Goodwin model has neutral equilibrium No tendency towards or away from equilibrium, hence limit cycle behaviour generalisations (endogenous technical change, variable capacity utilisation) make model unstable equilibrium a repeller Addition of debt changes nature of model: © Steve Keen 2005, www.debunking-economics.com

Debt Accumulation and Crisis Combination of Nonlinear investment function Capitalists invest more than profit during booms, less than profit during slumps Finance sector which lends at interest to finance investment Generates system which points out fundamental asymmetry in capitalism firms commit to debt during booms have to pay back debt during slumps cyclical tendency towards debt accumulation and debt-induced Depression © Steve Keen 2005, www.debunking-economics.com

Debt Accumulation and Crisis Model “bifurcates” Close to equilibrium rate of debt accumulation tapers till debt/output ratio a constant income distribution and employment also converge to equilibrium values Far from equilibrium “exponential” component of debt relation overwhelms economy a debt-induced depression in practical terms, no escape without debt moratoria, etc. Declining workers’ share of output (Minsky 1986) an emergent property of model © Steve Keen 2005, www.debunking-economics.com

Inflation and Deflation Two price levels (next lecture) commodity prices, set by markup on prime cost equipment prices, diverge from commodity prices rise relatively during boom fall relatively during slump Complex price dynamics and distributional effects General tendency towards deflation Deflationary impact of technical progress on wages share Tendency for employment to fall below level at which money wage rise exceeds growth in productivity Surges of (cost-push) inflation during times of high employment © Steve Keen 2005, www.debunking-economics.com

Inflation and Deflation At low interest rate High profit share causes high investment output and debt grow High level of output causes high employment High employment causes high wage demands profit share eroded BUT wage-push inflation reduces real debt burden System cycles as inflation erodes accumulation of debt A chaotic limit cycle in wages share, employment, debt and prices © Steve Keen 2005, www.debunking-economics.com

Inflation and Deflation At high(er) interest rate Faster growth of debt obligations reduces long run rate of growth Lower long run rate of growth amplifies tendency to deflation Deflation during slumps dramatically increases accumulation of debt Model collapses in high debt/deflation/unemployment spiral as per Fisher 1933 Much longer cycles similar to period of financial crises (10-20 years) shorter cycles (2-4 years) may be inventory (Metzler), etc. © Steve Keen 2005, www.debunking-economics.com

Inflation, Distribution, Growth and Crisis Capitalists beneficiaries of inflation inflation reduces real debt burden Bankers/rentiers losers/beneficiaries inflation erodes real value of debt; but avoidance of deflation means borrowers remain solvent Workers losers/beneficiaries erodes real value of wage; but wage can be restored via wage bargains during booms avoidance of depression maintains employment Inflation improves growth by sidestepping debt-deflation Matches 19th Century trade cycle (minimal government): © Steve Keen 2005, www.debunking-economics.com

Inflation, Distribution, Growth and Crisis Procyclical prices Frequent wage falls Financial crises roughly every 20 years © Steve Keen 2005, www.debunking-economics.com

Government and Reflation Big Government crucial difference post-WWII Fisher, Keynes & Minsky concur Reflation only feasible means of escape from Depression Keynes and Minsky Government spending provides means to reflation during depression financial basis for counter-cyclical stabilisation policy Government counter-cyclical spending stabilises economy, not by eliminating cycles but by preventing a debt-deflation Modelling this firstly without prices... © Steve Keen 2005, www.debunking-economics.com

Stabilising an unstable economy Government spending rises if unemployment exceeds 5% falls if less than 5% © Steve Keen 2005, www.debunking-economics.com

Government and Reflation Government spending a function of employment: Add to real model to get: © Steve Keen 2005, www.debunking-economics.com

Government and Reflation Generates four-dimensional system: Wages share of output Employment ratio Debt to output ratio Subsidy to output ratio Profit is now net of government: © Steve Keen 2005, www.debunking-economics.com

Government and Reflation Generates system with cycles but not instability: © Steve Keen 2005, www.debunking-economics.com

Stabilising an unstable economy Behaviour of model supports Minsky’s contention: Government stabilises economy But system remains cyclical Stability means not the absence of cycles, but The prevention of systemic collapse: A “strange attractor”? Technically, no: different initial conditions converge But an unstable equilibrium © Steve Keen 2005, www.debunking-economics.com

Government and Reflation A “chaotic limit cycle” at the heart of sustained cyclical behaviour in the long run The “long run” does not have to be an equilibrium position (c.f. Keynes “in the long run we are all dead...”) © Steve Keen 2005, www.debunking-economics.com

Leads to rising government spending (and vice versa) Government and Reflation Government spending counteracts tendency of private sector to accumulate excessive debt A homeostatic stabiliser: (with a phase difference) Leads to rising government spending (and vice versa) Falling employment © Steve Keen 2005, www.debunking-economics.com

Government and Reflation “Real” model with government confirms Minsky on “Big Government” Anti-cyclical spending and taxation of government enables debts to be repaid Renewal of cycle once debt levels reduced What about “the full Monty”?: Cyclical economy + Debt Prices Government A more complex picture still… but first, analysing the “simpler” one… © Steve Keen 2005, www.debunking-economics.com

Stabilising an unstable economy Mathematically, adding government sector to the model converts it from one with locally stable equilibrium but globally unstable behaviour to one with locally unstable equilibrium but globally stable behaviour public debt counterbalances private increased “taxation” during booms reduces size of booms increased “subsidies” during slump finances repayment of private debt © Steve Keen 2005, www.debunking-economics.com

Summing Up How does FIH & simple dynamic model compare to US data? 1929 private debt/output ratio 60% (Fisher 1932) Private non-financial debt/output ratio now 160+% 1929 employment high, economy booming, inflation low 2000 employment high, economy booming, inflation low… Pattern of US debt to fiat money, debt to output ratio eerily resembles 3 dimensional model debt dynamics: © Steve Keen 2005, www.debunking-economics.com

Summing Up Cyclical “ratcheting up” of debt over time Debt incurred during boom Repaid during slump with less than expected cash flows © Steve Keen 2005, www.debunking-economics.com

Next Lecture Introducing prices and time lags Overall systemic dynamics Future development Then introduction to econophysics… © Steve Keen 2005, www.debunking-economics.com