M. R. Grasselli. 9th World Congress of The Bachelier Finance Society New York City, July 18, 2016

Transcription

1 Keen Mathematics and Statistics, McMaster University and The Fields Institute Joint with B. Costa Lima (Morgan Stanley) and A. Nguyen Huu (CREST) 9th World Congress of The Bachelier Finance Society New York City, July 8, 26

2 Rational Keen Consider a representative agent solving sup E t β j t u(c j ) c j= for exogenously given (e t, d t ). The general solution for this problem is of the form p t = F t + B t where F t = β j [ E t dt+j u (e t+j + d t+j ) ] j= is the fundamental price and B t is a bubble term satisfying E t [B t+ ] = β B t ()

3 Consequences Keen B t for all t. Any nonzero rational bubble must start with B >. If T <, B t = for all t T, and this result is robust with respect to diverse information (Tirole 982). If T =, can exit in a myopic rational expectations equilibrium. Rational cannot exist in a fully dynamic REE with finitely many infinitely lived agents. They can exit in an overlapping generations s provided < r < g, where r is the asymptotic real interest rate and g is the rate of growth of the economy (Tirole 985).

4 Alternative s (Shiller, 984) Keen Consider a where sophisticated investors have a demand function (portion of shares) of the form Qt i = E t[r t+ ] α. (2) φ In addition, suppose there are noise traders who react to fads Y t through a demand function Q n t = Y t /p t. In equilibrium we have Q t + Yt p t =. Inserting this into (2) and solving recursively leads to p t = j= E t [d t+j ] + φe t [Y t +j ] ( + α + φ) j. (3) This is also with prices being not very forecastable.

5 Other sources of inefficiencies Keen Noise trader risk (DeLong, Shleifer, Summers and Waldmann 99): prices deviate from fundamentals due to uncertainty created by noise traders, who can earn higher expected returns than sophisticated investors. Limits of arbitrage (Shleifer and Vishny 997): fund managers leaving the market exactly when they are needed to restore fundamental value. No short-sales and diverse beliefs (Miller 977, Harrison and Kreps 978): pessimists stay on sidelines and optimists overbid Overconfidence (Scheinkman and Xiong 23): mean reverting confidence levels lead to prices that contain an option to re-sell the asset at a later time. These are all microeconomic s. What about macro?

6 Dynamic Stochastic General Equilibrium (DSGE) Keen Seeks to explain the aggregate economy using theories based on strong microeconomic foundations. Collective decisions of rational individuals over a range of variables for both present and future. All variables are assumed to be simultaneously in equilibrium. Equilibrium is only disrupted by exogenous shocks. 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.

7 SMD theorem: something is rotten in GE land Keen

8 Stock-Flow Consistent s Keen Stock-flow s emerged in the last decade as a common language for many heterodox schools of thought in economics. They consider both real and monetary factors simultaneously. Specify the balance sheet and transactions between sectors. Accommodate a number of behavioural assumptions in a way that is with the underlying accounting structure. Reject the RARE individual (representative agent with rational expectations) in favour of SAFE (sectoral average with flexible expectations) ling. See Godley and Lavoie (27) for the full framework.

9 Model - SFC matrix Keen Balance Sheet Households Firms Sum current capital Capital +pk pk Sum (net worth) Vf pk Transactions Consumption pc +pc Investment +pi pi Acct memo [GDP] [py ] Wages +W W Profits Π +Πu Sum Flow of Funds Capital +pi pi Sum Πu pi Change in Net Worth pi + ṗk pδk ṗk + p K Table: SFC table for the.

10 Model - Differential equations Define ω = wl py = w pa (wage share) Keen λ = l N = Y an It then follows that (employment rate) ω ω = ẇ w ṗ p ȧ a = Φ(λ, i, i e ) i α λ λ = ω α β δ ν In the original, all quantities were real (i.e divided by p), which is equivalent to setting i = i e =.

11 Example : Keen λ w =.8, λ = ω

12 Testing on OECD countries Keen Figure: Harvie (2)

13 Correcting Harvie (97 to 29) Keen Figure: Grasselli and Maheshwari (26, in progress)

14 SFC table for Keen (995) Keen Balance Sheet Households Firms Banks Sum current capital Deposits +D D Loans L +L Capital +pk pk Sum (net worth) Vh Vf pk Transactions Consumption pc +pc Investment +pi pi Acct memo [GDP] [py ] Wages +W W Interest on deposits +rd rd Interest on loans rl +rl Profits Π +Πu Sum Sh Sf pi Flow of Funds Deposits +Ḋ Ḋ Loans L + L Capital +pi pi Sum Sh Πu pi Change in Net Worth Sh (Sf + ṗk pδk) ṗk + p K Table: SFC table for the Keen.

15 Keen - Investment function Keen Assume now that new investment is given by K = κ(π)y δk where κ( ) is a nonlinear increasing function of profits π = ω rd. This leads to external through debt evolving according to Ḋ = κ(π)y πy The economy grows at a rate g(π) := Ẏ Y = κ(π) δ. ν

16 Keen - Differential Equations Keen Denote the debt ratio in the economy by d = D/Y, the can now be described by the following system ω = ω [Φ(λ) α] λ = λ [g(π) α β] (4) ḋ = κ(π) π dg(π)

17 Example 3: convergence to the good equilibrium in a Keen Keen ω λ Y d Y 8 x ω ω =.75, λ =.75, d =., Y = λ d ω Y λ d time Figure: Grasselli and Costa Lima (22)

18 Example 4: explosive debt in a Keen Y ω λ ω =.75, λ =.7, d =., Y = Y d ω λ Y d λ x d Keen ω time Figure: Grasselli and Costa Lima (22)

19 Basin of convergence for Keen 8 Keen 6 d λ..5 ω

20 Example 3 (continued): explosive debt in a Keen.9 ω =.75, λ =.7, d =. 9 Keen dd/dt λ d time

21 Corporate Debt share in the US Nonfinancial Business; Credit Market Instruments; Liability, Level/Gross Domestic Product Keen (Bil. of $/Bil. of $) Shaded areas indicate US recessions - 24 research.stlouisfed.org

22 Keen To introduce the destabilizing effect of purely speculative investment, we consider a modified version of the previous with L = pi + r L L κ L L + F Ḋ f = py W + r f D f κ L L + F Ḟ = Ψ(g(π))F where Ψ( ) is an increasing function of the growth rate of economic output g(ω, d) = κ(π) δ. ν

23 - Differential equations Keen Setting κ L = r L and defining c = r L b + (r L r f )d f and f = F /(py ), where d f = D f /(py ), the dynamical system becomes ω = ω [Φ(λ) α] λ = λ [g(π) α β] (5) ċ = r L κ(π) r f π c [g(π)] + (r L r f )f (6) ḟ = f [Ψ (g(π)) g(π)]

24 Example 4: effect of Keen ω ω =.95, λ =.9, d =, p =., Y = No Speculation Financing λ

25 Example 4 (continued): effect of Keen d Y ω =.95, λ =.9, d =, p =., Y = No Speculation Financing t 8 x t No Speculation Financing x d with Y with p t

26 Credit and Keen In Manias, Panics, and Crashes, Kindelberger and Aliber (2) state that most increases in the supply of credit do not lead to a mania - but nearly every mania has been associated with rapid growth in the supply of credit to a particular group of borrowers. Recall the Quantity Theory of Money equation MV = py, (7) where M is the money supply and V the velocity of circulation. In Werner (997), this is replaced by M R V R = py (8) M F V F = SQ F, (9) where R and F denote real and financial transactions respectively.

27 The monetary roots of and crashes Keen In Corsi and Sornette (22), this is through dm F t = µ F S t M F t dt + σ M M F t dw F t () ds t = µ S M F t S t dt + σ S S t dw S t, () which exhibits super-exponential behaviour. In our notation, the deterministic version of this is F = dm F = µ F SM F (2) dt ds dt = µ SM F S (3) and exhibits finite-time singularity (FTS).

28 Stock price dynamics Keen Instead of (), we consider a stock price process of the form ds t = r b dt + σdw t + jµ t dt dj t S t where J t is an inhomogenous Poisson process with intensity µ t = M(f (t)) and jump sizes distributed on (, ) with mean j. The interest rate for private debt is led as r t = r b + r p (t) where ρ r p (t) = (S t + ρ 2 ) ρ 3 for positive constants ρ, ρ 2, ρ 3.

29 Example 5: stock prices, explosive debt, zero speculation ω λ Keen p d 5 5 S t

30 Example 6: stock prices, explosive debt, explosive speculation Keen p d S t ω λ

31 Example 7: stock prices, finite debt, finite speculation Keen p d ω λ 3 S t

32 .7 Stability map Keen d Stability map for ω =.8, p =., S =, T = 5, dt =.5, # of simulations = λ

33 Concluding remarks Keen We provided a stock-flow for real-financial interactions as an extension of the -Keen labour, investment, and debt dynamics. The ling framework is an alternative to the dominant microfounded DSGE paradigm in s. It incorporates insights from endogenous money theory, sectoral balances, and Minskian financial instability. Opens up new avenues for the application of modern dynamical systems techniques to economics.

34 Keen Thank you!