fragility M. R. Grasselli Quantitative Finance Seminar Series - Fields Institute, October 26, 2011

Transcription

1 Keen Sharcnet Chair in Financial Mathematics Mathematics and Statistics - McMaster University Joint work with B. Costa Lima Quantitative Finance Seminar Series - Fields Institute, October 26, 2011

2 Outline 1 Dynamic General Equilibrium views Minskyian views 2 3 Keen 4 5 Keen

3 Dynamic General Equilibrium views Seeks to explain the aggregate economy using theories based on strong microeconomic foundations. Dynamic General Equilibrium views Minskyian views Keen

4 Dynamic General Equilibrium views 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. Dynamic General Equilibrium views Minskyian views Keen

5 Dynamic General Equilibrium views Dynamic General Equilibrium views Minskyian views 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. Keen

6 Dynamic General Equilibrium views Dynamic General Equilibrium views Minskyian views 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. The only way the economy can be in disequilibrium at any point in time is through decisions based on wrong information.

7 Dynamic General Equilibrium views Dynamic General Equilibrium views Minskyian views 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. 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.

8 Dynamic General Equilibrium views Dynamic General Equilibrium views Minskyian views 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. 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.

9 Voices of discontent Dynamic General Equilibrium views Minskyian views M. Morishima (1984): If economists successfully devise a correct general equilibrium (...) should it lack the institutional backing to realize an equilibrium solution, then [it] will amount to no more than a utopian state of affairs which bears no relation whatsoever to the real economy. Keen

10 Voices of discontent Dynamic General Equilibrium views Minskyian views Keen M. Morishima (1984): If economists successfully devise a correct general equilibrium (...) should it lack the institutional backing to realize an equilibrium solution, then [it] will amount to no more than a utopian state of affairs which bears no relation whatsoever to the real economy. A. Kirman (1989): [DSGE is] empty in the sense that one cannot expect it to house the elements of a scientific theory, one capable of producing empirically falsifiable propositions.

11 Voices of discontent Dynamic General Equilibrium views Minskyian views Keen M. Morishima (1984): If economists successfully devise a correct general equilibrium (...) should it lack the institutional backing to realize an equilibrium solution, then [it] will amount to no more than a utopian state of affairs which bears no relation whatsoever to the real economy. A. Kirman (1989): [DSGE is] empty in the sense that one cannot expect it to house the elements of a scientific theory, one capable of producing empirically falsifiable propositions. K. Arrow (1986): In the aggregate, the hypothesis of rational behavior has in general no implications.

12 Voices of discontent Dynamic General Equilibrium views Minskyian views Keen M. Morishima (1984): If economists successfully devise a correct general equilibrium (...) should it lack the institutional backing to realize an equilibrium solution, then [it] will amount to no more than a utopian state of affairs which bears no relation whatsoever to the real economy. A. Kirman (1989): [DSGE is] empty in the sense that one cannot expect it to house the elements of a scientific theory, one capable of producing empirically falsifiable propositions. K. Arrow (1986): In the aggregate, the hypothesis of rational behavior has in general no implications. R. Solow (2006): Maybe there is in human nature a deep-seated perverse pleasure in adopting and defending a wholly counterintuitive doctrine that leaves the uninitiated peasant wondering what planet he or she is on.

13 Minsky s alternative interpretation of Keynes Neoclassical economics is based on barter paradigm: money is convenient to eliminate the double coincidence of wants. Dynamic General Equilibrium views Minskyian views Keen

14 Minsky s alternative interpretation of Keynes Dynamic General Equilibrium views Minskyian views Neoclassical economics is based on barter paradigm: money is convenient to eliminate the double coincidence of wants. In a modern economy, firms make complex portfolios decisions: which assets to hold and how to fund them. Keen

15 Minsky s alternative interpretation of Keynes Dynamic General Equilibrium views Minskyian views Neoclassical economics is based on barter paradigm: money is convenient to eliminate the double coincidence of wants. In a modern economy, firms make complex portfolios decisions: which assets to hold and how to fund them. Financial institutions determine the way funds are available for ownership of capital and production. Keen

16 Minsky s alternative interpretation of Keynes Dynamic General Equilibrium views Minskyian views Keen Neoclassical economics is based on barter paradigm: money is convenient to eliminate the double coincidence of wants. In a modern economy, firms make complex portfolios decisions: which assets to hold and how to fund them. Financial institutions determine the way funds are available for ownership of capital and production. Uncertainty in valuation of cash flows (assets) and credit risk (liabilities) drive fluctuations in real demand and investment.

17 Minsky s alternative interpretation of Keynes Dynamic General Equilibrium views Minskyian views Keen Neoclassical economics is based on barter paradigm: money is convenient to eliminate the double coincidence of wants. In a modern economy, firms make complex portfolios decisions: which assets to hold and how to fund them. Financial institutions determine the way funds are available for ownership of capital and production. Uncertainty in valuation of cash flows (assets) and credit risk (liabilities) drive fluctuations in real demand and investment. Economy is fundamentally cyclical, with each state (boom, crisis, deflation, stagnation, expansion and recovery) containing the elements leading to the next in an identifiable manner.

18 Minsky s Financial Instability Hypothesis Start when the economy is doing well but firms and banks are conservative. Dynamic General Equilibrium views Minskyian views Keen

19 Minsky s Financial Instability Hypothesis Start when the economy is doing well but firms and banks are conservative. Most projects succeed - Existing debt is easily validated: it pays to lever. Dynamic General Equilibrium views Minskyian views Keen

20 Minsky s Financial Instability Hypothesis Dynamic General Equilibrium views Minskyian views Start when the economy is doing well but firms and banks are conservative. Most projects succeed - Existing debt is easily validated: it pays to lever. Revised valuation of cash flows, exponential growth in credit, investment and s. Keen

21 Minsky s Financial Instability Hypothesis Dynamic General Equilibrium views Minskyian views Start when the economy is doing well but firms and banks are conservative. Most projects succeed - Existing debt is easily validated: it pays to lever. Revised valuation of cash flows, exponential growth in credit, investment and s. Highly liquid, low-yielding instruments are devalued, rise in corresponding interest rate. Keen

22 Minsky s Financial Instability Hypothesis Dynamic General Equilibrium views Minskyian views Keen Start when the economy is doing well but firms and banks are conservative. Most projects succeed - Existing debt is easily validated: it pays to lever. Revised valuation of cash flows, exponential growth in credit, investment and s. Highly liquid, low-yielding instruments are devalued, rise in corresponding interest rate. Beginning of euphoric economy : increased debt to equity ratios, development of financier.

23 Minsky s Financial Instability Hypothesis Dynamic General Equilibrium views Minskyian views Keen Start when the economy is doing well but firms and banks are conservative. Most projects succeed - Existing debt is easily validated: it pays to lever. Revised valuation of cash flows, exponential growth in credit, investment and s. Highly liquid, low-yielding instruments are devalued, rise in corresponding interest rate. Beginning of euphoric economy : increased debt to equity ratios, development of financier. Viability of business activity is eventually compromised.

24 Minsky s Financial Instability Hypothesis Dynamic General Equilibrium views Minskyian views Keen Start when the economy is doing well but firms and banks are conservative. Most projects succeed - Existing debt is easily validated: it pays to lever. Revised valuation of cash flows, exponential growth in credit, investment and s. Highly liquid, low-yielding instruments are devalued, rise in corresponding interest rate. Beginning of euphoric economy : increased debt to equity ratios, development of financier. Viability of business activity is eventually compromised. financiers have to sell assets, liquidity dries out, asset market is flooded.

25 Minsky s Financial Instability Hypothesis Dynamic General Equilibrium views Minskyian views Keen Start when the economy is doing well but firms and banks are conservative. Most projects succeed - Existing debt is easily validated: it pays to lever. Revised valuation of cash flows, exponential growth in credit, investment and s. Highly liquid, low-yielding instruments are devalued, rise in corresponding interest rate. Beginning of euphoric economy : increased debt to equity ratios, development of financier. Viability of business activity is eventually compromised. financiers have to sell assets, liquidity dries out, asset market is flooded. Euphoria becomes a panic.

26 Minsky s Financial Instability Hypothesis Dynamic General Equilibrium views Minskyian views Keen Start when the economy is doing well but firms and banks are conservative. Most projects succeed - Existing debt is easily validated: it pays to lever. Revised valuation of cash flows, exponential growth in credit, investment and s. Highly liquid, low-yielding instruments are devalued, rise in corresponding interest rate. Beginning of euphoric economy : increased debt to equity ratios, development of financier. Viability of business activity is eventually compromised. financiers have to sell assets, liquidity dries out, asset market is flooded. Euphoria becomes a panic. Stability - or tranquility - in a world with a cyclical past and capitalist institutions is destabilizing.

27 Model (1967) - Assumptions Assume that N(t) = N 0 e βt a(t) = a 0 e αt Y (t) = νk(t) = a(t)l(t) (total labour force) (productivity per worker) (total yearly output) Keen where K is the total stock of capital and L is the employed population.

28 Model (1967) - Assumptions Assume that N(t) = N 0 e βt a(t) = a 0 e αt Y (t) = νk(t) = a(t)l(t) (total labour force) (productivity per worker) (total yearly output) Keen where K is the total stock of capital and L is the employed population. Assume further that ẇ = Φ(λ)w K = (Y wl) δk (Phillips curve) (Say s Law)

29 Model - Differential equations Define ω = wl Y = w a (wage share) λ = L N = Y an (employment rate) Keen

30 Model - Differential equations Keen Define ω = wl Y = w a λ = L N = Y an It then follows that (wage share) (employment rate) ω = ω(φ(λ) α) (1) ( ) 1 ω λ = λ α β δ (2) ν

31 Model - Properties If we take Φ to be linear, (1) reduces to the Lotka-Volterra equations for the predator-prey. Keen

32 Model - Properties If we take Φ to be linear, (1) reduces to the Lotka-Volterra equations for the predator-prey. To ensure λ (0, 1) we take Φ(λ) = φ 1 (1 λ) φ 0 Keen

33 Model - Properties Keen If we take Φ to be linear, (1) reduces to the Lotka-Volterra equations for the predator-prey. To ensure λ (0, 1) we take Provided Φ(λ) = φ 1 (1 λ) φ 0 1 ν α β δ > 0 α + φ 0 ρ 1 > 0 (3) are satisfied, the trivial equilibrium (0, 0) is a saddle point.

34 Model - Properties Keen If we take Φ to be linear, (1) reduces to the Lotka-Volterra equations for the predator-prey. To ensure λ (0, 1) we take Provided Φ(λ) = φ 1 (1 λ) φ 0 1 ν α β δ > 0 α + φ 0 ρ 1 > 0 (3) are satisfied, the trivial equilibrium (0, 0) is a saddle point. Moreover the only other equilibrium ( ) (ω, λ) = is non-hyperbolic. 1 ν(α + β + δ), 1 φ 1 α + φ 0 (4)

35 Example 1: Keen

36 Example 1 (continued): ω 0 = 0.8, λ 0 = Keen GDP 3000 ω time 0.88

37 Model - Extensions, structural instability, and empirical tests Desai 1972: Inflation leads to a stable equilibrium. Keen

38 Model - Extensions, structural instability, and empirical tests Desai 1972: Inflation leads to a stable equilibrium. Ploeg 1985: CES production function leads to stable equilibrium. Keen

39 Model - Extensions, structural instability, and empirical tests Desai 1972: Inflation leads to a stable equilibrium. Ploeg 1985: CES production function leads to stable equilibrium. 1991: Pro-cyclical productivity growth leads to explosive oscillations. Keen

40 Model - Extensions, structural instability, and empirical tests Keen Desai 1972: Inflation leads to a stable equilibrium. Ploeg 1985: CES production function leads to stable equilibrium. 1991: Pro-cyclical productivity growth leads to explosive oscillations. Solow 1990: US post-war data shows three sub-cycles with a bare hint of a single large clockwise sweep in the (ω, λ) plot.

41 Model - Extensions, structural instability, and empirical tests Keen Desai 1972: Inflation leads to a stable equilibrium. Ploeg 1985: CES production function leads to stable equilibrium. 1991: Pro-cyclical productivity growth leads to explosive oscillations. Solow 1990: US post-war data shows three sub-cycles with a bare hint of a single large clockwise sweep in the (ω, λ) plot. Harview 2000: Data from other OECD confirms the same qualitative features and shows unsatisfactory quantitative estimations.

42 Testing on OECD countries Keen

43 Introducing a sector (Keen 1995) Assume now that new investment is given by K = κ(1 ω rd)y δk (5) where κ( ) is an increasing function the net profit share π = 1 ω rd. Keen

44 Introducing a sector (Keen 1995) Keen Assume now that new investment is given by K = κ(1 ω rd)y δk (5) where κ( ) is an increasing function the net profit share π = 1 ω rd. This leads to external through debt evolving according to Ḋ = κ(1 ω rd)y (1 ω rd)y

45 Introducing a sector (Keen 1995) Keen Assume now that new investment is given by K = κ(1 ω rd)y δk (5) where κ( ) is an increasing function the net profit share π = 1 ω rd. This leads to external through debt evolving according to We take for constants Ḋ = κ(1 ω rd)y (1 ω rd)y κ(x) = κ 0 + κ 1 e κ 2x, κ 0 < ν(α + β + δ), κ 1 > 0, κ 2 > 0. (6)

46 Keen - Differential Equations Keen Denote the debt ratio in the economy by d = D/Y, the can now be described by the following system [ ] φ 1 ω = ω (1 λ) 2 (α + φ 0) [ ] κ(1 ω rd) λ = λ α β δ (7) ν [ ] κ(1 ω rd) ḋ = d r + δ + κ(1 ω rd) (1 ω) ν

47 Keen - good equilibrium If we define π 1 = κ 1 (ν(α + β + δ)) = 1 ( ) ν(α + β + δ) κ0 log κ 2 we see that one possible equilibrium for (7) is κ 1 Keen ω 1 = 1 π 1 r ν(α + β + δ) π 1 α + β λ 1 = 1 φ 1 α + φ 0 d 1 = ν(α + β + δ) π 1 α + β (8)

48 Keen - Irrelevant equilibria Other equilibrium points are given by (ω 2, λ 2, d 2 ) = (0, 0, d 2 ) (9) where d 2 is any solution of the equation [ ] κ(1 rd) d r + δ + κ(1 rd) 1 = 0 ν Keen

49 Keen - Irrelevant equilibria Keen Other equilibrium points are given by (ω 2, λ 2, d 2 ) = (0, 0, d 2 ) (9) where d 2 is any solution of the equation [ ] κ(1 rd) d r + δ + κ(1 rd) 1 = 0 ν Another set of equilibrium points are provided 1 rd 1 = π 1, that is (ω 3, λ 3, d 3 ) = (0, λ, d 1 ) (10) 1 r ν(α + β + δ) κ 1 (ν(α + β + δ)) α + β = κ 1 (ν(α+β+δ))

50 Example 2 : convergent Keen Keen

51 Example 2 (continued): convergent Keen Keen GDP 8 x ω ω 0 = 0.75, λ 0 = 0.75, d 0 = time

52 Keen - Explosive debt Keen If we rewrite the system with the change of variables u = 1/d, we obtain [ ] φ 1 ω = ω (1 λ) 2 (α + φ 0) [ ] κ(1 ω r/u) λ = λ α β δ (11) ν [ ] κ(1 ω r/u) u = u r δ u 2 [κ(1 ω r/u) (1 ω)] ν

53 Keen - Explosive debt Keen If we rewrite the system with the change of variables u = 1/d, we obtain [ ] φ 1 ω = ω (1 λ) 2 (α + φ 0) [ ] κ(1 ω r/u) λ = λ α β δ (11) ν [ ] κ(1 ω r/u) u = u r δ u 2 [κ(1 ω r/u) (1 ω)] ν We now see that (0, 0, 0) is an equilibrium of (11) corresponding to the point for the original system. (ω 4, λ 4, d 4 ) = (0, 0, + ) (12)

54 Example 3: divergent Keen Keen

55 Example 3: divergent Keen Keen GDP ω ω 0 = 0.75, λ 0 = 0.7, d 0 = time 0

56 Example 3 (continued): divergent Keen ω 0 = 0.75, λ 0 = 0.7, d 0 = Keen GDP ω time

57 Data detour: debt Keen

58 Data detour: debt and employment Keen

59 Keen - Local stability Analyzing the Jacobian of (7) and (11) we obtain the following conclusions. Keen

60 Keen - Local stability Analyzing the Jacobian of (7) and (11) we obtain the following conclusions. The good equilibrium (ω 1, λ 1, d 1 ) is stable if and only if [ ( ) ] r κ π1 (π 1 ) ν δ (α + β) > 0. Keen

61 Keen - Local stability Keen Analyzing the Jacobian of (7) and (11) we obtain the following conclusions. The good equilibrium (ω 1, λ 1, d 1 ) is stable if and only if [ ( ) ] r κ π1 (π 1 ) ν δ (α + β) > 0. The second equilibrium (0, 0, d 2 ) is unstable whenever κ(1 rd 2 ) > κ(1 ω 1 rd 1 )

62 Keen - Local stability Keen Analyzing the Jacobian of (7) and (11) we obtain the following conclusions. The good equilibrium (ω 1, λ 1, d 1 ) is stable if and only if [ ( ) ] r κ π1 (π 1 ) ν δ (α + β) > 0. The second equilibrium (0, 0, d 2 ) is unstable whenever κ(1 rd 2 ) > κ(1 ω 1 rd 1 ) The third equilibrium (0, λ, d 1 ) is structurally unstable.

63 Keen - Local stability Keen Analyzing the Jacobian of (7) and (11) we obtain the following conclusions. The good equilibrium (ω 1, λ 1, d 1 ) is stable if and only if [ ( ) ] r κ π1 (π 1 ) ν δ (α + β) > 0. The second equilibrium (0, 0, d 2 ) is unstable whenever κ(1 rd 2 ) > κ(1 ω 1 rd 1 ) The third equilibrium (0, λ, d 1 ) is structurally unstable. The point (0, 0, 0) is a stable equilibrium for (11) if and only if κ 0 < min(α + β + δ, r + δ) ν

64 Basin of convergence for Keen Keen

65 Keen To introduce the destabilizing effect of purely speculative investment, Keen (2009) considers a modified version of the previous with Ḋ = κ(1 ω rd)y (1 ω rd)y + P Ṗ = Ψ(g)Y where Ψ( ) is an increasing function of the growth rate of economic output g = κ(1 ω rd) ν δ.

66 - Differential equations Keen ( ) Denoting Ψ(ω, d) = Ψ κ(1 ω rd) ν δ, the system now becomes [ ] φ 1 ω = ω (1 λ) 2 (α + φ 0) [ ] κ(1 ω rd) λ = λ α β δ ν [ ḋ = d r κ(1 ω rd) ] +δ +κ(1 ω rd) (1 ω)+p ν ( ) κ(1 ω rd) ṗ = Ψ(ω, d) δ p ν

67 - Equilibria Keen The finite debt equilibrium for the new system has π 1 and λ 1 as before, but ω 1 = 1 π 1 rd 1 d 1 = ν(α + β + δ) π 1 + p 1 α + β Ψ(α + β) p 1 = α + β and is now unstable for typical parameters.

68 - Equilibria Keen The finite debt equilibrium for the new system has π 1 and λ 1 as before, but ω 1 = 1 π 1 rd 1 d 1 = ν(α + β + δ) π 1 + p 1 α + β Ψ(α + β) p 1 = α + β and is now unstable for typical parameters. On the other hand, introducing u = 1/d and v = 1/p we find that (ω, λ, d, p) = (0, 0, +, ) (ω, λ, d, p) = (0, 0, +, + ) correspond to stable equilibria for the modified system.

69 Example 4: effect of Keen D/Y time 1.5 No speculation Finance Speculation/GDP: P K /Y Finance time

70 Introducing a sector A final extension proposed by Keen (echoing Minsky) consists of adding spending and taxation into the original system according to Ġ = Γ(λ)Y Ṫ = Θ(π)Y Keen

71 Introducing a sector Keen A final extension proposed by Keen (echoing Minsky) consists of adding spending and taxation into the original system according to Ġ = Γ(λ)Y Ṫ = Θ(π)Y Defining g = G/Y and t = T /Y, the net profit share is now π = 1 ω rd + g t

72 Introducing a sector Keen A final extension proposed by Keen (echoing Minsky) consists of adding spending and taxation into the original system according to Ġ = Γ(λ)Y Ṫ = Θ(π)Y Defining g = G/Y and t = T /Y, the net profit share is now π = 1 ω rd + g t The new 5-dimensional system displays more local fluctuations, but no breakdown for the same initial conditions as before.

73 Example 5: stabilizing w 0 = 0.75, λ 0 = 0.7, d 0 = 0.1, g 0 = 0.14, t 0 = 0.39, r = 0.03 wage share employment capitalists debt spending taxation Keen

74 Example 5 (continued): stabilizing w 0 = 0.75, λ 0 = 0.7, d 0 = 0.1, g 0 = 0.14, t 0 = 0.39, r = 0.03 Keen λ ω

75 Next steps Model prices for capital goods P k and commodities P c explicitly (Kaleckian mark-up theory, inflation, etc) Keen

76 Next steps Keen Model prices for capital goods P k and commodities P c explicitly (Kaleckian mark-up theory, inflation, etc) Introduce noise (stochastic interest rates, risk premium, etc)

77 Next steps Keen Model prices for capital goods P k and commodities P c explicitly (Kaleckian mark-up theory, inflation, etc) Introduce noise (stochastic interest rates, risk premium, etc) Calibrate to macroeconomic time series.

78 Concluding thoughts Solow (1990): The true test of a simple is whether it helps us to make sense of the world. Marx was, of course, dead wrong about this. We have changed the world in all sorts of ways, with mixed results; the point is to interpret it. Keen

79 Concluding thoughts Keen Solow (1990): The true test of a simple is whether it helps us to make sense of the world. Marx was, of course, dead wrong about this. We have changed the world in all sorts of ways, with mixed results; the point is to interpret it. Schumpeter (1939): Cycles are not, like tonsils, separable things that might be treated by themselves, but are, like the beat of the heart, of the essence of the organism that displays them.