macroeconomics M. R. Grasselli University of Technology Sydney, February 21, 2013

Transcription

1 of of Sharcnet Chair in Financial Mathematics Mathematics and Statistics - McMaster University Joint work with B. Costa Lima, X.-S. Wang, J. Wu University of Technology Sydney, February 2, 23

2 What is wrong with this picture? of Figure:

3 What is the fiscal cliff? of

4 Why is this a problem? of

5 In other words: Keynes roolz! of

6 Really bad economics: hardcore (freshwater) DSGE of The strand of DSGE economists affiliated with RBC made the following predictions after 28: Increases borrowing would lead to higher interest rates on debt because of crowding 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).

7 Wrong prediction number of Figure: Government borrowing and interest rates.

8 Wrong prediction number 2 of Figure: Monetary base and inflation.

9 Wrong prediction number 3 of Figure: Fiscal tightening and GDP.

10 Better (but still bad) economics: soft core (saltwater) DSGE of The strand of DSGE economists affiliated with New Keynesian got all these predictions right. They did so by augmented DSGE with imperfections (wage stickiness, asymmetric information, imperfect competition, etc). 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, 2, pp. 2-3)

11 Then we can safely ignore this... of Figure: Private and public debt ratios.

12 Really? of Figure: Change in debt and unemployment.

13 Much better economics: SFC models of Stock-flow consistent models emerged in the last decade as a common language for many heterodox schools of thought in economics. Consider both real and monetary factors from the start Specify the balance sheet and transactions between sectors Accommodate a number of behavioural assumptions in a way that is consistent with the underlying accounting structure. Reject silly (and mathematically unsound!) hypotheses such as the RARE individual (representative agent with rational expectations). See Godley and Lavoie (27) for the full framework.

14 An example of a (fairly general) Godley table of

15 Godley table for monetary of

16 Accounting relations for monetary of From the first, second and third columns of the transactions and flow of funds matrices we obtain that W + r M M h C = S h = Ṁ h, () C W + r M M f r L L = F fu I = Ṁ f L (2) r L L r M M = S b = L Ṁ. (3) An example of firm behaviour satisfying () (3) is L = I R (4) Ṁ f = I R + C W + r M M f r L L (5) for chosen levels of investment I and repayment R.

17 Behavioral relationships of Investment and repayment can then be modelled as a function of any firm-related variable: I = κ(ω, L, M f ), (6) R = ρ(ω, L, M f ). (7) Consumption is a function of household-related variables: Wage changes are given by C = ξ(ω, M h ). (8) ẇ = Φ(ω, λ, p). (9) Finally, the price level evolves according to ṗ = P(ω, λ, p). ()

18 Differential Equations of Collecting these definitions and that K R = νy R for a constant ν, we arrive at the following closed 6-dimensional system ẇ =Φ(ω, λ, p) [ κ(ω, L, Mf ) λ =λ ν ] α β δ ṗ =P(ω, λ, p) L =κ(ω, L, M f )Y ρ(ω, L, M f ) Ṁ f =κ(ω, L, M f )Y ρ(ω, L, M f ) + ξ(ω, M h ) wl + r M M f r L L Ṁ h =wl + r M M h ξ(ω, M h ), () where l = λn and Y = al, for total population N and productivity a.

19 Special case: Keen (995) of Let D = L M f and assume that p = p, r M = r F = r. Supposing further that Φ = Φ(λ) and I = κ(π)y, where π = ω rd, leads to ω = ω [Φ(λ) α] [ κ( ω rd) λ = λ ν ḋ = d ] α β δ ] [ κ( ω rd) r + δ ν (2) + κ( ω rd) ( ω) Observe that the equation for M h separates as Ṁ h = wl + r M M h ξ(ω, M h ), (3) and only depends on the rest of the system through w.

20 Equilibria of The system (2) has a good equilibrium at with ν(α + β + δ) π ω = π r α + β λ = Φ (α) d = ν(α + β + δ) π α + β π = κ (ν(α + β + δ)), which is stable for a large range of parameters It also has a bad equilibrium at (,, + ), which is stable if κ( ) δ < r (4) ν

21 Example : convergence to the good equilibrium in a of ω λ Y d Y 8 x ω ω =.75, λ =.75, d =., Y = λ d ω Y λ d time

22 Example 2: explosive debt in a of ω =.75, λ =.7, d =., Y = ω λ Y d.9.8 x Y 3 ω 2 λ Y d.6.5 λ.5 d ω time

23 Basin of convergence for of d λ..5 ω.5

24 on mathematical biology of Which species, in a mathematical model of interacting species, will survive over the long term? In a mathematical model of an epidemic, will the disease drive a host population to extinction or will the host persist? Can a disease remain endemic in a population? Reference: H. Smith and H. R. Thieme, Dynamical Systems and Population, Graduate Studies in Mathematics, 8. AMS, 2.

25 definitions of Let Φ(t, x) : R + X X be the semiflow generated by a differential system with initial values x X. For a nonnegative functional ρ from X to R +, we say Φ is ρ strongly persistent (SP) if lim inf ρ(φ(t, x)) > for any x X with ρ(x) >. t Φ is ρ weakly persistent (WP) if lim sup ρ(φ(t, x)) > for any x X with ρ(x) >. t Φ is ρ uniformly strongly persistent (USP) if lim inf ρ(φ(t, x)) > ε for any x X with ρ(x) >. t Φ is ρ uniformly weakly persistent (UWP) if lim sup ρ(φ(t, x)) > ε for any x X with ρ(x) >. t

26 Example: Goodwin model of (Goodwin 967) Predator-prey system of wage share (ω) and employment rate (λ), with π = ω: ω = ω[φ(λ) α] λ = λ[π/ν α β δ]. This is e π SP and e π UWP, but not e π USP. employment wage

27 a sector of Following Keen (and echoing Minsky) we add discretionary spending and taxation into the original system in the form where G b = η b (λ)y T b = Θ b (π)y G = G b + G s T = T b + T s G s = η s (λ)g s T s = Θ s (π)t s Defining g = G/Y and τ = T /Y, the net profit share is now π = ω rd + g τ, and debt evolves according to Ḃ = rb + G T.

28 Differential equations - full system of Denoting γ(π) = κ(π)/ν δ, a bit of algebra leads to the following eight dimensional system: ω = ω[φ(λ) α] λ = λ[γ(π) α β] ḋ = κ(π) π dγ(π) ġ b = η (λ) g b γ(π) (5) ġ s = g s [η s (λ) γ(π)] τ b = Θ b (π) g Tb γ(π) τ s = τ s [Θ s (π) γ(π)] ḃ = b[r γ(π)] + g b + g s τ b τ s

29 Differential equations - reduced system of Notice that π does not depend on b, so that the last equation in (5) can be solved separately. Observe further that we can write π = ω rḋ + ġ τ (6) leading to the five-dimensional system ω =ω [Φ(λ) α], λ =λ [γ(π) α β] ġ s =g s [η s (λ) γ(π)] (7) τ s =τ s [Θ s (π) γ(π)] π = ω(φ(λ) α) r(κ(π) π) + ( ω π)γ(π) + η b (λ) + g s η s (λ) Θ s (π) τ s Θ s (π)

30 Good equilibrium of The system (7) has a good equilibrium at ν(α + β + δ) π ω = π r + η b(λ) Θ b (π) α + β α + β λ = Φ (α) π = κ (ν(α + β + δ)) g s = τ s = and this is locally stable for a large range of parameters. The other variables then converge exponentially fast to d = ν(α + β + δ) π α + β g b = η b(λ) α + β τ b = Θ b(π) α + β

31 Bad equilibria - destabilizing a stable crisis of Recall that π = ω rd + g τ. The system (7) has bad equilibria of the form (ω, λ, g s, τ s, π) = (,,,, ) (ω, λ, g s, τ s, π) = (,, ±,, ) If g s () >, then any equilibria with π is locally unstable provided η s () > r. On the other hand, if g s () < (austerity), then these equilibria are all locally stable.

32 e π persistence of Proposition : Assume g s () >, then the system (7) is e π -UWP if either λη b (λ) is bounded below as λ, or 2 η s () > r.

33 λ persistence of Proposition 2: Assume g s () >, then the system (7) is λ-uwp if either of the following conditions is satisfied: τ s () and λη b (λ) is bounded below as λ, or 2 τ s () and η s () > max{r, α + β}, or 3 τ s (), r < η s () α + β and r(κ(x) x) + ( x)γ(x) + η () Θ (x) > for γ(x) [η s (), α + β]. 4 η s () > max{r, α + β}, Θ s ( ) <, Θ s (π) < α + β, and Θ s is convex.

34 Example 3: Good initial conditions of d K d g g B +g S ω() =.9, λ() =.9, d k () =., g b () =.5, g s () =.5, τ b () =.5, τ s () =.5, d g () =, r =.3, η s () =.2 ω Keen Model + Unified Government time time λ τ B +τ S

35 Example 4: Bad initial conditions of d K d g ω() =.7, λ() =.7, d k () =.5, g b () =.5, g s () =.5, τ b () =.5, τ s () =.5, d g () =, r =.3, η s () =.2 ω g B +g S Keen Model + Unified Government time time λ τ B +τ S

36 Example 5: Really bad initial conditions with timid of d K d g 2.5 x ω() =.5, λ() =.3, d k () = 5, g b () =.5, g s () =.5, τ b () =.5, τ s () =.5, d g () =, r =.3, η s () = x g B +g S ω Keen Model + Unified Government time time λ τ B +τ S

37 Example 6: Really bad initial conditions with responsive of d K d g ω g B +g S ω() =.5, λ() =.3, d k () = 5, g b () =.5, g s () =.5, τ b () =.5, τ s () =.5, d g () =, r =.3, η s () = Keen Model + Unified Government time time λ τ B +τ S

38 Example 7: in good times: harmless of d K d g ω() =.9, λ() =.9, d k () =.5, g b () =.5, g s () = ±.5, τ b () =.5, τ s () =.5, d g () =, r =.3, η s () =.2 ω g B +g S.5 g s ()> g s ()< time time λ τ B +τ S

39 Example 8: in bad times: a really bad idea of d K x 26 ω() =.6, λ() =.6, d k () =.5, g b () =.5, g s () = ±.5, τ b () =.5, τ s () =.5, d g () =, r =.3, η s () = g s ()>.5 g.6 s ()< 2.4 ω λ d g g B +g S time 2 3 x time τ B +τ S

40 Hopft bifurcation with respect to spending. of OMEGA eta_max

41 Next steps of Introduce equities and household wealth effects in the consumption function (portfolio choices, etc) Extend to a stochastic model (stochastic interest rates, monetary policy, correlated market sectors, etc) Extend to an open economy model (exchange rates, capital flows, etc) Calibrate to macroeconomic time series

42 Jedi economics of