macroeconomics M. R. Grasselli UMass - Amherst, April 29, 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 UMass - Amherst, April 29, 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 associated 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 a perfect 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 associated with New Keynesianism got all these predictions more or less right. Works by augmenting DSGE with imperfections (sticky wages, asymmetric information, imperfect competition, frictions in financial markets,... ). 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 No, says Krugman of We can then model a crisis like the one we now face as the result of a deleveraging shock. For whatever reason, there is a sudden downward revision of acceptable debt levels a Minsky moment. This forces debtors to sharply reduce their spending. If the economy is to avoid a slump, other agents must be induced to spend more, say by a fall in interest rates. But if the deleveraging shock is severe enough, even a zero interest rate may not be low enough. So a large deleveraging shock can easily push the economy into a liquidity trap. Paul Krugman, Debt, deleveraging, and the liquidity trap, 2. (emphasis added).

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

14 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.

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

16 Godley table for a simplified monetary model of

17 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 () + κ( ω rd) ( ω) Observe that the equation for M h separates as Ṁ h = wl + r M M h C(ω, M h ), (2) and only depends on the rest of the system through w.

18 Equilibria of The system () 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 (3) ν

19 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

20 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

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

22 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.

23 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

24 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

25 Godley table for model with of

26 Modified with of Following Keen (and echoing Minsky) we model spending and taxation as Ġ = Γ(λ)Y Ṫ = Θ(π)Y and add subsidies to firms as G s = Γ s (λ)g s Defining g = G/Y, g s = G s /Y and τ = T /Y, the net profit share is now π = ω rd + g s τ, and debt evolves according to Ḃ = r B B + G + G s T.

27 Differential equations - full system of Denoting µ(π) = κ(π)/ν δ, a bit of algebra leads to the following seven dimensional system: ω = ω[φ(λ) α] λ = λ[µ(π) α β] ḋ = κ(π) π dµ(π) ġ = Γ(λ) gµ(π) (4) ġ s = g s [Γ s (λ) µ(π)] τ = Θ(π) τµ(π) ḃ = b[r B µ(π)] + g + g s τ

28 Differential equations - reduced system of Notice that π does not depend on b, so that the last equation in (4) can be solved separately. Observe further that we can write π = ω rḋ + ġ s τ (5) leading to the five-dimensional system ω =ω [Φ(λ) α], λ =λ [µ(π) α β] ġ s =g s [Γ s (λ) µ(π)] (6) π = ω(φ(λ) α) r(κ(π) π) + ( ω π)µ(π) + Γ(λ) + g s Γ s (λ) Θ(π)

29 Good equilibrium of The system (6) has a good equilibrium at ν(α + β + δ) π ω = π r Θ(π) α + β α + β λ = Φ (α) π = κ (ν(α + β + δ)) g 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 = Γ b(λ) α + β τ = Θ(π) α + β

30 Bad equilibria - destabilizing a stable crisis of Recall that π = ω rd + g s τ. The system (6) has bad equilibria of the form (ω, λ, g s, π) = (,,, ) (ω, λ, g s, π) = (,, ±, ) If g s () >, then any equilibria with π is locally unstable provided Γ s () > r. On the other hand, if g s () < (austerity), then these equilibria are all locally stable.

31 of Proposition : Assume g s () >, then the system (6) is e π -UWP provided Γ s () > r. Proposition 2: Assume g s () >, then the system (6) is λ-uwp if either of the following conditions is satisfied: Γ s () > max{r, α + β} 2 r < Γ s () α + β and r(κ(x) x) + ( x)µ(x) + Γ() Θ(x) > for µ(x) [Γ s (), α + β].

32 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

33 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

34 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

35 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

36 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

37 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

38 Hopft bifurcation with respect to spending. of OMEGA eta_max

39 Next steps of Introduce equities and portfolio choices (Tobin demand, 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

40 Jedi economics of