Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Three examples of new approaches tomacroeconomic modelling

M. R. Grasselli

Professor and Chair, Mathematics and Statistics - McMaster UniversityDirector, Centre for Financial Industries - Fields InstituteLeader, Systemic Risk Analytics Lab - Fields/CQAM

Based on joint work with Aditya Maheshwari, Patrick Li, Gael Giraud andAlex Lipton

New Analytical Tools and Techniquesfor Economic Policy Making

OECD-NAEC and Baillie GiffordApril 16, 2019

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Dynamic Stochastic General Equilibrium (DSGE)

Seeks to explain the aggregate economy using theoriesbased on strong microeconomic foundations.

Collective decisions of rational individuals over a range ofvariables for both present and future.

All variables are assumed to be simultaneously inequilibrium.

Equilibrium is only disrupted by exogenous shocks.

The only way the economy can be in disequilibrium at anypoint in time is through decisions based on wronginformation.

Money is neutral in its effect on real variables.

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

SMD theorem: something is rotten in GE land

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Stock-Flow Consistent models

Stock-flow consistent models emerged in the last decadeas a common language for many heterodox schools ofthought in economics.

They consider both real and monetary factorssimultaneously.

Specify the balance sheet and transactions betweensectors.

Accommodate a number of behavioural assumptions in away that is consistent with the underlying accountingstructure.

Reject the RARE individual (representative agent withrational expectations) in favour of SAFE (sectoral averagewith flexible expectations) modelling.

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Goodwin Model - SFC matrix

Balance Sheet HouseholdsFirms

Sum

current capital

Capital +pK pK

Sum (net worth) 0 0 Vf pK

Transactions

Consumption −pC +pC 0

Investment +pI −pI 0

Acct memo [GDP] [pY ]

Depreciation −pδK +pδK 0

Wages +W −W 0

Sum 0 Sf p(I − δK ) 0

Flow of Funds

Change in Capital +p(I − δK ) p(I − δK )

Sum 0 Sf p(I − δK )

Change in Net Worth 0 Sf + pK pK + pK

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Trajectories in the Goodwin model

0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92Wage Share

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

0.76

0.78

0.80

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Employment Rate

Boom Recession

DepressionRecovery

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Testing the Goodwin Model

Figure: Grasselli and Maheshwari (2017) and (2018)

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Stochastic orbits of a Goodwin model withproductivity shocks

Figure: Figure 3 in Nguyen Huu and Costa Lima (2014)

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

SFC table for Keen (1995) model

Balance Sheet HouseholdsFirms

Banks Sum

current capital

Deposits +∆ −∆ 0

Loans −Λ +Λ 0

Capital +pK pK

Sum (net worth) Xh 0 Xf Xb pK

Transactions

Consumption −pC +pC 0

Investment +pI −pI 0

Acct memo [GDP] [pY ]

Wages +W −W 0

Depreciation −pδK +pδK 0

Interest on deposits +rd∆ −rd∆ 0

Interest on loans −rΛ +rΛ 0

Sum Sh Sf −p(I − δK ) Sb 0

Flow of Funds

Change in Deposits +∆ −∆ 0

Change in Loans −Λ +Λ 0

Change in Capital +p(I − δK ) pI

Sum Sh 0 Sf Sb p(I − δK )

Change in Net Worth Sh (Sf + pK ) Sb pK + pK

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Convergence to the good equilibrium in a Keenmodel

0.7

0.75

0.8

0.85

0.9

0.95

1

λ

ωλYd

0

1

2

3

4

5

6

7

8x 10

7

Y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

d

0 50 100 150 200 250 300

0.7

0.8

0.9

1

1.1

1.2

1.3

time

ω

ω0 = 0.75, λ

0 = 0.75, d

0 = 0.1, Y

0 = 100

d

λ

ω

Y

Figure: Grasselli and Costa Lima (2012)

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Goodwin model

Keen model

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Explosive debt in a Keen model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

0

1000

2000

3000

4000

5000

6000

Y

0

0.5

1

1.5

2

2.5x 10

6

d

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

time

ω

ω0 = 0.75, λ

0 = 0.7, d

0 = 0.1, Y

0 = 100

ωλYd

λ

Y d

ω

Figure: Grasselli and Costa Lima (2012)

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

SFC table for the dual Keen model

Workers Investors Firms Banks Sum

Balance sheetCapital stock +pK pKDeposits +Mw +Mi +Mf −(Mw + Mi + Mf ) 0Loans −Lw −Li −Lf +(Lw + Li + Lf ) 0Equities +peE −peE 0

Sum (Net worth) Xw Xi Xf Xb X

Transactions Current CapitalConsumption −pCw −pCi +pC −pCb 0Investment +pI −pI 0Accounting memo [GDP] [pY ]Wages +w` −w` 0Depreciation −pδK +pδK 0Interest on loans −rLw −rLi −rLf +r(Lw + Li + Lf ) 0Interest on deposits +rMw +rMi +rMf −r(Mw + Mi + Mf ) 0Dividends +rkpK + ∆b −rkpK −∆b 0

Financial balances Sw Si Sf −pI + pδK Sb 0

Flows of fundsChange in capital stock +p(I − δK ) p(I − δK )

Change in deposits +Mw +Mi +Mf −(Mw + Mi + Mf ) 0

Change in loans −Lw −Li −Li +(Lw + Li + Lf ) 0

Change in equities +pe E −pe E 0

Column sum Sw Si Sf Sb p(I − δK )

Change in net worth Xw = Sw Xi = Si + peE Xf = Sf − peE + pK Xb = Sb X = pK + pK

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Bounded oscillations with stable income ratios

Figure: Giraud and Grasselli (2019)

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Explosive debt and increasing inequality

Figure: Giraud and Grasselli (2019)

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

SFC table for Keen model with monetary policy

Households Firms Banks Gov Sum

Balance SheetCapital stock +pK +pKDeposits +∆ −∆ 0Loans −Λ +Λ 0Bills +B −B 0

Sum (net worth) Xh Xf Xb Xg pK

Transactions current capitalConsumption −pC +pC 0Gov Spending +pG −pG 0Capital Investment +pI −pI 0Accounting memo [GDP] [pY ]Wages +W −W 0Taxes −pT +pT 0Depreciation −pδK +pδK 0Interest on deposits +rd∆ −rd∆ 0Interest on loans −rΛ +rΛ 0Interest on Bills +rgB −rgB 0Dividends +Πb −Πb 0

Financial Balances Sh Sf −p(I − δK ) Sb Sg 0

Flow of FundsChange in Capital Stock +p(I − δK ) +p(I − δK )

Change in Deposits +∆ −∆ 0

Change in Loans −Λ +Λ 0

Change in Bills −B +B 0

Column sum Sh Sf Sb Sg p(I − δK )

Change in net worth Sh Sf + pK Sb Sg pK + pK

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Convergence in a Keen model with monetary policy(moderate initial debt)

Figure: `0 = 0.6, g = 0.2, t = 0, δr = 0.03, ηr = 0.1 and ηg = 0.2.

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Stabilizing monetary policy (high initial debt)

Figure: `0 = 6, g = 0.2, t = 0, δr = 0.03, ηr = 0.1 and ηg = 0.2.

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

A model with two types of firms and two types ofhouseholds

Let z = 1, 2 denote aggressive and conservative firms withinvestment for firm n given by

int+1 = (αznt π + β)p · qnt − γ · bnt ,

where αz ≥ 0, β ≥ 0, λ ≥ 0 are known parameters, andα1 > α2, and π is the profit share (see next page).

Consider also two types of households, workers andinvestors, characterized by their consumption

ch1,t+1 = (1− sy1 )yht+1 + (1− sv1 )vht

ch2,t+1 = (1− sy2 )yht+1 + (1− sv2 )vht

Assume that sy1 ≤ sy2 and sv1 ≤ sv2 , which implies thatworkers save less than investors.

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Mean-Field approximation

Consider the ansatz

Xt = Nm(t) +√Nst ,

where m(t) = E [Xt ] and st is a stochastic spread.

Expanding the Master Equation and collecting terms oforder N−1/2 and N−1 lead to the following system ofcouple equations

dm

dτ= λ− (λ+ µ)m

∂Q

∂τ= (λ+ µ)

∂

∂s[sQ(s, τ)] +

λ(1−m) + µm

2

∂2Q(s, τ)

∂s2

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

ABM versus MF - number of firms

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Example 1: stable stock market

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Example 2: unstable stock market

Threeexamples of

newapproaches tomacroeco-nomic

modelling

M. R. Grasselli

Introduction

Inequality

NegativeInterest Rates

Mean-Fieldand ABM

Conclusions

Concluding remarks

Macroeconomics is too important to be left tomacroeconomists.

Banking, money, and finance should not be treated asfrictions in an ideal barter system.

Intermediation between saving households and borrowingentrepreneurs is only a small portion of banking andfinancial activity.

Equilibrium models are based on ludicrous assumptions,have serious problems of internal consistency (see SMDtheorems) and poor empirical performance.

SFC-ABM models, complemented by networks, mean-fieldapproximations and other techniques (including mean-fieldgames), have the potential to redefine the role ofmathematics in macroeconomics.