EC6012 International Monetary Economics Lecture 3

Cool Models of Business CyclesEC6012 2009 Lecture 3

Stephen Kinsella

Dept. Economics,University of Limerick.stephen.kinsella@ul.ie

February 8, 2009

Stephen Kinsella (University of Limerick) Cool Models of Business Cycles February 8, 2009 1 / 44

1 Introduction

2 Data

3 Multiplier-Accelerators

4 Goodwin’s Growth Model

5 MinskyGenerating a crisis

6 Summary

IntroductionStuff you’ll learn today

You’ll see some BC data

You’ll see 4 cool models1 Samuelson’s Multiplier Accelerator

2 Goodwin’s Nonlinear Accelerator3 Goodwin’s Growth Cycle4 Minsky’s Financial Fragility Model5 Obstfeld & Rogoff’s Redux (time allowing)

I’ll show you why they are cool as we go.

You’ll see 4 cool models1 Samuelson’s Multiplier Accelerator

2 Goodwin’s Nonlinear Accelerator3 Goodwin’s Growth Cycle4 Minsky’s Financial Fragility Model5 Obstfeld & Rogoff’s Redux (time allowing)

You’ll see 4 cool models1 Samuelson’s Multiplier Accelerator2 Goodwin’s Nonlinear Accelerator

3 Goodwin’s Growth Cycle4 Minsky’s Financial Fragility Model5 Obstfeld & Rogoff’s Redux (time allowing)

You’ll see 4 cool models1 Samuelson’s Multiplier Accelerator2 Goodwin’s Nonlinear Accelerator3 Goodwin’s Growth Cycle

4 Minsky’s Financial Fragility Model5 Obstfeld & Rogoff’s Redux (time allowing)

You’ll see 4 cool models1 Samuelson’s Multiplier Accelerator2 Goodwin’s Nonlinear Accelerator3 Goodwin’s Growth Cycle4 Minsky’s Financial Fragility Model

5 Obstfeld & Rogoff’s Redux (time allowing)

You’ll see 4 cool models1 Samuelson’s Multiplier Accelerator2 Goodwin’s Nonlinear Accelerator3 Goodwin’s Growth Cycle4 Minsky’s Financial Fragility Model5 Obstfeld & Rogoff’s Redux (time allowing)

Ireland’s Real Output has been all over the place since1970

1980 1990 2000

GDP Growth, %

Figure: Ireland’s Year on Year Percentage Real GDP Growth, 1970-2007.

But Look at Fixed Investment

1980 1990 2000

2 ´ 109

5 ´ 109

1 ´ 1010

2 ´ 1010

5 ´ 1010

Fixed Investment HlogsL

Figure: Logged Fixed Investment in Ireland, 1970–2008.

Or Government Consumption

1980 1990 2000

1 ´ 109

2 ´ 109

5 ´ 109

1 ´ 1010

2 ´ 1010

Government Consumption HlogsL

Figure: Logged Government consumption in Ireland, 1970–2008.

Or Inventory Changes

1980 1990 2000

2 ´ 107

5 ´ 107

1 ´ 108

2 ´ 108

5 ´ 108

1 ´ 109

Inventory Change HlogsL

Figure: Logged changes in inventory for Ireland, 1970–2008. Missing data is areporting error.

Or Total Consumption

1980 1990 2000

1.0 ´ 1010

1.0 ´ 1011

5.0 ´ 1010

2.0 ´ 1010

3.0 ´ 1010

1.5 ´ 1010

7.0 ´ 1010

Total Consumption HlogsL

Figure: Logged total consumption in the Irish economy, 1970-2008.

Different Explanations for these changes.Real Business Cycle People Say

(1, pg. 1):

[t]he economy is viewed as being in continuous equilibrium in thesense that, given the information available, people makedecisions that appear optimal for them, and so do not makepersistent mistakes. This is also the sense in which behaviour issaid to be rational. Errors, when the occur, are said to beinformation gaps, such as unanticipated shocks to the economy.

Different Explanations for these changes.Non mainstream People Say

(? , pg. 199):

The financing of investment by means of new techniques meansthe generation of demand in excess of that allowed for by theexisting tranquil state. The rise in spending upon investmentleads to an increase in profits, which feeds back and raises theprice of capital assets and thus the demand price of investment.Thus, any full-employment equilibrium leads to an expansion ofdebt-financing—weak at first because of the memory ofpreceding financial difficulties—that moves the economy toexpand beyond full employment. Full employment is a transitorystate because speculation upon and experimentation with liabilitystructures and novel financial assets will lead the economy to aninvestment boom. An investment boom leads to inflation, and,by processes still to be described, an inflationary boom leads to afinancial structure that is conducive to financial crises.

Multiplier Accelerators

Y = C + I + G + X −M, (1)

Which Says] National output (Y ) is the sum of Consumption, C ,Investment, I , Government expenditure, G , and exports minus imports,X −M

Multiplier-AcceleratorsConsumption

Yt = c0 + c1Yt−1. (2)

Multiplier-AcceleratorsInvestment

It = I0 + I (r) + b(Ct − Ct−1). (3)

Caution

We need b > 0. We can assume a constant interest rate pretty easily,which is the same thing as saying let’s drop it entirely. AssumingI (r) = 0...

Multiplier-Accelerators

It = I0 + b(Ct − Ct−1). (4)

AD Becomes

Ytd = Ct + It = c0 + I0 + cYt−1 + b(Ct − C + t − 1). (5)

Assume Y d = Y at time t

Multiplier-Accelerators

Yt = c0 + I0 + cYt−1 + b(Ct − Ct−1) (6)

Know values of Ct and Ct−1 are Ct = c0 + cYt−1, and Ct−1 = c0 + cYt−2.Subbing these in:

Yt = c0 + I0 + cY t − 1 + b(c0 + cYt−1 − c0 − cYt−2) (7)

Write equation 7 as a second order linear difference equation:

Yt − (1 + b)cYt−1 + bcYt−2 = (c0 + I0) (8)

Solution

We solve difference equations by finding their equilibrium or steady states,where Yt = Yt−1 = Yt−2 = Y ∗. Putting this into equation 8 andrearranging, we get

Y ∗ =(c0 + I0)

(1− c)(9)

Lessons from Samuelson

1 Cycle theory is tricky, normally requires a bit of differential calculus;

2 Some cycles are more ‘cyclical’ than others. Some will explode, somewill dampen, some will oscillate, some will focus in on one point. Thisis where phase plots and arrow diagrams come in really handy;

3 The economy is highly dependent on past values of itself for itscurrent levels of output, employment, etc., so lagged effects are alwaysgoing to matter in these models. (ARMA/ARIMA/etc modeling)

1 Cycle theory is tricky, and normally requires a bit of differentialcalculus;

Non Linear Accelerator

Call capital stock k, ψ is the desired capital stock proportional to incomeor output, C is consumption, y is income, c0, c1 and b are constants.Assuming a linear consumption function which relates consumer spending,C , to income, Y , such as c = c0 + c1YD, we have

ψ = by , (10)

C = c0 + c1y , (11)

y = C + k, (12)

Buildup: k∗; scrapping rate k∗∗

k∗, ψ > k,0, ψ = k,k∗∗, ψ < k.

Nearly there...

Now combine equations 10, 11, 12 and 13, to obtain

1− c1k +

1− c1. (14)

Looks like this

Investment over time

k, k*, k**0

Figure: Phase Diagram of the Non Linear Multiplier/Accelerator.

Very CoolWhy?

Even though it is really simple, this model is cool for at least four reasons.

1 The final result is independent of initial conditions;

2 The oscillation maintains itself without any stochastic shockswhatsoever;

3 The equilibrium exists, is attainable, but is unstable, which makessense to us intuitively;

4 No lags are required for this model to work, unlike Samuelson’s.

Very CoolWhy?

Predator Prey Interaction

Model is coupled equations where the interesting dynamics came fromfeedbacks and interactions

Two homogeneous, non-specific factors of production, labour, L andcapital, K , where all quantities are real and net, and all wages areconsumed with all profits being reinvested into the system.

A steady growth rate β of the labour force N according to N = N0eβt

Steady technical progress, α so that the capital-labour ratio evolvesaccording to y/l = α = α0e

The capital-output ratio k = Y /L is assumed constant and the realwage rises in the neighbourhood of full employment

The workers accrue to themselves a portion of the output of theeconomy, u and the capitalists receive v for their efforts

k− (α+ β)

]− 1

}v (15)

u = − [(α+ γ) + ρv ] u (16)

Goodwin on his cycle:

When profit is greatest, u = u, employment is average,. . . , andthe high growth rate pushes employment to its maximum v2,which squeezes the profit rate to its average value. . . thedeceleration in the growth employment (relative) to its averagevalue again, where profit and growth are again at their nadir u2.This low growth rate leads to a fall in output and employment towell below full employment, thus restoring profitability to itsaverage value because productivity is now rising faster than wagerates . . . . The improved profitability carries the seed of its owndestruction by engendering a too vigorous expansion of outputand employment, thus destroying the reserve army of labour andstrengthening labour’s bargaining power.

Figure: Evolution of capitalist/Worker interactions as they share the products ofthe economy. We see here that the motion is cyclical and bounded, implying thedynamics of the system exhibit limit cycle behaviour.

Why is this cool?

This is a cool model for three reasons.

1 The model generates a feedback-driven limit cycle, showing workersdependent on capitalists and vice versa

2 It shows Marxian macrodynamics in an interesting light

3 The model can be extended to include search and selection,endogenising the values of the parameters used in the model, seewww.stephenkinsella.net/research for details.

Minsky Setup

Debt structure of firms matters.

Market is naturally unstable

Presence of Big Bank and Big Government dampen cycles

Booms and busts are the inevitable result of institutionally-legitimisedhigh risk lending practices

Minsky Setup

A constant markup τ over the wage bill w , and the labour/output ratio isb. The price level P is determined by

P = (1 + τ)wb. (17)

The profit rate, r , is given by adding up the contributions to profit fromthe various sectors of the economy:

r =PX − wbX

(1 + τ)wbX=

1 + τ

K, (18)

Big Idea

The core of Minsky’s theory revolves around how expected returns relateto the capital stock, K .

Investment Decision

Pk = (r + ρ)P/i , (19)

Pk − P = (r + ρ− i)P/i . (20)

Investment Demand = PI = [g0 + h(r + ρ− i)]PK . (21)

Saving Supply = srPK = sτwbX . (22)

Equilibrium Conditions

g0 + h(r + ρ− i)− sr = 0. (23)

Solve equation 23 for r , plug it into the investment demand function, andwe get an expression for the capital stock growth rate, g(= I/K ).

g = = s[g0 + h(ρ− i)]s − h. (24)

Cool equation

Equation 24 is very cool: a fall in the interest rate, or an increase inanticipated profits leads to higher growth, since g = sr from the savingfunction, so the profit rate and capacity utilization go up as well.

Financial Side of the Economy

Fiscal debt: F . Can be converted into money, M, or short term bonds B,bond is held by rentiers. Value of all plant and equipment isPkK = (r + ρ)PK/i . Firms have equity, E , which has a market price atPe . The difference between capital stock and equity is firms’ net worth, N.The differential of the firms’ balance sheets is

Pk I + Pk = PkK = Pe E + PeE + N. (25)

The total wealth of all rentiers is

W = PeE + M + B = PeE + F . (26)

The rentiers’ wealth changes over time according to

W = PeE + Pe E + M + B = PeE + srPK . (27)

Which says...

Rentiers get rich from increases in capital gains and financial saving.

At each point in time, rentiers have to decide to allocatetheir wealth across assets according to these balancingrules:

µ(i , r + ρ)W = M = 0, (28)

ε(i , r + ρ)

PeW − E = 0, (29)

− β(i , r + ρ)W + B = 0. (30)

Here µ+ ε+ β = 1. The asset demand equations given above determinethe interest rate and the anticipated rate of profit on physical capital,r + ρ.

We can think of r + ρ as representing returns to equity. Higher returns willbid up the value of firm’s capital stock in this economy.

Combining 26 and 29, we have

1− ε(i , r + ρ). (31)

Equation 31 says that increasing r or ρ will drive up ε, and so share pricesand financial prices will rise.

Macro policies determine micro net worth

Rentier’s net worth is determined macroeconomically by their valuation ofanticipated profits, which feeds demand for asset supplies and demands inthe current period.

Pe = (ε/(1− ε))(F/E ); (32)

In turn Pe will determine the changes in firms’ net worth, given theirinvestment levels and issuance of new equity, and excess demand in themoney markets will be the sum of

µ(i , r + ρ) =M

F[1− ε(i , r + ρ)], (33)

Equations 33 and 24 pick out an ISLM relation whichlooks like this:

Financial Mkt

Commodity Mkt

Interest Rate, i

Profit Rate, r

Figure: Response of Interest and Profit rates to an increase in expected profit rateρ.

Adjustment Dynamics

Let the change in expected profits be given by

ρ = −β(i − i). (34)

When the rate of interest exceeds its normal long run level, i , expectedprofits will begin to fall, and fall sharply.

Government Policy

Call the money debt ratio α, and write

), (35)

where f is the ratio of outstanding fiscal debt to the capital stock. Fixgovernment expenditure as a proportion of the capital stock (and taxes ofexpenditures). Now f is fixed. The money debt ratio evolves according to

α = M − g . (36)

This says as money grows, α falls as g increases.

Stability Check

We can check for stability close to an equilibrium point (i = i , g = M)using the Jacobian: [

−βiρ −βiα−(gi iρ + gρ) −gi iα

Any increase in ρ, investor confidence, will lower the interest rate, andraise the derivative of ρ in equation 34. This is positive feedback. Therecan of course be negative feedback. Crises can come at any moment.

Adjustment Dynamics

A MINSKY CRISIS 881

Anticipated incremental

profit rate p

Ratio of money to outside assets a

FIGURE II

Adjustment Dynamics When a Fall in the Expected Incremental Profit Rate p from an Initial Equilibrium at A Leads Finally to a Return to Steady State

where the subscripts on i stand for derivatives through the

IS/LM system, (7) and (18), and the growth rate derivatives come

from (8). Equations (20) and (21) are potentially unstable. From Figure

I, an increase in p lowers the interest rate and thus raises the

derivative p in (20). This positive feedback does not necessarily

dominate the system, since the Jacobian determinant - i,,figp is

easily seen to be positive (signaling possible stability).

The phase diagram appears in Figure II, with arrows showing

directions of adjustment in the different quadrants. To explore

the possibilities, assume that the economy is initially in a com-

plete steady state equilibrium at point A. A momentary lapse of

confidence would cause p to jump down from A to a point like B.

Equally, a one-shot market operation to reduce the money supply

would cause i to rise. For a newly set (lower) value of a-, (20) shows

that p would start to fall from A, setting off a dynamic process

like the one beginning to B.

If the authorities hold to a constant money supply growth

M when the economy is away from steady state, then a below-

equilibrium value of p is associated with slow capital stock growth

and a rising money-debt ratio a from (21). This increase would

Figure: Adjustment Dynamics

Debt Deflation Dynamics

The economy starts at point A. Any drop in investor confidence will moveit to point B, where authorities will try, through policy, to increase M andhence the money debt ratio, α. This would move the economy to point C,and back to equilibrium. If the economy does not turn the corner at C,then the economy enters a debt-deflation scenario.

1 Introduction

2 Data

3 Multiplier-Accelerators

4 Goodwin’s Growth Model

5 MinskyGenerating a crisis

6 Summary

References

[] Hyman P. Minsky. Stabilising and Unstable Economy. McGraw-Hill,New York, 1986.

[1] Michael Wickens. Macroeconomic Theory. Princeton University Press,Princeton, New Jersey and Oxford, 2008.

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