Liquidity-Profit Rate Cycles and Chaos: An Harrodian Circuit of Capital ModelXiao Jiang

jiangx@denison.edu

Denison University

GDP Growth Rate, USA

Modified Circuit of Capital Model

Productive /

Commercial

Capital (X)

Central Bank Financial

Capital (M)

C Capital

Outlays

Sales

Lending

Borrowing

Interest

Payments

Firm’s Behavior1. Making the investment decision. A financially-extended

Harrodian investment function that depends on: the firm’s liquidity ratio (m=M/X), and the difference between the on-going uniform interest rate (i) and this firm’s profit rate (r).

2. The capital outlays of one firm are the sales of some other firms. As soon as new capital outlays are determined, new sales are determined as long as we know the distribution of the capital outlays to sales – we call it Matrix A

Ci,t+1 =Ci,t + a[mi,t, it - ri,t ]Ci,t

Cm > 0,Ci-r < 0

S = A ×C

4. Capital outlays increases productive capital and sales (profit discounted) reduces productive capital.

5. Capital outlays reduces financial capital and sales increases financial capital.

6. Profit rate is configured given sale, productive capital, and profit margin.

7. The central bank determines the interest rate for next round of interaction, the circuit closes for this particular round.

Xi,t+1 = Xi,t +Ci,t+1 - Si,t+1 ×(1- qi )

Mi,t+1 = (Mi,t -Ci,t+1 + Si,t+1)×(1+ i)

ri,t+1 = (qi ×Si,t+1) / Xi,t+1 = qui,t+1

it+1 = f[mt ], im < 0

The case of identical firms

• We make a hard assumption about the relation between capital outlays and sales…

• Firms receive equal proportions of capital outlay from each individual firm.

• In the end, we have C = S for each firm, that is, firms accumulate financial capital at the same rate.

• All firms are identical

A =

0 13

13

13

13

0 13

13

13

13

0 13

13

13

13

0

é

ë

êêêêê

ù

û

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Profit Rate Simulation

0 100 200 300 400 500Time

0.06

0.08

0.10

0.12

0.14

r

Liquidity-Profit Rate Simulation

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

m

r

Liquidity-Profit Rate Cycle

Low r -> higher lending -> higher m -> r

Too high of r chokes m off

Low m -> high I -> Shortage of Finance

r falls enough below i -> Recovery of liquidity

Implications• This model produces cycles and fluctuations endogenously.

• Via stability and bifurcation analyses, the source of the instability of the system is found be the size of a crucial parameter:

• This parameter is a(i-r): The marginal propensity to invest with respect to interest and profit rates differential.

• Larger the |a(i-r)| is, more aggressive the firm is with its investment in financial products.

• The system goes through Hopf bifurcation precisely when ap becomes large enough.

• The high |a(i-r) | is essentially the Harrodian instability in this model.

• However, the resulting exploding trajectory of accumulation is contained by the negative effect of liquidity constraint and interest rate.

The case of heterogeneous firms• Relax the assumption that all firms receive equal proportion

of capital outlays from each firm as their sales.

0 0.85 0.75 0.05

0.20 0 0.45 0.03

0.54 0.10 0 0.92

0.26 0.04 0.47 0

é

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ù

û

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Col[A] = {1,1,1,1}

Row[A] = {0.98,0.68,1.56,0.78}

Ci ¹ Si

Firms accumulate financial capital at different ratesFirms become heterogeous

The Principal Eigenvectors

• Gibrat (1931), Cabral and Mata (2003), Simon and Bonnini (1958), Ijiriand Simon (1965)

0.05 0.10 0.15 0.20 0.25

5

10

15

Profit Rate

100 200 300 400 500Time

0.096

0.098

0.100

0.102

0.104

0.106

r

Liquidity-Profit Rate

0.095 0.100 0.105 0.110r

0.160

0.165

0.170

0.175

0.180

m

GDP Growth Rate

GDPt = kCt-1 + qSt

200 400 600 800Ticks

0.100

0.105

0.110

0.115

0.120

0.125

Rate of Growth

Individual Profit Rates

100 200 300 400

0.09

0.10

0.11

0.12

Is there an equilibrium for this system?

Average and Equilibrium Growth Rates of Productive Capital

Evolution of Firm Size Distribution Measured in X

Implications

• The instability of this “capitalist economy” lies in the Harrodian instability caused by the firm’s “aggressiveness” regarding the size of the “rent” between investment in the financial market and the goods market.

• The chaotic fluctuations in the case of heterogeneous firms are caused by some sort of mis-coordination between different firms.

• The existence of chaotic movement might suggest a basic social coordination problem. With such a closed monetary system, one firm’s spending has the external effect of relieving other firms’ financial constraints, and no firm has a reason to take these external effects into account in choosing its own spending.