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Liquidity-Profit Rate Cycles and Chaos: 
An Harrodian Circuit of Capital Model 
Xiao Jiang 
jiangx@denison.edu 
Denison Un...
GDP Growth Rate, USA
Modified Circuit of Capital Model 
Productive / 
Commercial 
Capital (X) 
Central Bank 
C 
Capital 
Outlays 
Financial 
Ca...
Firm’s Behavior 
1. Making the investment decision. A financially-extended 
Harrodian investment function that depends on:...
4. Capital outlays increases productive capital and sales (profit 
discounted) reduces productive capital. 
Xi,t+1 = Xi,t ...
The case of identical firms 
• We make a hard assumption about the relation between 
capital outlays and sales… 
• Firms r...
Profit Rate Simulation 
0 100 200 300 400 500 
Time 
0.14 
0.12 
0.10 
0.08 
0.06 
r
Liquidity-Profit Rate Simulation 
0.0 0.1 0.2 0.3 0.4 0.5 
0.5 
0.4 
0.3 
0.2 
0.1 
0.0 
m 
r
Liquidity-Profit Rate Cycle 
Low r -> higher lending -> higher m -> r 
Too high of r chokes m off 
Low m -> high I -> Shor...
Implications 
• This model produces cycles and fluctuations endogenously. 
• Via stability and bifurcation analyses, the s...
The case of heterogeneous 
firms 
• Relax the assumption that all firms receive equal proportion 
of capital outlays from ...
The Principal Eigenvectors 
0.05 0.10 0.15 0.20 0.25 
15 
10 
5 
• Gibrat (1931), Cabral and Mata (2003), Simon and Bonnin...
Profit Rate 
100 200 300 400 500 
Time 
0.106 
0.104 
0.102 
0.100 
0.098 
0.096 
r
Liquidity-Profit Rate 
0.095 0.100 0.105 0.110 
r 
m 
0.180 
0.175 
0.170 
0.165 
0.160
GDP Growth Rate 
200 400 600 800 
GDPt 
= kCt-1 + qSt 
Ticks 
Rate of Growth 
0.125 
0.120 
0.115 
0.110 
0.105 
0.100
Individual Profit Rates 
100 200 300 400 
0.12 
0.11 
0.10 
0.09
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 “aggr...
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Liquidity Profit Rate Cycles and Chaos

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Models of Economic Policy session at 12th International Conference

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Liquidity Profit Rate Cycles and Chaos

  1. 1. Liquidity-Profit Rate Cycles and Chaos: An Harrodian Circuit of Capital Model Xiao Jiang jiangx@denison.edu Denison University
  2. 2. GDP Growth Rate, USA
  3. 3. Modified Circuit of Capital Model Productive / Commercial Capital (X) Central Bank C Capital Outlays Financial Capital (M) Sales Lending Interest Payments Borrowing
  4. 4. Firm’s Behavior 1. 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). Ci,t+1 = Ci,t + a[mi,t, it - ri,t ]Ci,t Cm > 0,Ci-r < 0 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 S = A×C
  5. 5. 4. Capital outlays increases productive capital and sales (profit discounted) reduces productive capital. Xi,t+1 = Xi,t +Ci,t+1 - Si,t+1 ×(1- qi ) 5. Capital outlays reduces financial capital and sales increases financial capital. Mi,t+1 = (Mi,t -Ci,t+1 + Si,t+1)×(1+i) 6. Profit rate is configured given sale, productive capital, and profit margin. ri,t+1 = (qi × Si,t+1) / Xi,t+1 = qui,t+1 7. The central bank determines the interest rate for next round of interaction, the circuit closes for this particular round. it+1 =f[mt ], im < 0
  6. 6. 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. A = 0 1 3 1 3 1 3 1 3 0 1 3 1 3 1 3 1 3 0 1 3 1 3 1 3 1 3 0 é ê ê ê ê ê ë ù ú ú ú ú ú û • In the end, we have C = S for each firm, that is, firms accumulate financial capital at the same rate. • All firms are identical
  7. 7. Profit Rate Simulation 0 100 200 300 400 500 Time 0.14 0.12 0.10 0.08 0.06 r
  8. 8. Liquidity-Profit Rate Simulation 0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.4 0.3 0.2 0.1 0.0 m r
  9. 9. 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
  10. 10. 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.
  11. 11. 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 ê ê ê ê ë ù ú ú ú ú û 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 rates Firms become heterogeous
  12. 12. The Principal Eigenvectors 0.05 0.10 0.15 0.20 0.25 15 10 5 • Gibrat (1931), Cabral and Mata (2003), Simon and Bonnini (1958), Ijiri and Simon (1965)
  13. 13. Profit Rate 100 200 300 400 500 Time 0.106 0.104 0.102 0.100 0.098 0.096 r
  14. 14. Liquidity-Profit Rate 0.095 0.100 0.105 0.110 r m 0.180 0.175 0.170 0.165 0.160
  15. 15. GDP Growth Rate 200 400 600 800 GDPt = kCt-1 + qSt Ticks Rate of Growth 0.125 0.120 0.115 0.110 0.105 0.100
  16. 16. Individual Profit Rates 100 200 300 400 0.12 0.11 0.10 0.09
  17. 17. Is there an equilibrium for this system?
  18. 18. Average and Equilibrium Growth Rates of Productive Capital
  19. 19. Evolution of Firm Size Distribution Measured in X
  20. 20. 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.

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