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Financial Economics Lecture Ten

Financial Economics Lecture Ten. Dynamic Modelling & the Circuit. Dynamic modelling—an introduction. Dynamic systems necessarily involve time Simplest expression starts with definition of the percentage rate of change of a variable: “Population grows at 1% a year”

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Financial Economics Lecture Ten

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  1. Financial Economics Lecture Ten Dynamic Modelling & the Circuit

  2. Dynamic modelling—an introduction • Dynamic systems necessarily involve time • Simplest expression starts with definition of the percentage rate of change of a variable: • “Population grows at 1% a year” • Percentage rate of change of a variable y is • Slope of function w.r.t. time (dy/dt) • Divided by current value of variable (y) • So this is mathematically • This can be rearranged to… • Looks very similar to differentiation, which you have done… but essential difference: rate of change of y is some function of value of y itself.

  3. Dynamic modelling—an introduction • Dependence of rate of change of variable on its current value makes solution of equation much more difficult than solution of standard differentiation problem • Differentiation also normally used by economists to find minima/maxima of some function • “Profit is maximised where the rate of change of total revenue equals the rate of change of total cost” (blah blah blah…) • Take functions for TR, TC • Differentiate • Equate • Easy! (also wrong, but that’s another story…) • However differential equations…

  4. Dynamic modelling—an introduction • Have to be integrated to solve them: Rearrange Integrate Solve Take exponentials • Constant is value of y at time t=0:

  5. Dynamic modelling—an introduction • Simple model like this gives • Exponential growth if a>0 • Exponential decay if a<0 • But unlike differentiation technique (most functions can be differentiated)… • Most functions can’t be integrated: no simple solution can be found; and also • Models can also be inter-related • Two variables x & y (and more: w & z & …) • y can depend on itself and x • x can depend on itself and y • All variables are also functions of time • Models end up much more complicated…

  6. Dynamic modelling—an introduction • Simple example: relationship of fish and sharks. • In the absence of sharks, assume fish population grows smoothly: • “The rate of growth of the fish population is a% p.a.” Rearrange Integrate Solve Exponentials

  7. Dynamic modelling—an introduction • Simulating gives exponential growth if a>0

  8. Dynamic modelling—an introduction • Same thing can be done for sharks in the absence of fish: • Rate of growth of shark population equals c% p.a. • But here c is negative • But we know fish and sharks interact: • “The rate of change of fish populations is also some (negative) function of how many Sharks there are” • “The rate of change of shark population is also some (positive) function of how many Fish there are”

  9. Dynamic modelling—an introduction • Now we have a model where the rate of change of each variable (fish and sharks) depends on its own value and the value of the other variable (sharks and fish): • This can still be solved, with more effort (don’t worry about the maths of this!):

  10. Dynamic modelling—an introduction • But for technical reasons, this is the last level of complexity that can be solved • Add an additional (nonlinearly related) variable—say, seagrass levels—and model cannot be solved • But there are other ways… • Mathematicians have shown that unstable processes can be simulated • Engineers have built tools for simulating dynamic processes… similarly,

  11. Dynamic modelling—an introduction • (1) Enter (preferably empirically derived!) parameters: • (2) Calculate equilibrium values: Simulate! • (3) Rather than stopping there: Graph the results:

  12. Dynamic modelling—an introduction • System is never in equilibrium • Equilibrium “unstable”—repels as much as it attracts • Commonplace in dynamic systems • Systems are always “far from equilibrium” • (What odds that the actual economy is in equilibrium?…)

  13. Dynamic modelling—an introduction • That’s the “hard” way; now for the “easy” way… • Differential equations can be simulated using flowcharts • The basic idea… • Numerically integrate the rate of change of a function to work out its current value • Tie together numerous variables for a dynamic system • Consider simple population growth: • “Population grows at 2% per annum”

  14. Dynamic modelling—an introduction • Representing this as mathematics, we get • Next stage of a symbolic solution is • Symbolically you would continue, putting “dt” on the RHS; but instead, numerically, you integrate: • As a flowchart, you get: • Read it backwards, and it’s the same equation: • Feed in an initial value (say, 18 million) and we can simulate it (over, say, 100 years)

  15. Dynamic modelling—an introduction • MUCH more complicated models than this can be built…

  16. Dynamic modelling—an introduction • Models can have multiple interacting variables, multiple layers…; for example, a racing car simulation:

  17. Dynamic modelling—an introduction • “System dynamics” block has these components: • And this block has the following components…

  18. Dynamic modelling—an introduction • This is not toy software: engineers use this technology to design actual cars, planes, rockets, power stations, electric circuits…

  19. Dynamic modelling—an introduction • Let’s use it to build the Fish/Shark model • Start with population model, only • Change Population to Fish • Alter design to allow different initial numbers • This is equivalent to first half of • To add second half, have to alter part of model to LHS of integrator

  20. Dynamic modelling—an introduction • Sharks just shown as constant here • Sharks “substract” from fish growth rate • Now add shark dynamics

  21. Dynamic modelling—an introduction • Shark population declines exponentially, just as fish population rises • (numbers obviously unrealistic) • Now add interaction between two species

  22. Dynamic modelling—an introduction • Model now gives same cycles as seen in mathematical simulation. • Now to apply this to endogenous money!

  23. Modelling endogenous money dynamically • Remember our minimum 4 accounts? • Capitalist Credit (KC) • Money paid in here • Interest on balance • Repayments out of here to Banker • Capitalist Debt (KD) • Repayment of debt recorded here • Interest on balance • Banker Principal Account (BP) • Accepts repayment of debt by capitalist • Banker Income Account (BY) • Records incoming and outgoing interest payments • Starting at the very simplest level: compound interest…

  24. Modelling endogenous money dynamically • KD grows at rate of debt interest rd: • KC grows at rate of credit interest rc: • BY pockets the difference: • As Circuitists feared, this is “the road to ruin” for capitalists… • With Loan L=$100, rd=5%, rc=3%, after one year • Simulating this with equations…

  25. Modelling endogenous money dynamically

  26. Modelling endogenous money dynamically • Who’d be a capitalist? • But model isn’t complete yet: no repayment • First, same model as a flowchart…

  27. Modelling endogenous money dynamically • In flowchart format, equations • Become… • Simulating this in real time…

  28. Modelling endogenous money dynamically • Tidied up using compound blocks:

  29. Modelling endogenous money dynamically • How to model repayment? • Standard home loan style—regular payments over term of loan? • OK at micro level, but at macro: • Lots of loans, starting different times, ending different times, some extended, others paid off early… • Aggregate outcome very different to each individual loan • Fixed repayment commitment? • Like “exponential decay”: • Debt would fall towards zero as t • Sensible objective for capitalist borrowers; • How to achieve it?

  30. Modelling endogenous money dynamically • Currently debt has dynamics: • To convert this to • Need to add: • Repayment of interest • & Repayment of principal • Both amounts have to come out of capitalist credit account KC (capitalists’ only source of funds) • Repayment of principal to banker principal account BP • Interest to banker income account BY (already shown)

  31. Modelling endogenous money dynamically • So (still incomplete) system is: • How does this behave? • Simulate and find out… • Try R=1… • Picture looks a lot better for capitalists than without repayment…

  32. Modelling endogenous money dynamically But first... a word from our sponsor... • But being a capitalist still a losing proposition… • Because production isn’t yet modelled • Capitalists borrow to produce, sell, & make a profit…

  33. Modelling endogenous money dynamically • Observations on Circuitists vs Keynes • Graziani: “As soon as firms repay their debt to the banks, the money initially created is destroyed.” • Keynes: “Credit, in the sense of ‘finance,’ looks after a flow of investment. It is a revolving fund which can be used over and over again. It does not absorb or exhaust any resources. The same ‘finance’ can tackle one investment after another.” (247) • Keynes is right: the money created by the loan continues to exist & circulate as an asset of the banks. It is either • The outstanding debt of capitalists to banks; or • The balance in bankers principal account:

  34. Modelling endogenous money dynamically • Sum of capitalist debit account KD + banker principal account BP always equals value of initial loan L: • Debt destroyed by repayment; money is conserved as “a revolving fund of finance”

  35. Modelling endogenous money dynamically • Final step to close (simple) model: production • How to model? • Complicated! Input-output issues, pricing, stocks… • Leave till later & take essence: • Whole purpose of production to make $ profit • Must spend $ to make $: outflow from KC • Production requires workers • Must be paid wages (into new account WY) • Products sold to capitalists, workers, bankers • Transactions from sale flow to capitalist credit account (out of BY, WY) • Net revenue generated resolves into either profits or wages • in proportions of flow of income from production, wage share + profit share = 1

  36. Modelling endogenous money dynamically • Call proportion of outflow that finances production P: • Fraction (1-p) of this flows to workers as wages: • Fraction p flows back to capitalists as profits: • Workers earn interest too: • Paid by bankers from BY: • & pay $ to KC account • Workers & bankers consume:

  37. Modelling endogenous money dynamically • So the first complete (but still simple!) system is:

  38. Modelling endogenous money dynamically • Almost obvious why capitalists go into debt: • Does it matter that capitalists are in net debt?

  39. Modelling endogenous money dynamically • No: • Capitalist entrepreneurs are the debtors “par excellence” in capitalism: • “becoming a debtor arises from the necessity of the case and is not something abnormal, an accidental event to be explained by particular circumstances. What he first wants is credit. Before he requires any goods whatever, he requires purchasing power. He is the typical debtor in capitalist society.” (Schumpeter, Theory of Capitalist Development: 102) • Net debt tapers to zero over time:

  40. Modelling endogenous money dynamically • After ten years • Capitalist net debt position negligible • Almost all net money in BP account: • What about incomes? • KD, KC, WY, etc. record bank balances • Entries wWY, bBY, etc. record transactions between accounts • Incomes are subset of transactions…

  41. Modelling endogenous money dynamically • Workers: wages + interest = • Bankers: net interest = • Capitalists: profit + net interest = • These are flows • Aggregate income generated by initial loan L=$100 is stock: • Integral of these over time:

  42. Modelling endogenous money dynamically • So initial loan of $100 can generate: • $89 income to capitalists; • $2 to bankers; and • $215 to workers • (with example parameters) • Without relending • All money eventually accumulates in BP & activity ceases • Next extension: with re-lending so that we model credit as “a revolving fund which can be used over and over again.” (Keynes) • Bankers re-lend proportion B of existing BP • Amount paid into KC; recorded in KD:

  43. Modelling endogenous money dynamically • Final system is: • Initial loan L can now fund “one investment after another”, as Keynes argued:

  44. Modelling endogenous money dynamically • All accounts stabilise at constant level: • Flow of “revolving fund” through accounts generates sustained stream of income for all 3 classes from single initial loan:

  45. Modelling endogenous money dynamically • With parameter values used, $100 initial loan can finance: • $61 of profit • $143 of wages; and • $1 of bank income per year

  46. Modelling endogenous money dynamically • “How can capitalists borrow money & make a profit?” dilemma easily solved: • So long as income from production exceeds interest payments on borrowed money!: Income from production Interest on debt • Circuitist “how can M become M`?” dilemma a case of confusing stocks (initial loan) and flows (incomes, including profits)

  47. Additional insights from model • Endogenous money “works” • Don’t need deposits to make loans (exogenous money perspective) • instead “loans create deposits” • Loans(t) = KD(t) • Deposits(t) = KC(t)+WY(t)+BY(t) • Amounts identically equal over time • Bank creates money “ab initio” • No need for it to have any assets • Just need acceptance of its “IOUs” as money by third parties

  48. Additional insights from model • Deposits “destroyed” as Loans repaid, not Money—which is conserved • Money a bank asset: • Money(t) = KD(t) + BP(t) • Technically, Loans “destroyed” by returning Deposits to Banks

  49. Additional insights from model • Form of money is either • Debt by other parties to banks (“loans=deposits”); or • Banks unencumbered asset in BP account (“reserves”) • Reserves created by repayments of loans • Reverse of “exogenous money” perspective (“loans made possible by reserves”)

  50. Next extension: creation of new money… • Model so far has single injection of money KD(0) • In actual economy, new money created endogenously all the time… • How to model here? • Simplest step: banks create new money at rate nm% p.a. • New money generated in Bank Principal account • New money then loaned to firms • Recorded as positive entry in credit (money) and debt account • Subtracted from Bankers’ Principal account once paid to capitalists…

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