20.10 Modelling VoTa as a dynamic monetary system

VoTa's economic claims can be expressed as a stock-flow-consistent dynamic model. Professor Steve Keen's Ravel combines multidimensional data analysis with the Minsky system-dynamics engine. Its graphical equations, integrators and Godley tables make it suitable for modelling an economy in which every monetary payment must leave one account and enter another, private debt is explicit, and behaviour evolves outside equilibrium.

Such a model would not merely forecast GDP. It would test whether VoTa's accounting identities hold, identify its feedback mechanisms, compare it with conventional bank-created money, and expose the conditions under which decentralised lending produces growth or financial distress.

Conserved currency

Divide Central Bank currency among households, firms, lenders, the UBI pool and the tax-choice pools for public services. Let their respective balances be \(M_H\), \(M_F\), \(M_L\), \(M_U\) and \(M_{S_j}\). The total quantity of currency is

\[ M=M_H+M_F+M_L+M_U+\sum_j M_{S_j}. \]

VoTa's defining monetary constraint is

\[ \dot M=0. \]

Every payment changes the distribution and possibly the velocity of currency, but never its total quantity. A Godley table can enforce this identity by recording every flow once as a payment and once as a receipt.

Sectoral monetary flows

Let \(C\) be tax-inclusive household consumption, \(W\) wages, \(\Pi_H\) profits distributed to households, \(A\) new savings entrusted to lenders, \(X\) savings withdrawn from lenders, \(L\) new business lending, \(Q\) principal repayments, \(rD\) interest, \(U\) aggregate UBI payments, \(G\) purchases of public services, and \(\Pi_L\) lender costs and distributed profits. Let \(\tau_u\) and \(\tau_s\) be the UBI and public-service shares of consumption tax, with \(\tau=\tau_u+\tau_s\).

The monetary stocks can then be represented by differential equations such as

\[ \dot M_H=W+\Pi_H+U+X-C-A, \] \[ \dot M_F=(1-\tau)C+G+L-W-\Pi_H-Q-rD, \] \[ \dot M_L=A+Q+rD-L-X-\Pi_L, \] \[ \dot M_U=\tau_u C-U, \] \[ \dot M_{S_j}=\theta_j\tau_s C-G_j, \qquad \sum_j\theta_j=1. \]

Payments represented by \(\Pi_L\) enter the accounts of households or firms. Once those receipts and all other internal flows are included, adding every sectoral equation gives \(\dot M=0\). Consumption tax redistributes currency, UBI recirculates it, and neither operation changes the money supply.

Lending transfers existing currency

Let total private debt be \(D\), defaults and discharges be \(\Delta\), new lending be \(L\), and principal repayments be \(Q\). Debt changes according to

\[ \dot D=L-Q-\Delta. \]

Under VoTa, a loan transfers existing currency from a lender to a borrower:

\[ \dot M_L=-L, \qquad \dot M_F=+L. \]

Repayment reverses the transfer:

\[ \dot M_F=-Q, \qquad \dot M_L=+Q. \]

Consequently,

\[ \frac{\partial M}{\partial L}=0. \]

Neither lending nor repayment creates or destroys currency. This removes the conventional mechanism by which bank lending expands deposit money and widespread repayment contracts it. Lending can still affect aggregate demand through circulation: a borrower may spend money that its previous holder would otherwise have left idle, while repayments retained by lenders may reduce velocity.

Production and real growth

The model must connect monetary flows to physical production. Let \(K\) be productive capital, \(N\) employment, \(a\) labour productivity, \(Y\) real output, \(I\) real investment and \(\delta\) the depreciation rate. A simple production side is

\[ Y=aK^\alpha N^{1-\alpha}, \qquad \dot K=I-\delta K. \]

Lending can be separated into productive lending \(L_P\), which finances new capital and production, and asset lending \(L_A\), which finances transfers of existing assets:

\[ L=L_P+L_A. \]

Productive lending contributes to \(I\) and therefore to future output. Lending used to purchase existing assets does not itself add productive capacity and instead increases the monetary demand for those assets. Ravel can track both flows by lender, borrower, industry, region and purpose.

What long-term growth means

Real output \(Y\) is not the mass of material moved through the economy. It is a quality-adjusted measure of the goods and services produced. Growth can consist of producing more physical units, but it can also consist of longer-lasting products, better medical treatment, more capable machines, faster communication, less wasted time, greater reliability and entirely new capabilities.

This distinction becomes decisive over long periods. Real growth of ten percent a year sustained for a century gives

\[ 1.1^{100}\approx13{,}781. \]

It would be misleading to imagine the future economy producing 13,781 copies of the present economy's basket of goods. Many present products would disappear, new products would have no present equivalent, and apparently familiar products would embody different materials, production methods, quality and associated services. The growth would describe a transformation in human capability, not simply a multiplication of present physical quantities.

The same problem affects the interpretation of the price level. A computer can retain a similar sticker price through successive generations while its productive capability increases by orders of magnitude. Its observed price has remained stable, but its quality-adjusted price per unit of computation has collapsed. Future households may continue to see familiar nominal prices while receiving products that perform tasks which are currently extremely expensive or impossible.

Long-lived products need not follow the aggregate index either. A loaf of bread might retain a similar nominal price after a century because its remaining costs include land, energy, distribution and labour. It would nevertheless have become expensive relative to goods whose production was transformed by technology. The aggregate price level can fall greatly even though some persistent products retain their prices and some scarce goods rise in price.

Dividing nominal output \(PY\) between a price index \(P\) and real output \(Y\) therefore becomes increasingly dependent upon measurement conventions as the interval grows. Statistical agencies must account for quality changes, substitution, new products and products which cease to exist. Over a century of rapid innovation, an apparently precise real-growth multiple should not be mistaken for a directly observable physical quantity.

Growth and use of the Earth's resources

Economic value can grow while the use of energy and material resources falls. Miniaturisation, durability, recycling, precise manufacturing, efficient logistics and the replacement of physical products with digital services can produce greater capability from less material. A smartphone, for example, replaces or reduces the use of cameras, maps, standalone GPS navigators, music players, calculators, payment terminals, voting ballots, physical identity and driver's-licence cards, letters, postal services, tickets, printed restaurant menus, catalogues, shops, printed documents and travel previously required for banking, shopping, voting, administration, communication and meetings.

Let \(R\) be resource throughput and let \(\epsilon\) be the resource intensity of real output:

\[ R=\epsilon Y. \]

The proportional change in resource use is

\[ \frac{\dot R}{R} = \frac{\dot\epsilon}{\epsilon} + \frac{\dot Y}{Y}. \]

If real output grows by eight percent a year while resource intensity falls by twelve percent, total resource throughput falls by approximately four percent a year. This is absolute decoupling: the economy becomes more valuable and capable while placing less demand upon the Earth's resources. If resource use grows more slowly than output but still grows, the result is only relative decoupling.

Efficiency does not guarantee absolute decoupling. Making a resource cheaper to use can increase demand for it, offsetting some or all of the saving. At ten percent annual real growth, resource intensity must fall by more than ten percent a year for total throughput to decline. Digital goods can achieve extraordinary improvements for long periods, while food, housing and other activities retain irreducible physical requirements.

A complete model should therefore keep monetary value, real output and physical effects separate. It can add a renewable or exhaustible resource stock \(S\) and an environmental burden \(E\):

\[ \dot S=\operatorname{regeneration}-R, \] \[ \dot E=\operatorname{waste}(R)-\operatorname{recycling}-\operatorname{absorption}. \]

Ravel can then test whether a growth path merely uses resources more efficiently, actually reduces total extraction, or transfers pressure from one resource or ecosystem to another. Real output is also not identical to welfare: a larger \(Y\) does not by itself establish greater happiness, freedom, health or environmental quality. Those outcomes must be represented and assessed separately.

Prices under conserved money

The quantity identity \(MV=PY\) relates the conserved currency supply \(M\), its velocity \(V\), the price level \(P\), and real output \(Y\). Since \(\dot M\) is zero,

\[ \frac{\dot P}{P} = \frac{\dot V}{V} - \frac{\dot Y}{Y}. \]

When velocity is stable, increasing production causes the price level to fall. A five percent increase in real output produces approximately five percent deflation. The purchasing power of each conserved currency unit rises because it represents a larger quantity of goods and services.

A simulation can replace the identity with an explicit adjustment process. If \(C_r\), \(I\) and \(G_r\) are real private consumption, investment and public-service demand, one possible equation is

\[ \frac{\dot P}{P} = \lambda_P \left( \frac{C_r+I+G_r-Y}{Y} \right). \]

Competition among lenders

Ravel can use a lender dimension to represent many independent lenders. For lender \(i\), let \(D_i\) be outstanding loans, \(L_i\) new lending, \(Q_i\) repayments, \(d_i\) the default rate, \(F_i\) funds supplied by savers, \(\rho_i\) its realised return and \(s_i\) its share of entrusted savings. Its loan book changes according to

\[ \dot D_i=L_i-Q_i-d_iD_i. \]

A simple realised return is

\[ \rho_i = \frac{r_iD_i-d_iD_i-\operatorname{cost}_i}{F_i}. \]

Savers can move money towards lenders producing superior risk-adjusted results. This evolutionary selection can be represented by

\[ \dot s_i = \gamma s_i(\rho_i-\bar\rho), \qquad \bar\rho=\sum_i s_i\rho_i. \]

An above-average lender attracts a growing share of savings, while a below-average lender loses it. The model can therefore test VoTa's claim that profit and loss move capital towards people who are better at identifying productive businesses and pricing risk.

UBI as an automatic circulation mechanism

The UBI pool receives its fixed share of consumption tax and distributes the entire pool among eligible citizens. In continuous time,

\[ \dot M_U=\tau_uC-U. \]

With rapid distribution,

\[ U\approx\tau_uC. \]

Because UBI moves purchasing power towards people likely to spend it, it creates a stabilising circulation loop:

\[ C \longrightarrow \tau_uC \longrightarrow U \longrightarrow C. \]

Simulation can determine how strongly this loop supports consumption following a fall in employment, investment or velocity.

Tax choice as evolutionary allocation

Let \(\theta_j\) be the share of public-service tax directed to service \(j\), \(v_j\) its measured value to taxpayers, and \(\bar v=\sum_j\theta_jv_j\) the weighted mean. Movement of funding towards more highly valued services can be represented by

\[ \dot\theta_j = \eta\theta_j(v_j-\bar v), \qquad \sum_j\theta_j=1. \]

Tax choice therefore changes the composition of public-service demand without changing its total flow. Ravel can display the resulting allocation by taxpayer group, provider, service, region, rating, cost and outcome.

Debt under productivity-driven deflation

VoTa prevents repayment from destroying currency, but falling prices increase the real burden of a fixed nominal debt. Let the debt burden relative to nominal output be

\[ d=\frac{D}{PY}. \]

Its rate of change is

\[ \frac{\dot d}{d} = \frac{\dot D}{D} - \frac{\dot P}{P} - \frac{\dot Y}{Y}. \]

The model can compare fixed nominal loans, price-indexed loans, revenue-participation finance, equity investment, and loans of different duration. This would show how a free lending market adapts contracts and interest rates to expected deflation.

Comparison with conventional banking

The clearest experiment is to construct two otherwise identical models. In a conventional commercial-bank system, changes in bank debt change the deposit-money supply:

\[ \dot M=\dot D_{\mathrm{bank}}. \]

Under VoTa,

\[ \dot M=0, \qquad \dot D=L-Q-\Delta. \]

The same shocks can then be applied to both systems: a property crash, widespread defaults, a fall in consumption, an increase in saving, lender failures, rapid entry of small lenders, or a major productivity improvement. The comparison would distinguish instability caused by bank money creation from instability caused by debt, expectations, poor investment and changes in velocity.

Simple stability experiments

A small closed model can demonstrate stability or instability before the full stock-flow model is built. Let \(y\) be economic activity relative to its steady level and \(\ell\) productive lending relative to its steady level. Consider

\[ \dot y=-ay+b\ell, \] \[ \dot\ell=cy-d\ell. \]

Activity dissipates at rate \(a\), lending raises activity with strength \(b\), stronger activity encourages lending with strength \(c\), and repayments, losses and lender caution reduce lending at rate \(d\). The system's characteristic equation is

\[ \lambda^2+(a+d)\lambda+(ad-bc)=0. \]

For positive parameters, the steady state is stable when

\[ ad>bc. \]

For example, \(a=0.4\), \(d=0.3\), \(b=0.2\) and \(c=0.2\) give eigenvalues of approximately \(-0.144\) and \(-0.556\). Any small disturbance in activity or lending therefore decays. Lending can amplify activity temporarily, but its feedback is weaker than the forces returning both variables to their steady levels.

The same model is unstable when

\[ bc>ad. \]

Keeping \(a=0.4\) and \(d=0.3\), but using \(b=0.8\) and \(c=0.6\), gives eigenvalues of approximately \(0.345\) and \(-1.045\). The positive eigenvalue means that a disturbance containing that mode grows: lending raises activity, stronger activity attracts still more lending, and this feedback overwhelms repayment and adjustment. A realistic model would eventually restrain the expansion through limited profitable projects, rising prices, shortages, defaults or depleted lender balances.

A simple nonlinear model can also generate a persistent business cycle. Let \(y\) again be the activity gap and let \(v=\dot y\) represent its momentum, including the immediate effect of changes in investment and productive lending:

\[ \dot y=v, \] \[ \dot v=\mu(1-y^2)v-\omega^2y, \qquad \mu>0, \quad \omega>0. \]

Near the steady state, \((1-y^2)\) is positive, so lending and investment momentum amplify small movements rather than allowing them to disappear. At high or low activity, it becomes negative: capacity constraints, poor prospective returns, defaults and lender losses then brake the movement. The result is a stable limit cycle. Small fluctuations expand towards it and unusually large fluctuations contract towards it. The economy repeatedly passes through expansion, restraint, contraction and renewed lending.

None of these behaviours requires a changing currency supply. They arise from changes in the allocation and velocity of a conserved currency, expectations about business returns, and delayed responses by savers, lenders and firms. In Ravel or Minsky, sliders for \(a\), \(b\), \(c\), \(d\), \(\mu\) and \(\omega\) would make the transition between convergence, divergence and cycles immediately visible. These small experiments would then be joined to the Godley-table model so that every simulated lending and investment flow also moves currency between named accounts.

What stability means in practice

Stability does not mean prosperity, full employment or rapid growth. It means that a disturbance does not push the economy progressively farther from its existing path. A stagnant economy can be stable because every attempted expansion quickly disappears. A growing economy can also be stable when temporary departures converge back to a rising long-term path. The level and growth rate of that path are separate questions from its stability.

In the simple lending model, \(bc\) measures the strength of the reinforcing loop from lending to activity and back to lending. Productive lending finances investment, investment raises employment, output and profit, and those results encourage savers and lenders to finance another round. The product \(ad\) measures the forces that dissipate the expansion: projects finish, expenditure leaks into saving, loans are repaid, losses are recognised, lenders become cautious and the supply of attractive unfunded projects declines.

The condition \(bc>ad\) therefore has a direct economic meaning. An initial increase in productive lending generates enough additional activity to induce more new lending than is removed through repayment, losses and caution. Each passage around the loop is larger than the previous one. When \(bc<ad\), the same process begins, but each successive round is smaller and activity returns to its previous path.

Short-term instability can be productive

An unstable steady state is not automatically a bad result. A major invention, a new source of energy or the removal of an artificial restriction can make the old level of production obsolete. If lenders recognise the opportunity, lending, investment, employment and output can reinforce one another. The economy should move away from its former state because that state no longer represents what it can produce.

During such an expansion, the condition \(bc>ad\) describes entrepreneurial discovery spreading through the economy. Successful projects reveal related opportunities. Workers acquire new skills, suppliers expand, infrastructure becomes worthwhile and lenders learn which proposals work. A model that forced every disturbance to die away would treat this desirable structural transformation as a defect.

The practical question is not whether the economy ever becomes locally unstable. It is what stops the reinforcing process, where the transferred currency goes, what remains after the expansion and who bears the cost of mistakes. Productive instability leaves useful factories, tools, knowledge, networks and lower production costs. Destructive instability leaves inflated asset prices, unpayable debts and little additional productive capacity.

Why a VoTa expansion has limits

Under VoTa, lenders transfer existing Central Bank currency. They cannot create a matching deposit when they make a loan. A lender can expand only by attracting currency from savers, retaining its earnings or receiving repayments. More lending from one account therefore means less immediately available currency in another account. The monetary constraint is present throughout the boom, not imposed later by a central bank.

This does not prevent rapid growth. Currency that was being held idle can finance production and then circulate as payments to workers and suppliers. Its velocity rises, while new capital and better organisation raise real output. The same currency unit can support many successive transactions without being created anew.

The expansion becomes self-limiting as idle balances are committed and the best projects obtain finance. Lenders must offer savers better terms, investigate increasingly marginal proposals and allow for emerging shortages. At the same time, repayments replenish lender balances and failed projects impose losses. In the model, \(b\) and \(c\) decline while \(a\) and \(d\) rise. The economy moves from \(bc>ad\) to \(bc<ad\), ending the acceleration without requiring an authority to order a general credit contraction.

A complete simulation should make these coefficients endogenous rather than leaving them fixed. The return from an additional loan should decline as profitable opportunities are filled; default risk should rise when debt service becomes large relative to borrower revenue; and saver funding should respond to each lender's realised record. The linear equations describe the local tendency, while these nonlinear relationships determine where the expansion ends.

Downturns and lender failure

VoTa does not abolish bad judgment. Lenders can finance businesses that fail, savers can follow fashion, and many independent lenders can make the same mistake. Falling expected returns can cause new lending to contract quickly, producing unemployment and unused capacity in industries that had expanded too far.

The crucial monetary difference is that repayment and default do not automatically destroy the currency used for ordinary payments. Repayment transfers currency from borrower to lender. A default writes down a financial claim, but it does not erase currency already paid to workers, suppliers and other recipients. A failed lender distributes losses among its owners and participating savers without mechanically contracting the total currency supply.

Failures should therefore be common, visible and limited enough to perform an economic function. Poor lenders lose funds and influence; capable lenders and new entrants acquire them. The objective is not to prevent every failure. It is to prevent the failure of one institution from disabling the payments system or forcing every other lender to contract at once.

Decentralisation assists this process when lenders use different information, serve different industries and fail independently. It does not help when they all copy the same model, lend against the same assets or depend on the same source of funding. A Ravel model should therefore vary the correlation of lender decisions as well as the number of lenders. Ten thousand lenders behaving identically constitute one systemic strategy, not genuine decentralisation.

Circulation during a contraction

A fixed quantity of currency does not imply a fixed level of spending. If households and lenders hold larger idle balances, velocity falls and firms receive less revenue. Production and employment can contract even though no currency has disappeared. VoTa removes monetary destruction through loan repayment, but it does not remove instability caused by changes in circulation.

The UBI pool creates a continuous recirculation channel. Part of consumption expenditure returns to individuals and can become consumption again. This reduces the accumulation of currency in low-circulation accounts and gives every person some purchasing power during a downturn. Because the pool is funded by consumption tax, its receipts also fall when consumption falls; it cushions the circulation shock rather than providing an unlimited external stimulus.

Falling prices provide another adjustment. When production grows faster than monetary expenditure, each currency unit purchases more output. That rewards saving and allows a given nominal income to support greater real consumption. Prices, wages and contracts must be able to adjust for this mechanism to work without prolonged unemployment.

Debt and deflation

Productivity-driven deflation becomes dangerous when debts remain fixed in nominal currency. If prices and borrower revenue fall while required repayments do not, the real debt burden increases. Defaults can then reduce lender returns, frighten savers and curtail productive lending even while the physical capacity of the economy is improving.

An unregulated lending market has strong incentives to adapt its contracts to this environment. Lenders can use revenue-participation finance, equity, price-indexed principal, shorter maturities and repayment schedules tied to cash flow. Lenders that continue to price long nominal contracts as though persistent inflation were guaranteed will lose money. The speed with which selection removes such practices is a central question for simulation.

Long-term behaviour under VoTa

The long-run pattern is an evolutionary economy rather than a motionless equilibrium. New lenders enter with different knowledge and methods. Successful lenders attract savers and expand; poor lenders shrink or disappear. New firms test ideas, successful production methods spread, and failed projects release people and resources for other uses. Periods of rapid expansion alternate with consolidation and reallocation.

Competition does not guarantee permanent decentralisation. The lender-share equation rewards past success and can produce concentration as a few lenders accumulate reputation and funds. VoTa removes regulatory privileges that protect large institutions, and unrestricted entry allows challengers to form, but savers can still herd towards size. The model should test whether new entry and the diseconomies of managing a large loan book overcome this concentrating force.

VoTa's strongest stability claim is therefore not that cycles, defaults or recessions disappear. It is that these events occur without an elastic bank-created money supply during the boom and without the corresponding destruction of transaction money during repayment. Instability remains in investment, expectations, debt contracts and velocity, where profit, loss and entry can act on it directly.

The desirable long-term outcome is bounded instability: strong enough for new ideas to displace old arrangements, but constrained enough that errors remain local and the economy retains its capacity to transact. Whether VoTa produces that outcome depends on lender diversity, the quality of debt contracts, the mobility of saver funds, the response of prices and wages, and the strength of the UBI circulation loop. These are results for the dynamic model to establish, not assumptions to place inside it.

Questions for simulation

A complete Ravel model should test:

• Whether decentralised lender selection raises the proportion of savings invested productively.
• How quickly new lenders enter and unsuccessful lenders lose saver confidence.
• Whether the UBI circulation loop stabilises consumption during a downturn.
• How tax choice reallocates funding among competing public-service providers.
• How changes in saving and velocity affect employment when the money supply cannot expand.
• How lending contracts adapt to productivity-driven deflation.
• Whether defaults impair lending without causing a contraction of the currency supply.
• How VoTa responds to the same shocks that generate debt-deflation crises under conventional banking.

The result would be an auditable mathematical account of VoTa's economy. It would show where every currency unit is held, how every flow changes sectoral balances, how real production interacts with money, and which feedbacks produce stability, growth or contraction.