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Stock-Flow Norms and Systemic Stability

We offer a dynamical systems translation of the Godley and Cripps (1983) framework. Stability of an economy is shown to depend on the concatenation of four parameters: a steady-state money income ratio, a speed-of-adjustment-of-assets coefficient, an inventory accumulation index, and the share of government in aggregate income.

Stock-Flow Norms and Systemic Stability

We offer a dynamical systems translation of the Godley and Cripps (1983) framework. Stability of an economy is shown to depend on the concatenation of four parameters: a steady-state money income ratio, a speed-of-adjustment-of-assets coefficient, an inventory accumulation index, and the share of government in aggregate income.



he dynamics of the wealth of nations can be captured in one of two ways. The common procedure is the use of microeconomics on the grounds that systemic outcomes cannot be explained by recourse to means other than the attempts of individuals, albeit with their bounded rationalities, to maximise their ends. An alternative logic is founded on the precept that economies warrant consideration directly. The discipline of doubleentry bookkeeping ensures that items in the rows and columns of national income accounts sum and cancel out appropriately. In a sectoral disaggregation of the economy some ratios between categories will display a certain durability and others will change, often discontinuously, under specified conditions. While there must be behavioural drives underlying these ratios, we will take them as institutional data. Indeed, one advantage of manipulating macroeconomic identities is that certain sets of choices get deleted and some degrees of freedom are made available allowing, thereby, for the possibility of policy intrusions. An axiom here is that it is the ensemble of these institutional scalars that generate systemic outcomes Such an orientation encourages a pluralist perspective on monetary and financial institutional combinations. It is not true, for example, that societies are moving inexorably towards laissez-faire described by independent central banks and the logic of capital markets.

R Boyer (2005) has finely distinguished at least three other macroeconomic configurations. A statist regime would be one in which monetary and credit policies are rigidly circumscribed. The economy delineated below would not be different from a social-democratic regime, one in which full employment figures prominently in the objective function of the central bank governor. Our system is, however, best described as mesocorporatist, as an economy in which banks play a paramount role in lending and capital accumulation. The central bank is respectful of stable monetary and financial equations. Another practice from which an empirical orientation departs is the habit of thinking in terms of real variables and worrying, thereafter, about how best to incorporate monetary and financial variables. We contend with the capitalist economy as it exists, as a mechanism of interconnected monetary circuits. A respect for financial history would imply that at least the following components be included in any narrative: public finance, money, banking, central banking, securities markets, and corporations [Sylla 2005]. These elements interlock in surprising and not obvious ways. For instance, public debts often originated in wars and thereafter gave birth to securities markets. These markets then funded corporations. A subset of banks privileged to finance governments were picked to be central banks and now are expected to ensure the stability of entire monetary systems. Securities markets finance governments, corporations, and banks. Banks invest in securities, accept them as collateral for loans, and so on. The evidence, then, works in favour of the Chartalist view that modern money comes into existence alongside the ability of the state to levy taxes by fiat. Thus, an authoritarian system would militate against the evolution of commercial banks and credit money in the form of bills of exchange, bank notes, and deposits. Conversely, a country without a solid monetary authority was unlikely to generate a stable money space. Such a regime would be described by competing coinage, unstable prices, and chaotic monetary arrangements [Goodhart 2005:819-20].

The collapse of a strong central bank, it is believed, would lead to the downgrading of the quality of the mint and relapse into barter. It is the absence of an established money axis undergirded by a monetary authority that blocked the growth and development of private sector financial institutions. One outcome of Chartalist reasoning is a look askance at the European Union wherein constituent countries enjoy fiscal autonomy while having ceded their monetary authority to the European Central Bank. The prognosis is that the stability and growth pact is unstable.

The focus of the present essay is fluctuations and potential crises generated endogenously by a modern closed economy. In that case inventory behaviour must form an important constituent of the model. Students of business cycles are aware that the net investment in inventories by businesses is very volatile and accounts for most of the aggregate fluctuations in a recession [Solow 2005]. A plethora of not-mutually-excluded hypotheses are available. The obvious reason for a firm to hold inventories is that sales might fluctuate unpredictably. One empirical fact is that inventories of goods-in-process and raw materials and components are correlated with current production. Businessmen might speculate on the prices of raw materials and hold inventories for that purpose. They would purchase more than needed when the price is low, using the surplus when the price is high. Other theories pertain to relationships with suppliers and customers and financiers. The problem of modelling the accumulation of inventories is to weigh these different impulses so that they can be encapsulated in a formula. In general, a primary cause of systemic failures and crises lies in the possibility of chaos arising from the coupling between sectors in an economy, including especially the banking sector. In a review of models of financial fragility C A E Goodhart and L Zicchino (2005) find most of the stories wanting on the following grounds: The focus is on primary optimisation exercises conducted by agents and there is inadequate attention given to dynamic secondary interactions between them. Relatedly, much work focuses on single institutions and the frameworks do not permit the modelling of feedforward loops between institutions. In particular there is need to incorporate the connection between the banking and other sectors. We contend that the W Godley and F Cripps (1983) (hereafter GC) classic is a fruitful methodology to employ towards these ends. In the next few sections we proffer a dynamical systems treatment of the definitions and categories therein.

Economic and Political Weekly March 18, 2006

Godley and Cripps (1983):The Closed Economy with a Monetary-Fiscal Stance

A careful distinction between stocks and flows results in a serious consideration of the speed with which variables might change (GC:36-38). For example, the relationship between the stocks of a commodity bought by a trader and sold by her can vary. Then, if there is a continuous flow of “input” and “output” the length of time each unit of the commodity is held by the trader would be equal to the ratio of the purchases per unit of time to the total stock at each point of time. On the other hand, suppose the trader buys and sells a certain quantity of the commodity per unit of time but that at each point of time holds inventories equal to twice that quantity. This means that each unit of the product remains in the distribution pipeline for two months. Here the time lag between inflow and outflow is equal to the stock-flow norm. Still another possibility is that the trader does not follow an unchanging formula. Her sales might be constituted of a certain percentage of purchases in a certain period, the rest flowing out from inventories. In this situation, if the system is stationary, the same stock-output ratio would be observed. However, around this number some stocks of the commodity would pass through more rapidly while others would do so more slowly. The discussion leads to the concept of a “mean lag” which is the average time it takes for a commodity to pass through the businessman’s hands. The notion is important in an analysis between flow variables.

Writing the aggregate disposable income of all private institutions as YP, and the net income of government as YG, total national income in each period is expressed as (GC:28)

Yt = YGt + YPt

The change in the aggregate money value of inventories in each period must be added to the aggregate money value of final purchases so as to equalise total expenditure and total money income. In equations,

Yt = FEt + ΔI

where FE denotes the money value of final purchases, and ΔI is the change in the money value of inventories in the accounting period (GC:33).

Businesses are assumed to finance their holding of inventories by borrowing from banks. Then the total value of inventories will be equal to the aggregate value of the debt of the production and distribution sector to the financial institutions. Since there are no other types of debt, total loans by the banking system will be equal to the total value of inventories (GC:73). In terms of notations,

LI = I

where LI is total outstanding bank loans.

The steady state of final expenditures and stocks of money corresponds to a constant flow of money income. GC (p 54) derive the following proposition: A steady state ratio of money to income is equal to the mean lag of expenditures behind incomes. In the steady state the actual stock of money, FA, is equal to the desired stock FA* which is some proportion alpha of the total income flow Y (GC:84). That is,

FAt = αYt

Since the total stock of money equals the total stock of loans, the following relationship also holds in the steady state

Yt = LIt/α = It/α

Assume that the change in the stock of money which actually takes place in a certain period is a certain portion of the gap between the stock of money inherited from the previous period and the stock of money warranted by the steady state income flow. Formally (GC:88),

FAt – FAt-1 = φ(FA* – FAt-1)

In order to introduce the government budget in the model, final sales must now distinguish between government expenditure, G, and private sector purchases, PE. That is to say (GC:102),

Yt = Gt +PEt + ΔI

Combining the national income identity with the first equation, dividing aggregate money income into private disposable income, and net government income, we get the familiar relationship stating that the private sector surplus must be equal to the government deficit. That is (GC:105),

YPt – (PEt + ΔI) = Gt - YGt

The government deficit in each period is financed by government borrowing in the same period. Assuming that end-period government debt is supported by bank lending to the government, GD (GC:105-06), we have

Gt – YGt = GDt – GDt-1

The private financial surplus, in its turn, equals the change in the money stock minus private borrowing in the same period. Denoting end-period private borrowing from banks by PD,

YPt – (FEt + ΔI) = FAt – FAt-1 + PDt – PDt-1

Rearranging and rewriting, the flow-of-funds identity emerges as

FAt – FAt-1 = GDt – GDt-1 + PDt – PDt-1

The “further assumptions to complete the model” (GC:107-08) are that the flow of government expenditure is exogenous. Also, the government can, by choosing tax rates suitably, determine the share of national income it appropriates. Symbolically,

YGt = θYt

In order to make inventory accumulation endogenous, businessmen are assumed to be possessed of enough foresight and flexibility to keep their end-of-period inventories at a normal level relative to the flow of final sales in the previous period. Formally,

It =


A State Space Representation

The equations can be combined and presented as the following discrete-time dynamic system in two variables.

FAt= (1 – φ)FAt-1 + φγFEt-1

FEt = –[γ – (1-θ)γ/α]FEt-1 + FAt-1 + Gt

In vector-matrix form,

⎛ FA t ⎞⎛1 −φ φγ⎞⎛ FA ⎞⎛ 0 ⎞


⎜⎜ ⎟⎟=⎜⎜ ⎟⎟⎜⎜ ⎟⎟+⎜⎜ ⎟⎟

⎝ FE t ⎠⎝ 1 −[γ − (1 −θ)γ / α]⎠⎝ FE t −1 ⎠⎝ Gt ⎠

The above indecomposable system nicely underscores the Chartalist link made earlier between the sovereign power over the mint and the ability of the state to raise taxes. The connection is simultaneous and two-way [Goodhart 2004]. The ability of governments to have access to an immediate source of purchasing power has been invoked, especially in times of crisis. Without money, taxes would have to be levied on the production, transport, and trade of goods. When taxes are received in goods and labour,

Economic and Political Weekly March 18, 2006

the balance of goods and labour obtained will not be that required for public sector expenditure. In other words, fiat money reduces the transaction costs of both governments and the private sector.

The separation of the two items of the government budget constraint in the above model is eloquent. First, a word on the non-homogeneous portion with a positive coefficient of unity is on order. The macro system highlights the fallacy of composition enunciated by countercyclical policy [Baumol 2005]. Consider an economy in the pits of a depression wherein tax revenues have fallen because of reduced incomes and thus reduced expenditures (the negative coefficient on FEt-1). Just the second equation in the homogeneous part of the dynamical system might indicate a stance of fiscal consolidation. Typically, underlying this milieu would be rising debt obligations, leading to the conclusion that fiscal retrenchment was called for if a debt trap was to be avoided. Keynes and later Abba Lerner with his development of the concept of functional finance advocated the polar opposite of this position [Lerner 1951]. It is incorrect to treat fiscal instruments in combination. The well-known sequence is worth repeating: The more the government increases autonomous spending Gt without a corresponding increase in tax revenue, the better off society will be. The deficit spending puts purchasing power in the hands of the public raising thereby the demand for goods and services. Sales expand inducing further production and more jobs. The budget deficit is also pared by the addition to tax collections as private incomes rise, as well as the pruning of government expenditures such as unemployment benefits. The argument, grounded in an individual inter-temporal optimisation exercise, that the government deficit is a burden we bequeath to our grandchildren is another illustration of the fallacy of composition. Suppose a government in the present sharply increases its expenditure on goods and services, financing the programme through the issue of government bonds. The labour, steel, power, and so on are now unavailable for private use. This is a burden that falls on the public in period t and need not impact on the supply of goods and services to future generations. It is not even the case that government debt incurred today is a burden tomorrow when the debt has to be repaid. Suppose that government bonds that finance the debt are due for redemption 20 years hence. At that point of time, the government raises taxes by just the amount required to cover the nominal debt. The sum is a burden to those who must pay it in taxes but it is matched by exactly the same figure that is available as cash to bondholders. If, for instance, government bonds are held by individuals in accordance with their incomes, the wealthier an individual, the greater her holdings. In that case, if the tax is also a proportional income tax, the repayment process will not generate any significant transfer of purchasing power. The money will be taken from the wealthy only to be returned to the same individuals. The disjunction between components of the government budget and various concepts of deficits can be illustrated with a case study. In a study of the (in)famous Japanese problems with public finances in the 1990s, M Tsuri (2005) has distinguished between cyclical and discretionary components of the deficits. While there are no specific effects of public deficits or structural deficits on macroeconomic variables, discretionary deficits and discretionary expenditure have multiplier effects. The author finds that stimulus policies have multiplier effects while overall public deficits do not. Decreases in the revenue of both the discretionary as well as the cyclical component have had a greater impact on fiscal deficits than discretionary expenditures.

Denoting the coefficient matrix by A, we recall that a fixed point of the above system is locally asymptotically stable if the two eigenvalues of the Jacobian matrix, calculated at the fixed

point, are less than one in modulus. Necessary and sufficient conditions are [Medio and Lines 2001]: 1 + trA + detA > 0 (1) 1– trA + detA > 0 (2) 1– detA > 0 (3)

In a non-exhaustive typology, stability is ensured by the following two cases:

Case I: φ= 1, γ< 2

There is a vital difference between treating inventories as exogenous and endogenous (GC:86). The quantity of credit supplied by banks to finance inventories is sensitive to changes in income. A small change in γcould alter the income-expenditure flow dynamic, unleashing a cumulative expansion or contraction of borrowing. For example, if the growth in income causes loans to expand faster than people want to hold the money created corresponding to the income flow, there would be an explosive growth in aggregate income. Alternatively, if the demand for loans is weak relative to income, aggregate income would continuously fall. Stability is ensured by the speed of adjustment of asset stocks to their steady state values. Here asset stock adjustment is assumed to take place rapidly.

Case II: φ= 0, θ= α/γ (if α= 1, γ< 2)

In the polar opposite instance, the adjustment of money stocks is very sluggish. (GC:128 deal with a situation in which φ→0, delivering an exploding economy. A solution is provided by the tax system.) In our case, if only φ= 0, the system is indeterminate. A precise fiscal stance is called for to render the economy determinate. If the mean lag between expenditures and income is very short (the value of unity is only indicative), the earlier inventory accumulation norm can be met.

The stability requirements of the two cases can be enriched with the inclusion of government expenditure (GC:121). In sum, the stability of a fiscal system depends on the steady-state money to disposable income ratio and the ratio of inventories to sales and the ratio of the former to the latter. In recent years, there has been some discussion regarding the stability of demand for money functions. The LM curve has been jettisoned from some textbooks. However, some reason that a stable money demand function can be posited if the right monetary aggregate is sought [Teles and Zhou 2005]. In the US, from the late 1970s to the early 1980s estimated demand for narrow money functions turned unstable. However, if technological changes and the dilution of regulatory practices are considered, a larger monetary aggregate would deliver the stability required. The presumption, of course, is that the authorities are interested as they desire to provide liquidity at stable prices. In the case of the US, electronic payments breakthroughs exemplified by credit cards as well as a dispensation towards banking deregulation made other monetary assets as liquid as M1. In the case of the second norm, if the scale of the response of inventories to change in aggregate income is large, the responsiveness of the system is elastic. The third influence is the taxation rate. If θwere high, then the response of the system would be immediate because the addition to income would take place with little addition to disposable income. On the other hand, if there were no taxation and government expenditure were maintained indefinitely at a certain level, the response of total national income would be infinite. National income would grow indefinitely because the constant flow of government expenditure would generate indefinitely rising stocks of money and flows of private income.

We now look at situations when a change in the parameters leads an economy from evenly rotating to chaotic. A Flip Bifurcation

Economic and Political Weekly March 18, 2006 occurs when equation 1 is an equality, the other two inequalities holding. In our model, they occur under the following two circumstances.

Case III: φ = 1, γ < 1

The case mirrors Case I leading to the conclusion that inventory holding cannot be a negligible portion of the final sales in the previous period. Indeed, for stability in our model precise bounds are provided under a high adjustment-to-steady-state assets norm, that is, 1 < γ < 2.

Case IV: φ = 0, θ < 2α/γ

In this dual of Case II, the burden on taxation policy becomes onerous. It is not too difficult to appreciate that in a situation of arbitrarily low adaptation of asset holdings to steady state values, when the monetary economy itself is ill-defined, tax revenues should be high enough to deliver the requisite stability.

A Fold Bifurcation occurs when the left hand side of equation 2 this time is equal to zero, the other two inequalities remaining unchanged. In our framework, it is represented by, among other possibilities,

Case V: φ = 1, θ = α/γ

Note that fiscal policy under φ = 1 has been undefined until now. Fortunately, it turns out that a tax regime that is familiar from the other cases cannot do worse in ensuring the stability of a financially layered economy.

A Neimark-Sacker Bifurcation is a combination of the earlier two bifurcations. That is to say it occurs when detA = 1. In our case, after ignoring some economically impossible cases, it leads to

Case VI: φ = 0, θ = α/γ – α + 1

Viewed in tandem with Case IV, the suggestion, once again, is that the tax to income ratio has to be sensitive to the mean lag between income and expenditure as well as the inventory accumulation norm.

The model can be extended to include equity (GC, chapter 13). Once equity is added stocks of privately owned land, buildings, equipment enter; so do private insurance and pension institutions. The definition of assets now extends to the broad concept of wealth not excluding new instruments like derivatives, swaps, futures, options, and the like. As a simplification, GC exclude assets from the aggregate stock when they are held against debt. For instance, transactions in equity are conducted exclusively between private sector agents. They need not appear in the flow budget constraint for the private sector as a whole. The implications for the stock-flow system is that value of the stock of assets can now be altered by capital gains. Still, aggregate flows of private income and expenditure must adjust to levels at which the stock of assets is appropriate relative to income. Capital gains then play the same role as the accumulation of inventories and personal loans. All the same, it might be worthwhile to note here the concern expressed in some quarters about the “search for yield” by financial intermediaries and investors responding to the compression of yields by moving down the menu of credit risk offerings to higher-yielding riskier assets [Plantin, Sapra and Shin 2005]. The greater flow of funds into riskier asset classes further contributes to the compression of yield spreads inducing migration further down the risk spectrum in a potentially destabilising spiral. Mark-to-market activity will have a huge potential impact when extended to loan activity and hitherto illiquid assets. Financial innovation in the form of credit derivatives has led to the possibility of deriving surrogate prices for standardised loans. The answer of GC is that in the absence of a fiscal system this alarm is warranted. The introduction of the new instruments would make a crisisprone system more so. With a fiscal system stock-flow processes are dictated by fiscal instruments with a short mean lag.


The time dimension is a critical ingredient in modelling dynamics. Speeds of adjustment of different variables vary greatly. In particular, the new ordering of work and production is likely to dramatically reduce the need to hold huge stocks of inventories. Information and communication technologies (ICTs) are hastening just-in-time production and delivery and the process of arbitrary shortening of new service development and implementation cycles. The stability properties of such innovations are unclear and we suggest bounds within which inventory holding norms might be supported by the overall economy. While the money income norm has remained tacit, except on occasion when α = 1 has been used for pedagogic purposes, the implication is that the general level of employment depends upon the rate at which money is spent on purchasing currently produced goods and services. One of the corollaries of functional finance, the manner in which it departs from sound finance, is the proposition that taxes should not be imposed for the purpose of collecting revenues. While it is true that taxes make monies available to the government, there are other well-known ways to realise that objective. The essential objective of taxation in a social system is the effect it has on private norms and thereby behaviour. In contrast with a market fundamentalist model advocating a secular decline in state intervention across the fiscal board, the connection between our four norms is summarised in the following theorem: under all values of the speed-of-adjustment of assets coefficient, the share of government in aggregate income must not fall below the ratio of the steady-state money income ratio to the inventories to final sales ratio.




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Economic and Political Weekly March 18, 2006

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