Stock–Flow Analysis
Stock–Flow Analysis
The purpose of stock-flow analysis is to describe the formation of economic plans and the determination of market prices in an economy where one or more commodities (e.g., wheat, bonds, money) are traded simultaneously on both capital and current account. Traditional demand and supply analysis is not entirely silent on this subject, but it is uncomfortably vague. Walras and later general equilibrium theorists focused attention on the existence and stability of a set of market-clearing prices in pure stock and pure flow models, i.e., models in which no means exist whereby individuals can convert current income into present wealth, or present wealth into future expenditure. In a pure stock economy, assets can be exchanged only for other assets; in a pure flow economy, income can only be consumed. Explicit analysis of saving, investment, and growth processes is conceptually possible only in the context of a stockflow model.
Marshall and later partial equilibrium theorists did more justice to the special characteristics of a stock-flow economy. The familiar trichotomy of market equilibria into temporary, short-run, and long-run periods was conceived specifically to deal with transitory saving and investment processes. But this analytical schema was applied systematically only to business transactors. Thus, changes in long-run supply induced by business decisions to vary physical plant were investigated in detail, while changes in long-run demand induced by analogous saving decisions of households were largely ignored. In the end, therefore, Marshall and his followers contributed little more than did Walras and the neo-Walrasians toward the development of a coherent theory of price-quantity behavior in a stock-flow economy.
The existence of this gap in traditional value theory was gradually forced upon the attention of economists by the prolonged debate about the foundations of economic analysis which followed the publication of Keynes's General Theory in 1936. Even so, more than 15 years elapsed before the appearance, in 1954, of an explicit model of price determination in a stock-flow economy (see Glower 1954; Glower & Bushaw 1954). Subsequent contributions to stock-flow analysis (particularly Archibald & Lipsey 1958; Chase 1963; Hadar 1965; Smith 1961) have extended its boundaries to include, as special cases, both the general equilibrium theory of money and established microeconomic analysis. Most of this material lies outside the scope of the present discussion. The exposition that follows is intended to provide not a survey of, but an introduction to, the literature of stock-flow analysis.
Basic concepts. The rudiments of stock-flow analysis may be set forth most conveniently by considering an economy in which all commodities are traded in central auction markets at prices established by an independent market authority. In keeping with familiar procedure, we may suppose that individual transactors formulate tentative trading plans at the outset of any given market period, on the basis of given initial asset holdings and given rates of exchange (as reflected in provisional price announcements by the market authority). In general these plans will involve decisions about the quantity of each commodity to be purchased for current consumption, to be purchased to hold for future disposal, to be sold from current production, and to be sold from previously accumulated stocks.
Thus, for any commodity traded in the economy, e.g., the nth, and for any given market period, we may suppose that there are defined the following:
(1) An aggregate stock demand function, Dn, which indicates for any given vector of market prices, P , and any given matrix of individual asset holdings, S (indicating the holdings of each commodity by each individual), the gross quantity of a particular commodity that individuals plan to hold for future disposal at the end of the current market period: Dn = Dn (P, S ).
(2) An aggregate flow demand function, dn, which indicates for any given P and S the gross quantity of a particular commodity that individuals plan to consume during the current market period: dn = dn(p,S ).
(3) An aggregate flow supply function, sn, which indicates for any given P and S the gross quantity of a particular commodity that individuals plan to produce during the current market period: sn = sn(P,S ).
(4) An aggregate stock supply quantity, Sn, defined as the sum of individual holdings of a particular commodity at the outset of the current market period.
Given the “primitive” demand and supply relations (1) to (4), we may proceed immediately to define various “derived” relations that are relevant for describing market trading plans for each commodity. Specifically, we define planned net purchases on capital account—henceforth referred to as holder excess demand—as the difference between aggregate stock demand and aggregate stock supply: Zn ≡ Dn - Sn. Similarly, we define planned net purchases on current account—henceforth referred to as user excess demand—as the difference between aggregate flow demand and aggregate flow supply: zn ≡ dn-sn. Finally, we define market excess demand by the identity xn ≡ zn + Zn. Thus, if N different commodities are traded in the economy, there will in general be 3N market trading relations. Depending on the precise character of the commodities traded, however, certain user and holder excess demands may be ignored. Just as in established price theory, moreover, one of the market excess demand relations may be assumed to be defined in terms of the others, by virtue of Walras' law.
Trading equilibrium . The demand and supply relations of stock-flow analysis, like those of established price theory, are defined by underlying conceptual experiments in which all factors, other than prices, that might influence current economic plans are assumed to be fixed. Thus, the only requirement for individual trading plans to be mutually consistent is that the market authority establish a set of provisional prices such that market excess demand is zero for each and every commodity traded in the economy.
Accordingly, let us suppose that the finalization of individual trading plans in any given market period is preceded by a bargaining process in the course of which provisional market prices are varied in accordance with prevailing conditions of market excess demand. We shall not deal with the details of this process (on this, see Bushaw & Glower 1957; Hadar 1965; Negishi 1962); we shall simply assume that the bargaining process is globally stable and very heavily damped. We may then argue that the process leads rapidly to the announcement by the market authority of a set of market-clearing trading prices, at which binding exchange transactions may be concluded between individual market participants. Since individuals will then be able (at least in principle) to carry out their respective production, consumption, and asset-holding plans precisely as scheduled, it is natural to associate the establishment of such a set of trading prices with the attainment of a state of trading equilibrium.
In a pure stock economy, where individuals trade only on capital account, trading equilibrium will occur if and only if prices are such that holder excess demand is zero for every commodity; for in this case, user excess demand is identically zero in every market and x'tn a Ztn, where the superscript t denotes the market period. Similarly, in a pure flow economy, where individuals trade only on current account, trading equilibrium will occur if and only if prices are such that user excess demand is zero for every commodity; for in this case, holder excess demand is identically zero in every market and xtn ≡ z'tn. In a stock-flow economy, however, trading equilibrium requires only that market excess demand be zero for every commodity, and this condition may be satisfied even if user and holder excess demands are not zero, i.e., even if individuals in the aggregate are planning to save or dis-save. To be sure, the market clearance condition xtn ≡ ztn + Znt = 0 will automatically be satisfied if user and holder excess demands are both zero. In general, however, this requirement is merely a sufficient, not a necessary, condition for trading equilibrium in a stock-flow economy; for trading equilibrium will also occur if z'tn = -Ztn.
The exception to the last rule concerns what might be called a mixed stock-flow economy, in which some commodities are held for future disposal and some commodities are produced and consumed but no asset can be produced or consumed and no commodity other than an asset can be held for future disposal. An example of such a system is provided by the familiar production and exchange economy of contemporary monetary theory in which the only assets are fiat money and bonds. In such models, market excess demand for each commodity is identically equal either to user excess demand or to holder excess demand for the same commodity; hence, trading equilibrium cannot occur if the aggregate stock demand for any commodity differs from aggregate stock supply. However, trading equilibrium may occur even though some individuals plan to save, provided such plans are offset in each market by dissaving plans of other individuals.
Intertemporal equilibrium . The significance of stock-flow analysis does not lie in what it adds to existing accounts of market bargaining and the determination of equilibrium trading prices. The interest of the subject lies, rather, in the fact that it provides for the first time an explicit conceptual framework to analyze intertemporal saving and investment processes as market phenomena.
In order to indicate the force of these observations, we begin by distinguishing between the formation and the execution of individual economic plans. The bargaining process may be presumed to lead to the establishment of a specific vector of trading prices at the end of any given market period and so to the determination of a set of vectors of mutually consistent production, consumption, and asset-holding plans. However, the theory of bargaining does not itself say anything about actual trading; that is an entirely different subject, which requires separate analysis.
The easiest way to characterize the trading process is to suppose that quantities actually produced, consumed, and traded at the conclusion of the bargaining process are precisely as planned. This assumption is logically permissible, of course, only in special circumstances, namely, when no actual transactions take place except in trading equilibrium. Since this restriction is not peculiar to stock-flow analysis, however, we shall accept it without question here and proceed on the assumption that equilibrium trading plans are in fact carried out by individual transactors at the end of each market period. The question then arises: will completion of the trading process in one period and reopening of the bargaining process at the beginning of the next period lead to the establishment of a set of trading prices identical with or different from those established during the first market period?
If we grant the validity of accepted statical theories of household and business behavior, the answer to this question is fairly straightforward. In pure stock economies, the execution of plans at the end of one market period will not alter the real wealth or income of any transactor, nor will any change in the distribution of assets within individual portfolios alter existing asset-holding plans. Thus, the only effect of the trading process will be to eliminate any initial gaps between desired and actual holdings of various commodities; i.e., individual, as well as aggregate, holder excess demands will be zero for every commodity at the end of the trading process. Therefore, other things being equal, in a pure stock economy a once-over execution of economic plans will eliminate for all time the need for further trade.
A similar result is obtained for pure flow models. As before, the execution of plans does not alter the real wealth or income of any household or the physical assets of any business. The only effect of the trading process is to permit individuals to produce and consume as desired. Therefore, other things being equal, in a pure flow economy a single bargaining process will lead to the establishment of a set of trading prices and transactions quantities that will be maintained throughout all subsequent time.
Our conclusions regarding pure stock and pure flow models may be summarized by saying that, in such systems, trading equilibrium implies intertemporal equilibrium. Such models are not without interest as devices for analyzing elementary bargaining processes. Moreover, they may be made to generate nonstationary price and quantity time series by introducing price and income expectations, wage and interest-rate rigidities, trading at disequilibrium prices, etc. Under no circumstances, however, may such models be considered appropriate vehicles for any but preliminary analysis of price-quantity behavior in an asset-holding economy. For this purpose we must have recourse to stock-flow models.
In general, the execution of economic plans in a stock-flow economy will alter both the real income and real wealth of some households and also will lead to changes in the asset holdings of some businesses. Such effects are inevitable, indeed, if any transactor in the economy plans to save or dissave at the outset of the trading process. As a rule, therefore, the trading process will in itself lead some individuals to revise their production, consumption, and asset-holding plans. Hence trading equilibrium does not imply intertemporal equilibrium in a stock-flow economy.
The truth of the last remark is obvious in cases where the excess user demand for some asset is nonzero in trading equilibrium; for this means that planned production of the asset differs from planned consumption, and hence, that aggregate stocks of the asset will change from one market period to another if plans are executed as scheduled during the trading process. The truth of the remark is less obvious in the case of mixed stock-flow models, where excess user demand is identically zero for every asset and aggregate stocks are necessarily constant over time. Trading equilibrium then requires that holder excess demand be zero for every asset. However, this does not imply constancy over time in the asset holdings of individual transactors following a once-over redistribution of existing asset stocks. For in a stock-flow economy, unlike a pure stock economy, individuals may continue to save and dissave indefinitely, even though aggregate asset stocks never change.
Stability of intertemporal equilibrium . The distinction between trading equilibrium and inter-temporal equilibrium is an inherent and distinctive characteristic of stock-flow analysis. To describe individual economic plans in pure stock and pure flow models requires, as it were, just one analytical dimension—prices. All other determinants of individual conduct are specified in advance, and none can be altered by market trading. In such models logic does not compel us to develop separate theories of trading and intertemporal equilibrium, even though we may find it convenient to do so in certain instances. To describe economic plans in a stock-flow model, however, requires two analytical dimensions—prices and individual asset holdings—because individual asset holdings may be altered by market trading. In the case of stock-flow models, therefore, logic does indeed compel us to develop separate theories of trading and intertemporal equilibrium.
The problems posed by this characteristic of stock-flow analysis have to do mainly with the stability of intertemporal equilibrium. As remarked earlier, the stability of intertemporal equilibrium in pure stock and pure flow models is an immediate consequence of the stability of trading equilibrium. Intertemporal disequilibrium may occur in stock-flow systems, however, either because markets fail to clear or because individual transactors choose to save or dissave. Therefore, even if bargaining processes are inherently stable, a stock-flow economy may fail to converge to a state of intertemporal (stationary) equilibrium if individual asset-adjustment processes are unstable. This possibility will most certainly be realized if one or more individuals in the economy invariably save, and add to previously accumulated resources, some fraction of current income (as is presumed to be true, for example, in von Neumann and other linear models of economic growth and also in most theories of the consumption function). In general, however, the saving behavior of individuals will depend on market prices and asset holdings, as well as income. Therefore, even if individual asset-adjustment processes tend to be unstable at some initial set of market prices, intertemporal instability of the economic system may be avoided by appropriate intertemporal adjustments in market prices. Whether intertemporal instability deserves to be regarded as anything more than a theoretical curiosity is an open question at the present time. The answer is of obvious relevance for such practical problems as the lag effects of monetary policy, econometric forecasting of consumption and investment expenditures, and the existence and persistence of structural unemployment. To date, however, the derivation of intertemporal stability conditions for various possible stock-flow systems has received little explicit attention (Hadar 1965; Negishi 1962).
The preceding discussion does little more than scratch the surface of stock-flow analysis. Because stock-flow analysis involves an integration of balance-sheet with income-expenditure aspects of economic behavior, the subject directly embraces or indirectly bears upon virtually every other branch of contemporary economic analysis: value and monetary theory, the theory of income and employment, the theory of growth and economic development. What needs to be emphasized, however, is not so much the virtual scope of stock-flow analysis as the severely limited extent of actual knowledge about the properties of stock—flow systems.
First, we must recognize that most of the familiar weaknesses of established price theory, e.g., inadequate treatment of expectations phenomena and related problems of market organization, are shared by stock-flow analysis.
Second, we must note that the explicit inclusion of asset variables in the theory of business behavior forces us to think in terms of preference-maximization, rather than profit-maximization, models, which leads to numerous analytical complications and uncertainties not found in established theories. Hardly any work has been done so far in this area of stock-flow analysis.
Third, we should remark that the present literature on the dynamics of multiple markets—which, incidentally, includes nearly all modern treatments of the theory of income and employment—is concerned with the dynamics of bargaining, rather than the dynamics of bargaining and trade; i.e., it does not deal at all with problems of intertemporal equilibrium. The relevance of this literature for interpreting actual market behavior is dubious, to say the least. However, since a satisfactory account of the intertemporal dynamics of stock-flow systems has yet to be developed, these shortcomings of established theory provide no present grounds for congratulatory remarks about stock-flow analysis.
Finally, a comment is in order concerning a problem of fundamental importance that is only dimly foreshadowed in earlier discussion. The whole of stock-flow analysis and most of contemporary value and monetary theory rest on the assumption that market exchange is a complicated form of barter, involving multiple, rather than double, coincidence of wants. This is reflected in the proposition known as Walras‘ law, which asserts that units of any given commodity (goods or money) constitute effective means of payment for units of any other commodity. This can only be true in an economy where trading processes in all markets are rigidly synchronized, so that purchases and sales of different commodities can be set off against each other without having recourse to intermediate market transactions. If trading processes are not synchronized, we move from the barter economy of “classical” economics to the money economy of John Maynard Keynes; from a world where supply creates its own demand to a world where demands are directly constrained by current accruals of cash and cash substitutes and where supplies are directly constrained by current levels of factor employment. To investigate the dynamic properties of such systems clearly requires the use of stock-flow models. As of this time, however, stock–flow analysis provides nothing more than a foundation for future research in this area.
Robert W. Glower
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