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The Genuine Meaning of Keynes’ Multiplier* By Ezra Davar Abstract This paper shows that Keynes’ investment multiplier and Kahn’s employment multiplier are different: (1) Kahn’s approach is compatible with the second phase of the investment process, when fixed capital (investment) is determinant; while Keynes’ approach is compatible with the first phase when investment (saving) is determinate; (2) The source of an additional increment of investment in Kahn’s approach is exogenous, whilst in Keynes’ approach is endogenous. Also, Keynes’ investment “multiplier” is a reciprocal of the marginal propensity to invest (save); therefore, its genuine meaning is that of a requirement, which indicates the quantity of national income required to realize one unit of investment (saving). Keywords: Multiplier, Requirement, Walras, Keynes, Kahn, General Equilibrium Theory JEL Classifications: B, D5, and E2 Author: Dr, Ezra Davar, Ben-Gurion University of the Negev Yehuda Hanasi St. 15/17 Netanya 42444 Israel Telephone: 972-9-8349790 E-mail: [email protected] * The paper was written to the 2003 UK HET Conference, Leeds, 3-5 September The Genuine Meaning of Keynes’ Multiplier By Ezra Davar “The traditional analysis has been aware that saving depends on income but it has overlooked the fact that income depends on investment, in such fashion that, when investment changes, income must necessarily change in just that degree which is necessary to make the change in saving equal to the change in investment.” Keynes, The General Theory, p.1841 1. Introduction The multiplier is one of the main tools for establishing a relationship between income, investment, consumption and employment developed by Keynes in his “The General Theory”. Since its very first appearance, the attitude towards the multiplier has been ambiguous. One group of economists (mainly Keynes’ followers) state that the Keynesian multiplier is a new paradigm in economic theory and that it is a milestone in the development of macroeconomic theory. Blaug writes (p.189): ‘The principal novel prediction of Keynesian economics is that the value of the instantaneous multiplier is greater than unity’ (see also Pasinetti, p.40). Yet, some scholars stated that "This is, no doubt, in a general way, correct" (Pigou, 1936). And "In order to avoid any possible misunderstanding, let me conclude this paper by emphasizing that there can be no doubt about the importance of Kahn's 1931 multiplier article in the history of the development of macroeconomic theory. Nor can there be any doubt that his multiplier formula is logically equivalent to the theory of effective demand" (Patinkin, pp.324-5). But at the same time, there were economists who indicated doubt regarding the multiplier "... it seems to me doubtful whether, for the analysis of a fluctuating world, the "multiplier" constitutes much advance over more crudely "monetary" weapons of thought".(Robertson, 1936); Another group posed serious doubts concerning the multiplier’s correctness (Ahiakpor, 2001; Hazlitt, 1959). There are authors who considered multiplier as a dynamic phenomenon (process) (Goodwin, 1947; Hansen, 1953) and V. Chick claims that Keynes’ 1 multiplier has two views: equilibrium and dynamic (Chick, 1983). Many arguments were raised against it, such as: the multiplier is severely static (Schumpeter, p.1174) unrealistic, characterized by instability, and so forth. Keynes’ followers have been attempting to "improve" the theory (multiplier) of Keynes in order to silence such critics. A central argument, often used by them, is that the multiplier has rests on a solid mathematical foundation (Minsky, 1975). In addition, several papers show that ‘the balanced-budget expenditure has a multiplier of exactly one;’ (Samuelson, 1966; see also Samuelson, 1974; Salant, 1974 (1942); Haavelmo, 1943; Turvey, 1953 and so on). But this “discovery” does not solve the problem because according to Keynes’ theoretical definition and practical estimation of multiplier, the “indicator’s” magnitude is strongly greater than one and, investment does not produce income itself (vide infra). Recent debate between Ahiakpor (2000) and Dimand (2000) and Darity and Young (2000) have been showing continuing controversial interpretation of the multiplier and how much the solution of this problem is required. Yet, this unresolved argument has serious implications, such as its continued presence in textbooks despite its unresolved status. Nevertheless, "Like many other tools of economic theory, the multiplier must be used with care, but there is no case for its demise. It remains a very important relationship in post-Keynesian monetary macroeconomics" (Gilbert, 1982). It is necessary to stress that Keynes himself in the letter to Beveridge claimed that “The theory of the multiplier. You write here as though this was a matter on which I had only dogmatized and not discussed. But not only have I given a long chapter expressly to this subject, but about half the book is really about it.” (Keynes, 1973, p.57) This paper will present and discuss the arguments made by both sides, and also present a third, new perspective on the Keynes’ multiplier. It will be shown that 2 Keynes’ investment multiplier and Kahn’s employment multiplier are different not only as regards their theoretical meaning, but also with respect their practical magnitude because of the following reasons: (1) Kahn’s approach is compatible with the second phase of the investment process, when fixed capital (investment) is determinant; therefore, the calculation of Kahn’s multiplier is a dynamic process, albeit though the time-lag is not taking into account; while in Keynes’ approach, when investment (saving) is determinate, is compatible with the first phase of the investment process, therefore, the calculation is static; (2) The source of an additional increment of investment in Kahn’s approach is exogenous, whilst in Keynes’ approach is endogenous; (3) Kahn’s multiplier is a technological-economic phenomenon, while Keynes’ multiplier is a psychological phenomenon. Also, it will be shown that Keynes’ investment “multiplier” is a reciprocal of the marginal propensity to invest (save); therefore, its genuine meaning is that of a requirement, which indicates the quantity of national income required to realize one unit of investment (saving). The paper consists of five sections. Following the introduction, the second section describes a commonly used general equilibrium theory approach and discusses two possibilities to income increases: endogenous and exogenous. In the third section, Kahn’s original definition of multiplier is presented. The forth section reviews Keynes’ definition of the multiplier. In the final section the genuine meaning of Keynes’ multiplier will be presented, followed by conclusions. 2. Produced Income and Used Income For the reasons of convenience, general equilibrium theories state that in an equilibrium position the total value of all employed services (employed quantities of services multiplied by their according prices) equals the total value of demand 3 (consumption) of all commodities (demanded quantities of commodities multiplied by their according prices). The first is known as produced (created) national income or national income in terms of factors’ (services’) prices Yp, while the latter is known as used (consumed) national income or national income in terms of commodities’ market prices Y. This statement is known as Walras’ law, which all schools of thought of economics recognize and, in addition, and something which is very important – it is compatible with empirical accounts of national income. At the same time, it is necessary to point out that there is a difference between Walras’ own law and the post-Walrasian version of this law. Walras’ law means that the law is only satisfied for equilibrium prices established for each goods and services separately in the condition that the total demand is equal to the total supply for services and the cost of production (the supply price) is equal to the demand prices for goods. In addition, prices are strongly positive even some services are underemployed in equilibrium state. The Walras followers’ law is satisfied for any prices and if there is unemployed service, then its price is equal zero. Therefore, the post-Walras’ law is a by-product of his original law, but Walras’ own law is compatible to economical reality, whilst the post-Walras’ law is incompatible with reality, whereas this law is fundamental to the post-Walrasian approach (Davar, 1994). Hence, the Walras’ law can be presented as n m 1 1 2 1 Oi pi D j j Where pi – is equilibrium price of service i; Oi – is equilibrium employed (required) quantity of service i, which is either equal to or less than its available (existence) quantity Qi, i.e., Oi Qi; j – is equilibrium price of commodity j; 4 Dj – is equilibrium demanded quantities of commodity j. For the following discussing there are two things must be pointed out: (1) the employed (required) quantities of services are obtained on the basis of inverse coefficients of factors (Walras, 1954, pp.240-1; Leontief, 1986, pp.394-5; Davar, 1994, pp.17-8), which means that they include the direct and indirect inputs of services. So, these required (employed) quantities of services include the total (direct plus indirect) quantities of services required for the production of demand goods; (2) the list of demand commodities in general include all kinds of commodities, “public work”. If there are some commodities that are not included in the list of demand commodities, it has to be added and required quantities for its production would be obtained according to the previous note. The left side expression of (2-1) is national income in terms of factors’ (services’) prices Yp (produced income) and if we take into account the results of the individuals model’ solution it have divided into two components: commodities for consumption (Cd) and saving (S). The right side expression is national income in terms of commodities prices (used income) and it is also combined by two components: commodities for consumption (Cd) and new capital goods (investment - I). So, in the equilibrium situation there are: Yp = Oipi = Cd + S, (2-2) Y = Dii = Cd + I, (2-3) Yp = Y and Cd + S = Cd + I, therefore S = I, (2-4) Equation (2-4) indicates that in an equilibrium situation saving equals investment. But, individuals who save differ from individuals who invest. This means saving may differ from investment. Therefore, in order to establish equilibrium, saving must be 5 equal to investment by means of changing the interest rate (Chick, 1997, p.174; Sawyer, p.52). In addition, it has been shown that the classical and the Walrasian approach -- as well as Keynes approach -- assumes that in an equilibrium position, might be unemployed services (including labor) (Davar, 1994), It is necessary to stress that there is a difference between Walras’ and Keynes’ unemployment albeit that they are interwoven. The former is observed as voluntary unemployment, while the latter as involuntary underemployment (Davar, 2002). If the unemployed share of services is considerable, it will be desirable to attempt to shift equilibrium position, if it is possible, in order to decrease unemployment and, consequently to increase income (investment). 2.1 Investment (Saving) as Determinate and Fixed Capital (Investment) as Determinant Keynes stated that ‘Thus the traditional analysis is faulty because it has failed to isolate correctly the independent variables of the system. Saving and Investment are the determinates of the system, not the determinants. They are the twin results of the system’s determinants, namely, the propensity to consume, the schedule of the marginal efficiency of capital and the rate of interest (Keynes, 1936, pp.1834).’ This is a very important statement, which is based on Walras’ theory, but Keynes destroyed it in the multiplier theory, where investment becomes a determinant (vide infra). For the following discussion it is necessary to make clear whether new capital goods (investment) produced in a certain period (year) transform services (fixed capital) either in that year or the next year. It is, in general, assumed that new capital goods produced during a certain year become services the next year. One of the central 6 assumptions of Keynes’ analysis is ‘the existing quality and quantity of available equipment, the existing technique,’ (Keynes, 1936, p.245). This means that new capital goods do not create (produce) national income in the year when they are produced. So in this case there is no direct connection between investment and the creating of national income. There is a connection between them in such way so as to produce new capital goods (invest) so that according national income must be created. This is the first phase of the whole process of investment, i.e. when investment is created, therefore, in this phase, investment (saving) is determinate. It must be stressed that investment might be increased without any changes of created income, and moreover it might be increased if created income is decreased, by decreasing of the value of consumers’ commodities (vide infra). Even if assuming that new capital goods (investment) became services in the same year when they are produced, then created income might be increased according the magnitude of its employment and according to the required employed quantities of other services (especially labour) determined by the current technology. In the second phase, investment is transformed into fixed capital and produces income in combination with other services (labour, land, money) according to the technology of the sector (industry) where fixed capital (investment) is planned. So, in this phase fixed capital (investment) is determinant. The first round of producing income continues over the number of years that the fixed capital might be served (5 years in our case). The income produced, in turn, makes up investment according to the marginal propensity to consume (invest) (vide infra) in each year. The latter produces new income in the second round and continues on the serving years of fixed capital (investment). This process continues infinitely (see Appendix 1, where we confine ourselves to describing only three rounds) (see also Sawyer, p.52). 7 There are two things to be noted: (1) starting from the second round (beginning from second year) income has a cumulative character. The same is true for investment from the third round (beginning from third year); (2) It is assumed, for all rounds (years), that the new fixed capital will be fully employed and required quantities of other factors (labour, land, money) will be satisfied accordingly. 2.2 Endogenous and Exogenous Increment of Investment Let us to point out, firstly, that the problem of investment increment is compatible with the first phase of investment process, when investment (saving) is determinate. There are two sources for increasing investment: endogenous and exogenous. Let us to start with the first - endogenous source. On the basis of (2.3) we can conclude that there are three possibilities for investment increasing: 1) when value of consumption is decreased and income does not change; 2) value of consumption is constant (no change) and investment increase by increasing income; 3) both, consumption and investment, are increased by increasing income, either they are increased with variable marginal propensity to consume, as Keynes assumed in the General Theory (Keynes, 1936, p.114, see also Hawtrey, p.179n) or they are increased by constant marginal propensity to consume. Instead of that, Keynes considered the first version of investment increasing with others ‘Assuming that the decisions to invest become effective, they must in doing so either curtail consumption or expand income. Thus the act of investment in itself cannot help causing the residual or margin, which we call saving, to increase by a corresponding amount.’ (Keynes, 1936, p.64) let us discuss only two latter cases, because not only the first happens very rarely in reality, but also Keynes discussed only the following. Assume that the value of the consumption and investment commodities are increased, so that Cd1 (>Cd) and I1 (>I). 8 Assume also that there is a possibility to achieve the next state of equilibrium according for this new data. Hence, for this case we also have Cd1 + S1 = Cd1 + I1, (2-5) Comparing (2-2) and (2-5) we obtain ∆Cd + ∆S = ∆Cd +∆I, (2-6) From this condition to hold we must conclude that an increment in demand for commodities, either consumption or investment or both has occurred, i.e., the increment of used income stipulates an increment in produced income of the same magnitude, and vice versa. In other words the increment in investment or consumption must be financed by equivalent increment in produced income (increment in value by the additional employed services). In addition, the increment in investment must be equal the increment of saving. The relationship between consumption and investment (marginal propensity to consume) does not influence this conclusion. The latter influences only on the magnitude of the increase in income (vide infra). So, we can conclude that in all cases where the increment in investment (income) is a result of endogenous sources, the magnitude of the increment of consumed (used) income equals the magnitude of produced income and vice versa (Negishi, 1979, p.182; Samuelson, 1948 and 1974; Salant, 1974 (1942 and 1964); Haavelmo, 1943; Turvey, 1953). The second source for increasing investment - exogenous - means that the increment is obtained by means of borrowing from either domestic sources (banks and other similar institutions) or foreign sources (governments and banks), or both of them. In the case of exogenous increment of investment, the calculation of the national income created and the additional (produced) investment, generally, is divided into to stages: (1) this is the same as was described for the second phase of the investment process 9 i.e., in the next year investment is transformed into fixed capital, and starts the infinite process of income and investment creation (see 2.1, p.7); (2) because of this, the source of increment of investment is exogenous in the year of the investment, also create national income in the branches where capital products are produced; and the quantity of income created is equal to the value of investment minus the intermediate input of goods. But this additional produced income is not used to finance additional consumption and investment in the year of the investment as in the previous case, just because its source is exogenous. So, the additional produced income is “free” and might be use for the new additional purchase of commodities for consumption and investment. Thus, this created income divided between consumption and investment in according to propensity to consume (save) and following the calculation of additional income and investment, is the same as the previous cases. In addition to the two notes (see p.7) there is one more note, namely, the question of the efficiency of borrowing and the problem of loan repayment are not discussed here. Despite these obstacles, we can conclude that in the case of an exogenous increment in used investment, the increase of desirable produced income is to be greater than the increment in used income. In other words, here, the coefficient of produced income increment might be more than one, whereas in the case of an endogenous increment of investment, the coefficient always equals one. 3. Kahn’s definition of Multiplier In his well-knowing paper “The Relation of Home Investment to Unemployment”, published in the June 1931 edition of The Economic Journal, Kahn introduced the concept of the employment multiplier into the economic literature. At that time, almost all European countries were characterized by very high levels of unemployment. Therefore, economists were not the only ones trying to find ways to 10 overcome unemployment and modify its results. Kahn suggested increasing the rate of home investment by “public work” and described its effects as decreasing unemployment for throughout the economy: The increased employment that is required in connection actually with the increased investment will be described as the “primary” employment. It includes the “direct” employment, and also, of course, the “indirect” employment that is set up in the production and transport of the raw materials required for making the new investment. To meet the increased expenditure of wages and profits that is associated with the primary employment, the production of consumption-goods is increased. Here again wages and profits are increased, and the effect will be passed on, though with diminished intensity. And so on ad infinitum. The total employment that is set up in this way in the production of consumption-goods will be termed the “secondary” employment. The ratio of secondary to primary employment is a measure of these “beneficial repercussions” that are so often referred to (Kahn, 1931, p.173). A careful analysis of this statement helps us understand Kahn’s genuine meaning and show that this is compatible with the second phase of the investment process, because of the description of production of output (employment). In Kahn’s approach, investment is exogenous ‘throughout this article it will be supposed that the necessary funds are raised by means of borrowing. … For it is always within the power of the banking system to advance to the Government the cost of the roads without in any way affecting the flow of investment along the normal channels’ (Kahn, 1931, p.174). So, Kahn assumes that one sources of net (additional) investment is borrowing from banks (ibid. p.179). As to the second source of net investment, Kahn implicitly 11 determined that foreign sources may also be used in order to increase net investment either by a reduction in foreign lending or by an increment in foreign borrowing (ibid. p.189). At the same time, it is necessary to stress that Kahn categorically refused to finance the additional investment by increasing the quantity of money, as some modern authors have suggested. Kahn wrote: ‘There was no reason why additional expenditure on public works needed to be financed by the creation of additional money as against borrowing from the public. … The increase of employment was not the result of an increasing quantity of money.’ (Kahn, 1984, p.104) It appears then, that Kahn considered the problem of open economy and assumed that the source of the increase of investment is exogenous. Therefore, the multiplier, expressing the relation between primary and secondary employment stipulated by the exogenous additional investment, might be more than one. In order to calculate the exact magnitude of such a multiplier, much more complicated tools (models) are needed than those available to Kahn. But, for our purposes his method is sufficient. It is necessary to point out that Kahn indicated that ‘…the increase of efficiency should certainly be taken into account if one wants to consider how a policy of public works at the present time would affect the budgetary problems of the future’ (ibid. p.193). Yet, ‘I am here considering the position in the final position of equilibrium when everything has settled down. But some time will, of course, elapse between the point when the primary employment begins and the point when the secondary employment reaches its full dimensions became wages and profits are not spent quite as soon as they are earned. I do not enter the question of this time-lag.’ (ibid. p.183n) But Kahn could not take into account these problems when he calculated the multiplier’s magnitude because, to repeat, the limited tools used by him, prevented such a possibility. Nevertheless, the method describing the multiplier’s calculation by Kahn 12 shows that his employment multiplier is a technological-economic phenomenon (Kahn, 1931, pp.183-6). Let us now summarize the main characteristics of Kahn’s employment multiplier: (1) this is compatible with the second phase of the investment process, when investment (fixed capital) is determinant; (2) this is a special case, because the source of investment increment is external to the system; and (3) the additional new investment is directed into “public work” such as roads; (4) the calculation of the multiplier is dynamic process, although a time-lag is not take into account; and finally, (5) the employment multiplier is a technological-economic phenomenon. 4. Keynes’ Definition of Multiplier 4.1 Keynes’ Equilibrium Theory Before we define Keynes’ version of the multiplier, let us be clear as to what kind of equilibrium theory is incorporated in Keynes’ approach. This is a very crucial point, because a number of authors have stated that Keynes is not among Walras’ followers (Blaug, p.188; Clower, 1985, p.110; Kaldor, 1983, p.9; Leijonhufvud, 1998, pp.175-6; Meltzer, 1983, p.69; Negishi, 1979, p.1). Keynes himself remarked that ‘Walras’s theory and all others along those lines are little better than nonsense’ (Skidelsky, 1996, p.615). In addition, as was shown, Kahn’s conception of the multiplier is based on an equilibrium theory similar to Walras. Proposition (4) of the summation of Keynes’ theory in General Theory states: (4) Since D1 + D2 = D = (N), where is the aggregate supply function, and since, as we have seen in (2) above, D1 is function of N, which we may write (N), depending on the propensity to consume, it follows that (N) - (N) = D2 (Keynes, 1936, p.29) where D1 – is the amount which the community is expected to spend on consumption; 13 D2 – is the amount, which it is, expected to devote to new investment; D – is what we have called above the effective demand. (Ib.29) The first equation of this proposition (D1 + D2 = D = (N)) is a simplified version of Walras’ law (vide supra), based on the fact that the right side of this equation expresses total value of demanded commodities (used income) and the left side – total value of employed labour (produced income). It is necessary to stress that there are some controversial points associated with this model, but they do not impinge on the subject of this paper; therefore, we will not discuss them here. This means that Keynes’ theory is, generally, an equilibrium theory just like that classical theory and Walras’ theory. Hence, all the properties of classical and Walrasian equilibrium theory are compatible with Keynes’ theory, too. The second equation ((N) - (N) = D2), which is obtained from the first equation, is correct from the point of mathematics but incorrect from the point of view of economics. As we have shown previously though saving equals investment in equilibrium, each of them is determined independently of one another and generally differs from each other. When they equal one another then equilibrium is established and the first equation is obtained. This means that, in general, it is incorrect to determine the amount of investment by deducting the amount of consumption from the amount of income. Despite this statement, for our purposes, let us assume that the second equation is also correct for equilibrium state, as Keynes assumed. Adopting this assumption as authentic allows us to discuss Keynes’ definition of the multiplier. 4.2 Keynes’ Definition of the Multiplier The main argument of Keynes’ theory is that ‘when the genuine income of the community increases or decreases, its consumption will increase or decrease but not 14 so fast" (Keynes, 1936, p.114). From this we can conclude that income is equivalent to the cause and consumption, as well as the saving (investment), is effect. Keynes, in the following, stated that the previous statement "... can, therefore, be translated ... into the propositions that ∆Cw and ∆Yw have the same sign, but ∆Yw>∆Cw, ... Let us define, then, dCw/dYw as the marginal propensity to consume" (ibid. pp.114-5). Keynes continues "This quantity is of considerable importance, because it tells us how the next increment of output will have to be divided between consumption and investment" (ibid. p.115). Here Keynes confirms once more our statement (vide supra) that consumption and investment are effects and that income (output) is a cause. Keynes continues " For ∆Yw = ∆Cw + ∆Iw, where ∆Cw and ∆Iw are the increments of consumption and investment; so that we can write ∆Yw = k∆Iw, where 1 - 1/k is equal to the marginal propensity to consume. " Let us call k the investment multiplier. It tells us that, when there is an increment of aggregate investment, income will increase by an amount which is k times the increment of investment" (ibid. p.115). There are several things here that require explanation. First, income and investment have been reversed (replaced); investment now becomes the cause and income the effect2. Moreover, here investment is determinant, which is opposite to Keynes’ above cited statement that “Saving and Investment are determinates’. But, the theory of causality teaches that such a reverse (replacement) is generally incorrect. However, the relationship between cause and effect does allow us to stipulate the magnitude of the cause required achieving a predicted magnitude of the effect. This means that either Keynes' definition of the multiplier is incorrect, or the assumed causal relationship between income, investment and consumption) is incorrect (vide infra). 15 Second, a contradiction exists between the determinant of the marginal propensity to consume and its use. Since, marginal utility depends on the magnitude of the increase in income. This means that in the case where a discrete event is discussed, the multiplier depends not only on the point (Yw), (as a continuous case), but also on the magnitude of ∆Yw. Stated differently, this means that Keynes confused discrete and continuous cases in his model. Yet, the marginal propensity to consume means that each additional unit of income is divided between consumption and saving (investment) in a way different from the previous instance of an additional unit of income. Keynes, however, interprets the multiplier, as a tool to calculate income by means of investment, assuming that the propensity to consume is constant. Third, Keynes offers two different definitions for k: 1) k is the reciprocal of the marginal propensity to invest (vide infra); and 2) k is the investment multiplier. But this is impossible because these definitions contradict one another (vide infra). Fourth, Keynes’ “multiplier” is, by definition, a psychological phenomenon. This means that the magnitude of multiplier depends on the psychological behaviour of the community on the condition that technological-economic conditions are given, and do not change over the time-period under discussing. In addition, Keynes approach is compatible with the first phase of the investment process and, therefore it is static. Hence, we can conclude that k cannot be the multiplier. The question then becomes: “what is the genuine meaning of Keynes’ multiplier? Before answering this question, let us to discuss the relationship between Kahn’s and Keynes’ multipliers. 4.3 The Relationship between Kahn’s and Keynes’ Multipliers Keynes indicated that there are two types of multipliers: his own and that of Kahn. Keynes tried to show that these two multipliers are identical. In the opposite case, that 16 is, if can be shown that they are not identical, and then the theory of the investment multiplier is incorrect. Why? Understanding this fact Keynes tried to prove this identity (Keynes, 1936, p.115). From this we can conclude that for a specific level of investment there is a single multiplier k', other things being equal. This means that the magnitude of the multiplier for the specific level of investment might vary, depending on the other conditions. But when these conditions are given, the magnitude of the multiplier for a specific level of investment has a single-value (vide infra). Keynes tried to discuss the conditions under which these two different multipliers might be equal (ibid. p.116). What is the condition that applies to the simplified case of k = k'? In note 1, page 116, Keynes described these conditions. Together with this strong, unrealistic assumption, another flaw can be found. These multipliers are simply different.3 First, Kahn’s model differs from Keynes’ because the first is open and the second is closed (see also Chick, 1997, p.165). Second, in Keynes’ approach, the source of the additional investment is not clear. Because that the multiplier connecting income and investment is calculated under internal conditions by the way of consumption. So, the source of additional investment must be endogenous. In other words, an exogenous source is excluded. Despite that, Keynes as well as Kahn, assumed that: ‘It follows, therefore, if the consumption psychology of the community is such that they will choose to consume, e.g., ninetenth of an increment of income, then multiplier k is 10; and the total employment caused by (e.g.) increased public works will be ten times the primary employment provided by the public works themselves, assuming no reduction of investment in other directions’ (ibid. pp.116-117). The last statement indicates that Keynes also assumed that the source of investment might be exogenous source, which contradicts 17 his approach’s basic principles. Finally, Keynes’ multiplier is a psychological phenomenon and the unrealistic magnitude of the multiplier (10-times increase of income) in Keynes’ approach is the main indicator of its unrealistic character (vide infra) (see also Davidson, 1994, pp. 37-43). So, Keynes' “multiplier” differs from that of Kahn. The next section will discuss whether Keynes' “multiplier” is indeed a “multiplier”. 5. The Genuine Meaning of Keynes’ Multiplier Let us assume that national income is only divided between consumption and saving (investment), as in Keynes' hypothetical case, summarized above. Than, by Keynes' definition of marginal propensity to consume c+i=1 or c = 1 - i i=1–c or (5-1) where c – is the marginal propensity to consume and i - is the marginal propensity to save (invest) Substituting the latter into Keynes’ formula we obtain ∆Yw = ∆Cw +∆Iw = c∆Yw + ∆Iw and ∆Yw = 1/(1-c) ∆Iw = 1/i ∆Iw = k ∆Iw, (5-2) From this we can conclude that Keynes’ "multiplier" is the reciprocal (inverse) of the marginal propensity to invest (save) (Ahiakpor, 2001, p.746; Hawtrey, p.179; Patinkin, 323). But the reciprocal of the marginal propensity to invest (saving) indicates the required quantities of income for a unit increment of investment when marginal propensity of both does not change. Therefore, the genuine meaning of Keynes’ multiplier’s is a requirement, and not a multiplication. Hence, the requirement indicates the quantity of national income needed to realize one unit of investment (saving) when the marginal propensity to consume is constant. Keynes 18 himself wrote that “The traditional analysis has been aware that saving depends on income but it has overlooked the fact that income depends on investment, in such fashion that, when investment changes, income must necessarily change in just that degree which is necessary to make the change in saving equal to the change in investment.” (ibid. p.184, see also p.117) By this definition, again taking Keynes’ example, the equation must be expressed, as, in order to increase the investment by one, the income is required to increase by 10, where 9 units is set to increase consumption. In other words, since the relationship between consumption and investment is given, the increase of investment stipulates the increase in consumption according to a given proportion. Therefore, to obtain an increment at specific levels of investment, the corresponding quantities of income are required. That income is composed of the value of relevant additional employed services (not only available fixed capital, which generally differs from investment, but also labours and land). At the same time, it must be stressed that the corresponding increase of income might not be possible at all, because of limitations of labor, fixed capital, scarce raw materials, and so on (Hicks, 1974). This reflects the fact that in this case, the connection (relationship) between national income and investment is reciprocal, whereas in the previous case, the relationship is direct. Therefore, the difference between them is that the marginal propensity to save indicates a realistic possibility, while the requirement may not. This means that Kahn’s multiplier and Keynes’ requirement are fundamentally different, and can never be identical. It becomes clear, in our opinion, why Keynes did not use the marginal propensity to invest in general case for determining the multiplier, and preferred to use the partial case (vide infra). 19 Now, let us to consider the quantitative relationship between the requirement and the “multiplier”. In general, assuming that the one unit can be divided into k parts, then we obtain c = (k-n)/k and i = n/k, where n = 1,2,...,k-1; (5-3) This means that Keynes "multiplier" is k/n; therefore, the case considered by Keynes is the partial case of the general approach, when n = 1 (vide supra). Whereas, the magnitude of the employment multiplier is near to two according to Kahn's computation (Kahn, 1931, p.186)4. The inverse of marginal propensity to invest (Keynes’ requirement), however, might reach 20, 30, or even more (Hazlitt, p.139, p.151). Therefore, Keynes' requirement (the inverse of the marginal propensity to invest) differs from the Kahn's multiplier not only in substance, but also in magnitude. We should point out that these magnitudes might coincide by chance. At the same time, some circumstances intensify the difference between Keynes’ multiplier and Keynes’ requirement. First, the investment differs from the fixed capital, one of services, which composed produced income. Second, different sources for investment (credits from domestic and foreign banks) and the "mutation" problem of investments can influence the magnitude. Finally, new fixed capital (investment) might also be unemployed. Conclusions In this paper the genuine meaning of Keynes’ multiplier is discussed. It was shown that Keynes’ investment “multiplier” is essentially a requirement, not a multiplier, and: 1. The whole process of investment generally consists of two phases: first, when investment (saving) is produced (created), as a part of used (produced) income, i.e., when investment (saving) is determinate; and therefore, the causal relation here is from income to saving (investment), this is, the increment in income 20 causes an increment in saving (investment), while the reverse relation means a requirement, this is, in order to increase investment there is a required income increment accordingly; second, when investment is transformed into fixed capital and creates (produces) income together with other services (labour, land, money), i.e., when investment (fixed capital) is determinant; and therefore, the causal relation here is from fixed capital (investment) to income, this is, fixed capital (investment) increase causes an increment income, while the reverse relation is a requirement, this is, in order to increase income there is a required fixed capital (investment) increment accordingly; also, this is the infinite and cumulative process of creation, not only for income but also for investment. 2. There are two resources of investment increment: (1) endogenous to the system; and (2) exogenous to the system: domestic (banking and other financial institution) and foreign (governments and banking). It is necessary to stress that the investment increment emanating from the first, endogenous, source is characterized by both phases of the investment process, whilst the investment increment generated by the second, exogenous, source is characterized only by the second phase of the investment process. But, at the same time, in this case there is a creation of additional income in the branches where capital products are produced and the quantity of income created is equal to the value of investment minus the intermediate input of goods; and this additional income allows us to the additional stage for the new infinite and cumulative process of creation of income and investment to take place. 3. Kahn’s definition of the multiplier is compatible with the second phase of investment process, when investment (fixed capital) is determinant and the increment in investment comes from an exogenous source (banking, domestic and foreign, other governments), and it is a technological-economic phenomenon; therefore, its magnitude might be more than one. The calculation of Kahn’s multiplier is a dynamic process and in order to calculate the magnitude of the multiplier, Kahn introduced simplifying assumptions. 4. It was shown that Keynes’ “multiplier” is compatible with the first phase of investment process, when investment (saving) is determinate and is a reciprocal of 21 the marginal propensity to invest (save) and therefore, it is a psychological phenomenon and static. Therefore, its genuine meaning is that of a requirement, which indicates the quantity of national income required to realize one unit of investment. Hence, Kahn’s employment multiplier and Keynes’ requirement are different. Notes 1 “…the multiplier tells us what level of income will be necessary to make savings equal to that value of investment” (Hicks, 1937, p.153). 2 “Then, by some wild leap, this “functional” and purely formal or terminological relationship is confused with a causal relationship. Next, the causal relationship is stood on its head and the amazing conclusion emerges that the greater the proportions of income spent, and the smaller the fraction that represents investment, the more this investment must ‘multiply” itself to create the total income!” (Hazlitt, p.139) 3 As was already shown by Patinkin, though by another arguments. In his paper "Keynes and the Multiplier" (1978) Patinkin discuss two questions: "... first, Keynes's empirical estimates of the multiplier, second, the relation between Kahn's multiplier analysis and the General Theory" (p.322). His conclusions in both cases are negatives. Patinkin's own words: "... the estimates (of multiplierK.D.) become 2.9, 1.8, and 2.1, respectively. And it is hard to say that these results justify Keynes's conclusion that "the multiplier seems to have been less than 3 and probably fairly stable in the neighborhood of 2.5" (GT, p.128)." and "I said this is the way we teach the multiplier today. ... But this is not the way Richard Kahn first presented the multiplier ..." (pp.323-4). 4 It is necessary to point out that Keynes also considered the value of his multiplier closer to this magnitude: ‘If single years are taken in isolation, the results look rather wild. But if they are grouped in pairs, the multiplier seems to have been lass than 3 and probably fairly stable in the neighborhood of 2.5’. (p.128) References Ahiakpor, J.C.W. 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(1983) Keynes and the Modern World. Cambridge University Press, London, New York. Appendix 1 The process of calculation of income and investment in consequence of incremented investment ∆I in year 0* assuming that Keynes’ approach is used. The first round 1.1 Y11 = k1 K11; ∆I11 = i1Y11; where K11 = ∆I; 1.2 Y12 = k1 K11; ∆I12 = i1Y12; 1.3 Y13 = k1 K11; ∆I13 = i1Y13; 1.4 Y14 = k1 K11; ∆I14 = i1Y14; 1.5 Y15 = k1 K11; ∆I15 = i1Y15; The second round 2.2 Y22=k2K22(=∆I11); ∆I22 = i2 (Y12+Y22); 2.3 Y23=k2K23(=∆I11); ∆I23 = i2 (Y13+Y23); 2.4 Y24=k2K24(=∆I11); ∆I24 = i2 (Y14+Y24); 2.5 Y25=k2K25(=∆I11); ∆I25 = i2 (Y15 +Y25); 2.6 Y26=k2K26(=∆I11); ∆I26 = i2 Y26; The third round 3.3 Y33=k3K33(=∆I12+∆I22); ∆I33 = i3 (Y13+Y23+ Y33); 3.4 Y34=k3K34(=∆I13+∆I23); ∆I34 = i3 (Y14+Y24+ Y34); 3.5 Y35=k3K35(=∆I14+∆I24); ∆I35 = i3 (Y15+Y25+ Y35); 3.6 Y36=k3K36(=∆I15+∆I25); ∆I36 = i3 (Y26+ Y36); 3.7 Y37=k3K37(=∆I26); ∆I37 = i3Y37; * For simplification we assume that fixed capital is served 5 years for all rounds and that the discount factor is missing. Where k – is the capital-output coefficient and indicates the amount of income produced by one unit of fixed capital (investment); and i – is the marginal propensity to invest (save). 25