Keynes's multiplier in a two‐sectoral framework

Abstract

This paper endeavours to reinterpret one of the most fundamental concepts of macroeconomics: the Keynesian investment multiplier. The multiplier is not interpreted as a dynamic process (or quantity reaction of output) nor as a logical relation (or ratio) between income and investment expenditure, but as an equilibrium condition that prescribes the proportionality between the two ‘departments’ of the economy (the consumption‐goods and the investment‐goods sector) necessary for ‘completely successful reproduction’. The Marxian concept of reproduction schemes is combined with Keynes's ‘fundamental psychological law’ (which states that the marginal propensity to consume is positive and less than unity) to derive this result. This ‘structural’ view of the multiplier is then used to analyse questions relating to economic growth, capital accumulation and structural change.

Notes

Cf. Keynes (1973a, p. 122) for these two views of the multiplier.

Correspondence Address: Jochen Hartwig, University of St. Gallen and Swiss Institute for Business Cycle Research, Zürich, Varnbüelstr. 14, 9000 St. Gallen, Switzerland. E‐mail: [email protected]

Note that this ‘conventional view’ of the multiplier process ‘was provided to Keynes’ (cf. Dimand, 1988, p. 188) from Kahn and Hawtrey rather than it was his own intellectual achievement. In the General Theory, Keynes presents the multiplier somewhat differently as will be shown below in Section 8.

Milgate, for example, writes: ‘The formal proposition is that saving and investment are brought into equality by variations in the level of income (output). This is the principle of effective demand’ (Milgate, 1982, p. 78).

Cf., e.g. Weintraub (1958), Davidson & Smolensky (1964), Davidson (1978), Parinello (1980), Koenig (1980), Casarosa (1981), Chick (1983), Wells (1987), Vickers (1987), Amadeo (1989), Davidson (1994) and Deprez (1997).

The possible objection that the 45°‐line of the ‘Keynesian cross’ represents such a supply function is not convincing. Unlike Z, the 45°‐line is no autonomous supply function, it is a ‘helping line’ (Samuelson, 1948, p. 257). It is just there to find out which level of income is consistent with the aggregate demand it supports. For example, unlike Z, the 45°‐line has no distinguishable price‐ and quantity‐component.

A major reason why planning periods differ is that some goods are produced on demand (mainly investment goods) while others are produced in batches and then distributed by retailers.

Nell presents the same argument somewhat differently: ‘“Continuous output” should not be overstressed. Even under Mass Production the seasons, traditional holidays and social customs provide a framework that sets definitive marketing dates toward which manufactures aim. … So, while under continuous production there need be no common starting and finishing points, these will often exist, nevertheless’ (Nell, 1998, p. 205).

‘Daily here stands for the shortest interval after which the firm is free to revise its decisions as to how much employment to offer. It is, so to speak, the minimum effective unit of economic time’ (Keynes, 1973a, p. 47).

Under the assumption that labour is the only variable input in the production process, the supply price is given by:

with w = nominal wage rate, N = employment, and Y = output, cf. Chick (1983, p. 66).

Victoria Chick was right to point out: ‘(e)ffective demand is an unfortunate term, for it really refers to the output that will be supplied; in general there is no assurance that it will also be demanded’ (Chick, 1983, p. 65).

Wray (1999) claims that Keynes could have easily adopted a labour theory of value because it was consistent with the purpose of the General Theory.

The thoroughgoing acceptance of marginalist concepts distinguishes the approach of this paper from the works of Edward J. Nell who has produced results comparable to those reported here, but who favours different assumptions. Nell rejects diminishing marginal returns for the analysis of mass production economies and thus discards the D/Z‐diagram, which rests on that assumption (cf. Nell, 1998, pp. 618–621). Also, he denies the significance of ‘psychological propensities of households’ in the context of multiplier analysis (cf. Nell, 1998, p. 563) and thus prefers the ‘classical savings hypothesis’ (c_w = 1; c_E = 0). Nell presents the ‘multiplier sequence’ as a converging series of induced demands for consumption goods (i.e. quantity reactions with fixed prices) (cf. Nell, 1998, pp. 560–564); and this is an interpretation this paper is opposing. But if these differences are accounted for, then Nell's perception of the multiplier (cf. Nell, 1998, pp. 560–564) is similar to the one presented below.

Tsuru (1942) has translated the elements of Marx's reproduction schemes into the terms of Chapter 6 of the General Theory. Note that the value of constant capital transferred onto the output (or total depreciation) equals the sum of user cost and supplementary cost minus what modern production accounts (not Keynes) call ‘intermediate transactions’. Depreciation will be allowed for below in Section 6.

To keep things simple, it is assumed that there is no ‘autonomous consumption’ and no non‐linearity in the consumption function. In this case the marginal propensity to consume equals the average rate of consumption.

A remark is apposite about workers' savings. In the reproduction scheme it looks as if their savings would constitute a direct demand for investment goods. Of course, this is not the case. The transfer of workers' savings to the entrepreneurs is effected by what Keynes has called the ‘financial machine’ (Keynes, 1973b, p. 352). It is assumed here that no part of workers' savings is hoarded. (Indeed, this is a precondition for ‘completely successful reproduction’, see below.) If workers save part of their income (and do not hoard) they are, in effect, granting loans to the entrepreneurs. Then, they can participate in the distribution of the surplus‐value (which is divided between profit and interest on debt). It follows that it is an over‐simplification not to distinguish between functional and personal income distribution in the scheme. On the other hand, this will not be harmful as long as the marginal propensities to consume are assumed to be identical.

Of course, this equilibrium proportion need not imply full employment of labour.

For that, it is also necessary that they estimate the marginal propensity to consume (which is here equal to the fraction of income that is consumed).

Nell (1998, p. 119) defines a ‘structural model’ as ‘a model setting out the framework of the economy—determines actions that must be taken to maintain or to reproduce the system’. A ‘structural model’ has to be distinguished from the ‘structuralist’ models applied by Lance Taylor for studying the economics of developing countries. Both types of models are similar in stressing the importance of institutions as well as in opposing the standard neoclassical optimization approach. But whereas structural models aim at establishing relationships between economic institutions and remain silent on human behaviour, ‘structuralist models incorporate important technical and behavioral relationships’ (Taylor, 1983, p. 7). Taylor's structuralist models are mathematical tools designed for the study of specific problems that incorporate real‐world features that are usually ignored in orthodox economics.

Moore (1988, p. 312) has criticized the dynamic view of the multiplier by claiming that ‘the equality of planned investment and saving does not occur through the adjustment of income, as the Keynesian income‐multiplier approach asserts’ (which conforms with the analysis of the present paper), and that ‘the Keynesian multiplier analysis is thus fundamentally flawed’. In a similar vein, members of the French circuitist school deny that the multiplier can be greater than unity. To assess how these contributions relate to the ‘structural view’ of the multiplier introduced in this paper would lead us too far astray into the theory of (endogenous) money. Cf. Cottrell (1994), Moore (1994) and Dalziel (1996) for a discussion of Moore's view and Gnos & Rochon (forthcoming), for the arguments of the French circuitists.

Interpreted this way, the structural multiplier relation becomes a tool for the entrepreneurs of Department II. They can use it to calculate how much they should produce, given their expectations about the volume of total investment. But in practice, much uncertainty is involved with investment decisions. In terms of Keynes (1973a, pp. 46–47), when contemplating an investment the entrepreneur has to form short‐term expectations about the state of long‐term expectations (cf. also Keynes, 1973a, Chapter 12). This is a difficult task, given that an investor not only has to form expectations about total future demand but also about her own potential share in the market. Deficient investment may be a product of fears of too much competitive investment; and this will have consequences for the equilibrium proportion of departments. (I am indebted to a referee for this point, which will be taken up below in Section 7. Information problems surrounding investment decisions have been analysed by Richardson, 1990).

Bhaduri (1986, p. 40, equation (2.17)).

S = saving, Y = income, K = value of the capital goods. Harrod used the symbol C instead of v, but C is here reserved for consumption.

It seems Harrod overlooks this, because he writes: ‘It may be well to emphasise … that no distinction is drawn in this theory between capital goods and consumption goods. … Some trade‐cycle theorists concern themselves with a possible lack of balance between these two categories; no doubt that has its importance. The theory here considered is more fundamental or simple; it is logically prior to the considerations regarding lack of balance …’ (Harrod, 1939, pp. 18–19). The connection between the Harrod–Domar formula and sectoral balance has previously been established by Lowe (1976, pp. 91–94), and Morishima (1969, pp. 26–28).

Marx's expanded reproduction schemes, which can be regarded as the first models of balanced growth, have triggered off a body of literature too voluminous to be reviewed here. Suffice it to name as important advocates of this line of thought Morishima (1969), Lowe (1976), and Nell (1998).

Note that in Marx's examples in Chapter 21 of Capital, Volume 2, the marginal propensities to save are not equal for the capitalists of Departments I and II. (Workers do not save here.) While, in Marx's schemes, the propensity to save remains constant for the capitalists of Department I, it has to grow continually for the capitalists of Department II to finance accumulation. This seems to be hardly a realistic assumption.

As was mentioned above (in note 15) half of their purchases of capital goods are financed by borrowing money from the workers of their own department. Alternatively, the workers could buy new equities issued by ‘their’ firms.

Nell (2002, p. 523), argues that, as a minimum requirement for completing the circuit of payments in each period, only the wage bill of Department I needs to be advanced. The entrepreneurs of Department II need not start paying their own workers until they have received revenue from the sale of consumption goods to Department I. In this case the entrepreneurs of Department II can economize on borrowing costs.

Authors who aim at marrying the income–expenditure circuit, credit‐money, and the multiplier include Chick (1997) and Dalziel (2001). In contrast to the exposition in this paper, these authors keep the traditional view of the multiplier as a process (of ever‐diminishing waves of quantity reactions and associated payments taking place in successive rounds in logical time), and they do not distinguish explicitly between the two departments.

The issue of proportionality between the two departments often goes unnoticed in numerical illustrations of Marx's expanded reproduction schemes, cf., for example, Lianos (1979, p. 407).

The condition for a departure from equilibrium being self‐righting is

This implies a marginal propensity to consume below 0.2 for Harrod's proposed value: v = 4.

Note that, ‘if we wish to examine in pure and unobscured form the exchange between the two large classes of social production, the producers of means of production and the producers of articles of consumption’ (Marx, 1973, p. 516), all firms producing investment goods and all firms producing consumption goods have to be lumped together to one department respectively. Therefore, the analysis has to proceed on a different level than that of Goodwin (1949), who interpreted the multiplier as a matrix of input–output relations of n sectors (one of which is the household sector, the others being productive sectors) that all spend less than they earn. Structural change and (equilibrium) economic growth in such a multi‐sector model have been studied by Pasinetti (1981, 1993). Analysing ‘structural’ unemployment in terms of sectoral imbalances in a multi‐sectoral framework has a tradition that goes back to pre‐Keynesian writers such as, e.g. Pigou (cf. Casson, 1984, Chapter 6; Hartwig, 2000, Chapter 3).

Price reactions that clear goods markets can be expected mainly in case of underproduction, whereas an addition to stocks should be entrepreneurs' typical short‐run response to an overproduction—given the empirical finding that short‐run prices are typically not flexible downwards.

Hicks (1982, 1985) also envisages the economic process as a succession of production periods. He distinguishes between ‘single period theory’ and ‘continuation theory’. A disequilibrium in the ‘single period’ subsequently leads to different expectations on the part of the entrepreneurs and to adaptations of their decisions. ‘Continuation theory’ then traces out the succession of those periods.

P^D = demand price level, P^S =supply price level, O_r = real output, w = nominal wage rate, N = employment. For the transformation in the second step, cf. above, note 9.

Keynes wrote that ‘the logical theory of the multiplier … holds good continuously, without time‐lag, at all moments of time…’ (Keynes, 1973a, p. 122). In our example, this ‘logic’ could only work if the marginal propensity to consume were to drop to zero initially. In this case the logical multiplier relation continues to hold, because we have:

and at the same time the identity of saving and investment is ‘rescued’.

Hicks (1975, pp. 11–21) discusses the importance of inventories in allowing the (traditional) multiplier process to work.